cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A036892 Number of partitions of 5n such that cn(1,5) = cn(4,5) < cn(0,5) = cn(2,5) = cn(3,5).

Original entry on oeis.org

0, 1, 3, 7, 14, 30, 62, 133, 275, 562, 1109, 2145, 4035, 7457, 13509, 24115, 42405, 73667, 126420, 214681, 360778, 600625, 990756, 1620449, 2628504, 4230770, 6758916, 10721739, 16892541, 26443435, 41137558, 63618639, 97825383, 149605621, 227593695
Offset: 1

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Author

Keywords

Comments

Alternatively, number of partitions of 5n such that cn(2,5) = cn(3,5) < cn(0,5) = cn(1,5) = cn(4,5).
For a given partition, cn(i,n) means the number of its parts equal to i modulo n.

Formula

a(n) = A036891(n) - A036895(n)
a(n) = A036890(n) - A036894(n)
a(n) = A036889(n) - A046776(n)

Extensions

Terms a(10) onward from Max Alekseyev, Dec 11 2011

A036895 Number of partitions of 5n such that cn(1,5) = cn(4,5) < cn(2,5) = cn(3,5) < cn(0,5).

Original entry on oeis.org

0, 0, 1, 4, 12, 29, 67, 146, 313, 654, 1342, 2691, 5291, 10184, 19237, 35680, 65102, 116967, 207241, 362423, 626265, 1070100, 1809513, 3029902, 5026778, 8267109, 13484325, 21821771, 35051459, 55901678, 88550088, 139355911, 217949589, 338837468
Offset: 1

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Author

Keywords

Comments

Alternatively, number of partitions of 5n such that cn(2,5) = cn(3,5) < cn(1,5) = cn(4,5) < cn(0,5).
For a given partition, cn(i,n) means the number of its parts equal to i modulo n.

Formula

a(n) = A036891(n) - A036892(n)
a(n) = A036887(n) - A202086(n)

Extensions

Terms a(10) onward from Max Alekseyev, Dec 11 2011

A036882 Number of partitions of 5n such that cn(1,5) = cn(4,5) <= cn(2,5) = cn(3,5) <= cn(0,5).

Original entry on oeis.org

1, 1, 3, 8, 22, 54, 128, 282, 602, 1235, 2474, 4831, 9263, 17418, 32242, 58737, 105519, 186976, 327238, 565896, 967910, 1638175, 2745588, 4558864, 7503737, 12248234, 19835700, 31882617, 50881290, 80648122, 126998962, 198743334, 309163475, 478177505, 735522058
Offset: 0

Views

Author

Keywords

Comments

Alternatively, number of partitions of 5n such that cn(2,5) = cn(3,5) <= cn(1,5) = cn(4,5) <= cn(0,5).
For a given partition cn(i,n) means the number of its parts equal to i modulo n.

Programs

  • Maple
    mkl:= proc(i, l) local ll, x, j; j:= irem (i, 5); j:= `if` (j=0, 5, j); ll:= applyop (x->x+1, j, l); map (x-> x-min(ll[]), ll) end:
    g:= proc (n, i, t) local x; if n<0 then 0 elif n=0 then `if` (t[1]=t[4] and t[4]<=t[2] and t[2]=t[3] and t[3]<=t[5], 1, 0) elif i=0 then 0 elif i=1 then g (0, 0, applyop (x-> x+n, 1, t)) elif i=2 then `if` (t[2]>t[3], 0, g (n-2*(t[3]-t[2]), 1, subsop(2=t[3], t))) elif (i=3 or i=4) and t[i]>t[5] then 0 else g(n, i, t):= g (n, i-1, t) +g (n-i, i, mkl(i, t)) fi end:
    a:= n-> g(5*n, 5*n, [0, 0, 0, 0, 0]):
    seq(a(n), n=0..15);  # Alois P. Heinz, Jul 07 2009
  • Mathematica
    mkl[i_, l_List] := Module[{ll, x, j}, j = Mod[i, 5]; j = If[j == 0, 5, j]; ll = MapAt [#+1&, l, j]; ll - Min[ll]]; g[n_, i_, t_List] := g[n, i, t] = Which[n<0, 0, n == 0, If[t[[1]] == t[[4]] && t[[4]] <= t[[2]] && t[[2]] == t[[3]] && t[[3]] <= t[[5]], 1, 0], i == 0, 0, i == 1, g[0, 0, MapAt [#+n&, t, 1]], i == 2, If[t[[2]] > t[[3]], 0, g[n - 2*(t[[3]] - t[[2]]), 1, ReplacePart[t, 2 -> t[[3]]]]], (i == 3 || i == 4) && t[[i]] > t[[5]], 0, True, g[n, i, t] = g[n, i-1, t] + g[n-i, i, mkl[i, t]]]; a[n_] := a[n] = g[5*n, 5*n, {0, 0, 0, 0, 0}]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 32}] (* Jean-François Alcover, Dec 23 2015, after Alois P. Heinz *)

Formula

a(n) = A036889(n) + A036887(n)
a(n) = A202085(n) + A036891(n)

Extensions

a(10)-a(32) from Alois P. Heinz, Jul 07 2009
Edited by Max Alekseyev, Dec 11 2011
More terms from Alois P. Heinz, Dec 23 2015

A202085 Number of partitions of 5n such that cn(1,5) = cn(4,5) = cn(2,5) = cn(3,5) <= cn(0,5).

Original entry on oeis.org

1, 2, 4, 11, 28, 69, 153, 323, 647, 1258, 2380, 4427, 8092, 14601, 25991, 45724, 79469, 136604, 232235, 390806, 651132, 1074863, 1758595, 2853386, 4592952, 7337821, 11639376, 18337780, 28704122, 44653849, 69055688, 106188925, 162402533
Offset: 1

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Author

Max Alekseyev, Dec 11 2011

Keywords

Comments

For a given partition, cn(i,n) means the number of its parts equal to i modulo n.

Formula

a(n) = A046776(n) + A202086(n)
a(n) = A036882(n) - A036891(n)

A202091 Number of partitions of 5n such that cn(1,5) = cn(4,5) and cn(2,5) = cn(3,5).

Original entry on oeis.org

1, 3, 11, 32, 88, 221, 532, 1213, 2672, 5676, 11724, 23568, 46315, 89076, 168124, 311763, 569000, 1023128, 1814776, 3178000, 5499588, 9411392, 15938221, 26726372, 44402336, 73121988, 119418609, 193488816, 311150404, 496783420, 787753316
Offset: 0

Views

Author

Max Alekseyev, Dec 11 2011

Keywords

Comments

For a given partition, cn(i,n) means the number of its parts equal to i modulo n.

Crossrefs

Formula

a(n) = A046776(n) + A202086(n) + A202088(n) + 2*( A036886(n) + A036892(n) + A036893(n) + A036894(n) + A036895(n) )
a(n) = A202192(n) + 2*( A036886(n) + A036892(n) + A036893(n) + A036894(n) + A036895(n) )
Showing 1-5 of 5 results.