A290942 -id:A290942 - cp's OEIS frontend

A218380 Number of partitions of n into distinct pentagonal parts.

1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 2, 1, 0, 1, 2, 2, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 2, 2

Offset: 0

Antonio Roldán, Oct 27 2012

nonn

			A(98)=3 because 98 = 12 + 35 + 51 = 1 + 5 + 92 = 1 + 5 + 22 + 70 with 1, 5, 22, 70, 92 pentagonal numbers.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..100 from Antonio Roldán)

Cf. A000326, A218379, A290942.

{ for (n=1, 100, m=polcoeff(prod(k=1, truncate(1+sqrt(24*n+1))/6, 1+x^(k*(3*k-1)/2)), n);write("B218380.txt",n, " ",m)) }

a(0) = 1 prepended by Seiichi Manyama, Dec 09 2017

A095699 Number of partitions of n into generalized pentagonal numbers.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 8, 10, 12, 14, 18, 20, 25, 29, 34, 40, 45, 53, 60, 69, 80, 89, 103, 114, 131, 147, 165, 186, 207, 232, 258, 286, 319, 352, 392, 432, 477, 525, 578, 636, 699, 765, 839, 916, 1002, 1093, 1192, 1298, 1413, 1536, 1671, 1810, 1965, 2126, 2304

Offset: 0

Jon Perry, Jul 06 2004

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A001318, A218379, A290942.

nmax = 100; CoefficientList[Series[1/Product[(1-x^(k*(3*k-1)/2)) * (1-x^(k*(3*k+1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 10 2017 *)

b(n) = (3*n^2 + 2*n + (n%2) * (2*n + 1)) / 8; \\ A001318
N=66; x='x+O('x^N);
Vec(1/prod(k=1,N, (1-x^b(k))) )
\\ Joerg Arndt, Oct 13 2014

G.f.: 1/Product_{k>=1} (1-x^(k*(3*k-1)/2))*(1-x^(k*(3*k+1)/2)).

A294623 Number of partitions of n into distinct generalized heptagonal numbers (A085787).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 0, 2, 3, 1, 0, 3, 3, 1, 2, 2, 1, 1, 3, 3, 3, 2, 1, 2, 3, 4, 3, 2, 2, 3, 3, 3, 5, 3, 1, 3, 4, 3, 4, 5, 2, 3, 5, 4, 3, 4, 5, 4, 4, 3, 5, 5, 3, 5, 7, 5, 3, 6, 6, 6, 6, 5, 5, 6, 6, 5, 8, 7, 5, 5, 6, 7, 8, 8

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

			a(18) = 2 because we have [18] and [13, 4, 1].

Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A024940, A085787, A279280, A290942, A294621, A294624.

nmax = 100; CoefficientList[Series[Product[(1 + x^(k (5 k - 3)/2)) (1 + x^(k (5 k + 3)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^(k*(5*k-3)/2))*(1 + x^(k*(5*k+3)/2)).

A294624 Number of partitions of n into distinct generalized octagonal numbers (A001082).

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 2, 0, 1, 1, 1, 1, 0, 2, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 2, 3, 1, 1, 2, 2, 1, 0, 1, 3, 2, 2, 3, 1, 1, 1, 3, 5, 2, 2, 3, 2, 2, 1, 3, 5, 2, 1, 3, 3, 2, 1, 3, 6, 3, 1, 3, 4, 3, 1, 4, 7, 3, 0, 3, 6, 4, 1, 2, 7, 5, 2, 4, 5, 5, 2

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

			a(21) = 2 because we have [21] and [16, 5].

Eric Weisstein's World of Mathematics, Octagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A001082, A024940, A279281, A290942, A294622, A294623.

nmax = 100; CoefficientList[Series[Product[(1 + x^(k (3 k - 2))) (1 + x^(k (3 k + 2))), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^(k*(3*k-2)))*(1 + x^(k*(3*k+2))).

A296238 Expansion of Product_{k>0} (1 + x^(k(3k+1)/2)).

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 2, 0, 2, 1, 1, 1, 0

Offset: 0

Seiichi Manyama, Dec 09 2017

nonn

Cf. A218380, A290942, A296237.

nmax = 100; CoefficientList[Series[Product[(1 + x^(k*(3*k+1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 10 2017 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+x^(k*(3*k+1)/2))))

cp's OEIS Frontend

A218380 Number of partitions of n into distinct pentagonal parts.

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A095699 Number of partitions of n into generalized pentagonal numbers.

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A294623 Number of partitions of n into distinct generalized heptagonal numbers (A085787).

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A294624 Number of partitions of n into distinct generalized octagonal numbers (A001082).

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A296238 Expansion of Product_{k>0} (1 + x^(k*(3*k+1)/2)).

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A296238 Expansion of Product_{k>0} (1 + x^(k(3k+1)/2)).