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 1-6 of 6 results.

A035678 Number of partitions of n into parts 8k and 8k+7 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 3, 6, 0, 0, 0, 0, 1, 3, 7, 11, 0, 0, 0, 1, 3, 7, 14, 18, 0, 0, 1, 3, 7, 15, 25, 29, 0, 1, 3, 7, 15, 28, 43, 44, 1, 3, 7, 15, 29, 50, 69, 67, 3, 7, 15, 29, 53, 84, 110, 99, 7, 15, 29, 54, 91, 138, 168
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035677, A035679-A035699.

Programs

  • Maple
    np:= combinat:-numbpart:
NP:= proc(n,m) if m > n then np(n) else np(n,m) fi end proc;
f:= proc(n) local r0;
   r0:= (-n) mod 8;
   add(np(s)*add(NP((n-8*s-7*r)/8, r), r=r0 .. floor((n-8*s)/7), 8), s=1..floor((n-1)/8))
end proc:
seq(f(n),n=1..100); # Robert Israel, Apr 06 2016
  • Mathematica
    nmax = 86; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 7;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 86; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 7)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 13 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 7)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 13 2020

A035680 Number of partitions of n into parts 8k+1 and 8k+3 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 5, 6, 6, 8, 9, 9, 11, 12, 15, 18, 19, 23, 26, 27, 31, 34, 39, 45, 49, 56, 62, 66, 74, 80, 89, 101, 109, 123, 136, 144, 160, 173, 187, 210, 227, 249, 275, 293, 319, 346, 371, 408, 442, 480, 525, 562, 608, 655, 701, 763, 822, 887, 963
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035679, A035681-A035699.

Programs

  • Mathematica
    nmax = 64; s1 = Range[0, nmax/8]*8 + 1; s2 = Range[0, nmax/8]*8 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2], x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 64; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(8*k + 3))). - Robert Price, Aug 15 2020

A035442 Number of partitions of n into parts 8k or 8k+2.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 3, 0, 3, 0, 5, 0, 7, 0, 8, 0, 8, 0, 11, 0, 15, 0, 17, 0, 18, 0, 23, 0, 30, 0, 35, 0, 37, 0, 45, 0, 57, 0, 66, 0, 71, 0, 84, 0, 104, 0, 121, 0, 131, 0, 151, 0, 183, 0, 212, 0, 231, 0, 263, 0, 313, 0, 362, 0, 396, 0, 446, 0, 523, 0, 601, 0, 660, 0
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035679.

Programs

  • Mathematica
    nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+8))*(1 - x^(8k+2))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 50; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8}~Join~{Range[0, kmax]*8 + 2}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 03 2020 *)

Formula

If n is even, a(n) ~ 2 * exp(Pi*sqrt(n/6)) * Gamma(5/4) / (6^(3/8) * Pi^(3/4) * n^(7/8)). - Vaclav Kotesovec, Aug 26 2015

A035447 Number of partitions of n into parts 8k or 8k+7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 1, 2, 4, 3, 0, 0, 0, 1, 2, 5, 7, 5, 0, 0, 1, 2, 5, 9, 12, 7, 0, 1, 2, 5, 10, 17, 19, 11, 1, 2, 5, 10, 19, 28, 30, 16, 2, 5, 10, 20, 33, 47, 46, 24, 5, 10, 20, 35, 57, 74, 69, 35, 10, 20, 36, 62, 93, 116, 102, 52, 20, 36, 64
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035679.

Programs

  • Mathematica
    nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+8))*(1 - x^(8k+7))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 50; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8}~Join~{Range[0, kmax]*8 + 7}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 03 2020 *)

Formula

a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(7/8) / (2^(23/16) * 3^(11/16) * Pi^(1/8) * n^(19/16)). - Vaclav Kotesovec, Aug 26 2015

A035448 Number of partitions of n into parts 8k+1 or 8k+2.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 6, 8, 9, 11, 12, 14, 15, 17, 19, 23, 26, 31, 34, 39, 42, 47, 51, 58, 65, 74, 82, 92, 100, 110, 119, 132, 145, 163, 179, 199, 216, 237, 255, 279, 303, 334, 365, 401, 435, 473, 509, 552, 596, 650, 705, 770, 832, 902, 968, 1044, 1121, 1213
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035679.

Programs

  • Mathematica
    nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+1))*(1 - x^(8k+2))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 60; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8 + 1}~Join~{Range[0, kmax]*8 + 2}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 03 2020 *)

Formula

a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(1/4) * Gamma(1/8) * 3^(1/16) / (8 * 2^(11/16) * Pi^(13/8) * n^(7/16)). - Vaclav Kotesovec, Aug 26 2015

A035449 Number of partitions of n into parts 8k+1 or 8k+3.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 6, 8, 8, 9, 11, 11, 13, 16, 17, 20, 23, 25, 28, 31, 34, 38, 43, 48, 53, 59, 65, 70, 78, 86, 93, 105, 115, 125, 139, 150, 162, 179, 193, 211, 233, 251, 274, 298, 320, 348, 377, 407, 443, 480, 519, 561, 604, 651, 700, 755, 815, 876, 946
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035679.

Programs

  • Maple
    N:= 100: # for a(1)..a(N)
P:= 1/mul((1-x^(8*k+1))*(1-x^(8*k+3)),k=0..floor((N-1)/8)):
S:= series(P,x,N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Aug 28 2018
  • Mathematica
    nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+1))*(1 - x^(8k+3))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 60; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8 + 1}~Join~{Range[0, kmax]*8 + 3}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 03 2020 *)

Formula

G.f.: Product_{k>=0} 1/((1-x^(8*k+1))*(1-x^(8*k+3))). - Robert Israel, Aug 29 2018
a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(3/8) * Gamma(1/8) / (8 * Pi^(3/2) * sqrt(2*n)). - Vaclav Kotesovec, Aug 26 2015
Showing 1-6 of 6 results.

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