 | ECS #184: Denumerant
|
1. Description
number of ways to make n cents with coins of 2 5 10 20 50 cents
2. Specification
This
unlabelled
structure is specified as S in
\displaystyle
\left\{ S={\rm Prod} \left( {\rm Sequence} \left( {\rm Prod} \left( Z,
Z \right) \right) ,{\rm Sequence} \left( {\rm Prod} \left( Z,Z,Z,Z,Z
\right) \right) ,{\rm Sequence} \left( {\rm Prod} \left( Z,Z,Z,Z,Z,Z,
Z,Z,Z,Z \right) \right) ,{\rm Sequence} \left( {\rm Prod} \left( Z,Z,Z
,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z \right) \right) ,{\rm Sequence}
\left( {\rm Prod} \left( Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z
,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z \right)
\right) \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ 50965320-48000\,f \left( n+49 \right) +2412982\,n-45840\,f
\left( n+55 \right) -44760\,f \left( n+56 \right) -47976\,f \left( n+
50 \right) -33576\,f \left( n+20 \right) -41880\,f \left( n+24 \right)
-40080\,f \left( n+23 \right) -44760\,f \left( n+26 \right) -43440\,f
\left( n+25 \right) -45840\,f \left( n+27 \right) -47976\,f \left( n+
32 \right) -48000\,f \left( n+33 \right) -48000\,f \left( n+34 \right)
-16800\,f \left( n+69 \right) -14424\,f \left( n+70 \right) -12120\,f
\left( n+71 \right) -9936\,f \left( n+72 \right) -7920\,f \left( n+73
\right) -6120\,f \left( n+74 \right) -4560\,f \left( n+75 \right) -
3240\,f \left( n+76 \right) -2160\,f \left( n+77 \right) -1320\,f
\left( n+78 \right) -720\,f \left( n+79 \right) -336\,f \left( n+80
\right) -120\,f \left( n+81 \right) -24\,f \left( n+82 \right) -47664
\,f \left( n+52 \right) -47280\,f \left( n+53 \right) -46680\,f \left(
n+54 \right) -4560\,f \left( n+7 \right) -48000\,f \left( n+38 \right)
-48000\,f \left( n+37 \right) -48000\,f \left( n+47 \right) -48000\,f
\left( n+46 \right) -48000\,f \left( n+48 \right) -48000\,f \left( n+
42 \right) -48000\,f \left( n+43 \right) -48000\,f \left( n+44 \right)
-48000\,f \left( n+45 \right) -28800\,f \left( n+18 \right) -24000\,f
\left( n+16 \right) -26400\,f \left( n+17 \right) -35880\,f \left( n+
21 \right) -38064\,f \left( n+22 \right) -31200\,f \left( n+19 \right)
-47880\,f \left( n+51 \right) -40080\,f \left( n+59 \right) -33576\,f
\left( n+62 \right) -38064\,f \left( n+60 \right) -35880\,f \left( n+
61 \right) -47664\,f \left( n+30 \right) -47880\,f \left( n+31 \right)
-47280\,f \left( n+29 \right) -46680\,f \left( n+28 \right) +42839\,{n}
^{2}-24\,f \left( n \right) +338\,{n}^{3}+{n}^{4}-31200\,f \left( n+63
\right) -28800\,f \left( n+64 \right) -26400\,f \left( n+65 \right) -
24000\,f \left( n+66 \right) -21600\,f \left( n+67 \right) -19200\,f
\left( n+68 \right) -43440\,f \left( n+57 \right) -41880\,f \left( n+
58 \right) -14424\,f \left( n+12 \right) -16800\,f \left( n+13 \right)
-120\,f \left( n+1 \right) -6120\,f \left( n+8 \right) -12120\,f
\left( n+11 \right) -21600\,f \left( n+15 \right) -7920\,f \left( n+9
\right) -19200\,f \left( n+14 \right) -3240\,f \left( n+6 \right) -
9936\,f \left( n+10 \right) -720\,f \left( n+3 \right) -48000\,f
\left( n+36 \right) -336\,f \left( n+2 \right) -48000\,f \left( n+40
\right) -1320\,f \left( n+4 \right) -2160\,f \left( n+5 \right) -48000
\,f \left( n+35 \right) -48000\,f \left( n+39 \right) -48000\,f \left(
n+41 \right) =0,f \left( 0 \right) =1,f \left( 1 \right) =0,f \left( 2
\right) =1,f \left( 3 \right) =0,f \left( 4 \right) =1,f \left( 5
\right) =1,f \left( 6 \right) =1,f \left( 7 \right) =1,f \left( 8
\right) =1,f \left( 9 \right) =1,f \left( 10 \right) =3,f \left( 11
\right) =1,f \left( 12 \right) =3,f \left( 13 \right) =1,f \left( 14
\right) =3,f \left( 15 \right) =3,f \left( 16 \right) =3,f \left( 17
\right) =3,f \left( 18 \right) =3,f \left( 19 \right) =3,f \left( 20
\right) =7,f \left( 21 \right) =3,f \left( 22 \right) =7,f \left( 23
\right) =3,f \left( 24 \right) =7,f \left( 25 \right) =7,f \left( 26
\right) =7,f \left( 27 \right) =7,f \left( 28 \right) =7,f \left( 29
\right) =7,f \left( 30 \right) =13,f \left( 31 \right) =7,f \left( 32
\right) =13,f \left( 33 \right) =7,f \left( 34 \right) =13,f \left( 35
\right) =13,f \left( 36 \right) =13,f \left( 37 \right) =13,f \left(
38 \right) =13,f \left( 39 \right) =13,f \left( 40 \right) =22,f
\left( 41 \right) =13,f \left( 42 \right) =22,f \left( 43 \right) =13,
f \left( 44 \right) =22,f \left( 45 \right) =22,f \left( 46 \right) =22
,f \left( 47 \right) =22,f \left( 48 \right) =22,f \left( 49 \right) =
22,f \left( 50 \right) =35,f \left( 51 \right) =22,f \left( 52 \right)
=35,f \left( 53 \right) =22,f \left( 54 \right) =35,f \left( 55
\right) =35,f \left( 56 \right) =35,f \left( 57 \right) =35,f \left(
58 \right) =35,f \left( 59 \right) =35,f \left( 60 \right) =53,f
\left( 61 \right) =35,f \left( 62 \right) =53,f \left( 63 \right) =35,
f \left( 64 \right) =53,f \left( 65 \right) =53,f \left( 66 \right) =53
,f \left( 67 \right) =53,f \left( 68 \right) =53,f \left( 69 \right) =
53,f \left( 70 \right) =77,f \left( 71 \right) =53,f \left( 72 \right)
=77,f \left( 73 \right) =53,f \left( 74 \right) =77,f \left( 75
\right) =77,f \left( 76 \right) =77,f \left( 77 \right) =77,f \left(
78 \right) =77,f \left( 79 \right) =77,f \left( 80 \right) =108,f
\left( 81 \right) =77 \right\}
other formats
3.3. Closed form
\displaystyle
{\frac {3198151}{8000000}}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}
}^{40}+{{\rm \_Z}}^{30}+{{\rm \_Z}}^{20}+{{\rm \_Z}}^{10}+1 \right) }{
\frac {1}{250}}\, \left( -1-{\alpha}^{4}-{\alpha}^{24}-{\alpha}^{14}+{
\alpha}^{26}+{\alpha}^{36}+{\alpha}^{21}+{\alpha}^{31}-{\alpha}^{2}+{
\alpha}^{23}-{\alpha}^{10}+{\alpha}^{29}+{\alpha}^{39}+{\alpha}^{33}-{
\alpha}^{20}+{\alpha}^{28}+{\alpha}^{25}+{\alpha}^{35}-{\alpha}^{12}-{
\alpha}^{22}+{\alpha}^{27}+{\alpha}^{37}+{\alpha}^{38} \right) {\alpha}
^{-1-n}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{5}+2\,{{\rm \_Z}
}^{4}+2\,{{\rm \_Z}}^{3}+2\,{{\rm \_Z}}^{2}+2\,{\rm \_Z}+1 \right) }{
\frac {1}{6000000}}\, \left( 41\,{\alpha}^{4}+29+25\,{\alpha}^{2}+33\,{
\alpha}^{3}+37\,\alpha \right) {\alpha}^{-4-n} \left( n+1 \right)
\left( n+2 \right) \left( n+3 \right) +\sum _{\alpha={\rm RootOf}
\left( {{\rm \_Z}}^{5}+2\,{{\rm \_Z}}^{4}+2\,{{\rm \_Z}}^{3}+2\,{{\rm
\_Z}}^{2}+2\,{\rm \_Z}+1 \right) }-{\frac {1}{4000000}}\, \left( 2351\,
{\alpha}^{4}+1687+2027\,{\alpha}^{2}+1331\,{\alpha}^{3}+2659\,\alpha
\right) {\alpha}^{-3-n} \left( n+1 \right) \left( n+2 \right) +\sum _
{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{5}+2\,{{\rm \_Z}}^{4}+2\,{{
\rm \_Z}}^{3}+2\,{{\rm \_Z}}^{2}+2\,{\rm \_Z}+1 \right) }{\frac {1}{
4000000}}\, \left( 58631\,{\alpha}^{4}+24463+72751\,{\alpha}^{2}+34551
\,{\alpha}^{3}+45919\,\alpha \right) {\alpha}^{-n-2} \left( n+1
\right) +\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{5}+2\,{{\rm
\_Z}}^{4}+2\,{{\rm \_Z}}^{3}+2\,{{\rm \_Z}}^{2}+2\,{\rm \_Z}+1 \right)
}-{\frac {1}{8000000}}\, \left( 430871\,{\alpha}^{4}+58055+266551\,{
\alpha}^{2}-199929\,{\alpha}^{3}-512969\,\alpha \right) {\alpha}^{-1-n}
+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{10}+1 \right) }{\frac {
1}{320}}\, \left( \alpha-{\alpha}^{2}+{\alpha}^{3}-{\alpha}^{4}+{\alpha
}^{5}+{\alpha}^{6}+{\alpha}^{7}+{\alpha}^{8}+{\alpha}^{9}-1 \right) {
\alpha}^{-1-n}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{4}-{{\rm
\_Z}}^{3}+{{\rm \_Z}}^{2}-{\rm \_Z}+1 \right) }{\frac {1}{200000}}\,
\left( -2-{\alpha}^{2}+3\,{\alpha}^{3}-\alpha \right) {\alpha}^{-3-n}
\left( n+1 \right) \left( n+2 \right) +\sum _{\alpha={\rm RootOf}
\left( {{\rm \_Z}}^{4}-{{\rm \_Z}}^{3}+{{\rm \_Z}}^{2}-{\rm \_Z}+1
\right) }{\frac {1}{100000}}\, \left( -123+45\,{\alpha}^{2}+84\,{
\alpha}^{3}+43\,\alpha \right) {\alpha}^{-n-2} \left( n+1 \right) +
\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{4}-{{\rm \_Z}}^{3}+{{
\rm \_Z}}^{2}-{\rm \_Z}+1 \right) }{\frac {1}{100000}}\, \left( -1377-
648\,{\alpha}^{2}+2191\,{\alpha}^{3}+3094\,\alpha \right) {\alpha}^{-1-
n}+{\frac {9839}{2400000}}\,{n}^{2}+{\frac {6583}{80000}}\,n+{\frac {1}
{2400000}}\,{n}^{4}+{\frac {29}{400000}}\,{n}^{3}
other formats
3.4. Asymptotics
4. Ordinary
generating function
\displaystyle
-{\frac {1}{ \left( -1+{x}^{2} \right) \left( -1+{x}^{5} \right)
\left( -1+{x}^{10} \right) \left( -1+{x}^{20} \right) \left( -1+{x}^
{50} \right) }}
other formats
5. References
EIS A001319
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).