A095699 -id:A095699 - cp's OEIS frontend

A218379 Number of partitions of n into pentagonal parts.

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 9, 9, 10, 11, 11, 13, 13, 14, 15, 15, 17, 17, 19, 21, 22, 24, 24, 26, 28, 29, 31, 31, 34, 36, 38, 41, 42, 45, 47, 50, 53, 54, 57, 59, 63, 67, 69, 73, 76, 80, 84, 87, 91, 94, 99, 103, 107, 112, 118, 124

Offset: 0

Antonio Roldán, Oct 27 2012

nonn

			A(15)=5 because 15 = 12+1+1+1 = 5+5+5 = 5+5+1+1+1+1+1 = 5+1+1+1+1+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 with 12, 5, 1 pentagonal numbers.

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

Cf. A000326, A095699.

{for (n=1, 100,p=truncate((1+sqrt(24*n+1))/6);m=polcoeff(prod(k=1,p,q=(3*k-1)*k/2;sum(h=0,truncate(n/q+1),x^(h*q))),n);write("B218379.txt",n," ",m))}

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

A294591 Expansion of Product_{k>=1} 1/((1 - x^(2k-1))^(k(3k-1)/2)(1 - x^(2k))^(k(3*k+1)/2)).

Original entry on oeis.org

1, 1, 3, 8, 18, 40, 88, 184, 384, 783, 1573, 3110, 6087, 11745, 22450, 42466, 79597, 147890, 272632, 498696, 905846, 1634270, 2929804, 5220581, 9249440, 16297659, 28567571, 49825296, 86487331, 149438681, 257077485, 440378787, 751313413, 1276765557, 2161511352

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

Euler transform of the generalized pentagonal numbers (A001318).

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A000294, A001318, A095699, A278768, A294654, A294655, A294667, A294669.

nmax = 34; CoefficientList[Series[Product[1/((1 - x^(2 k - 1))^(k (3 k - 1)/2) (1 - x^(2 k))^(k (3 k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Ceiling[d/2] Ceiling[(3 d + 1)/2]/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A001318(k).

a(n) ~ exp(Pi * 2^(5/4) / (3*5^(1/4)) * n^(3/4) + 3*Zeta(3) * sqrt(5*n) / (2^(3/2) * Pi^2) + (Pi/48 - 45*Zeta(3)^2 / (8*Pi^5)) * (5*n/2)^(1/4) + 225*Zeta(3)^3 / (8*Pi^8) - 11*Zeta(3) / (64*Pi^2))/ (2^(95/48) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 07 2017

A290942 Number of partitions of n into distinct generalized pentagonal numbers (A001318).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 0, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 4, 3, 3, 3, 2, 5, 4, 5, 4, 2, 3, 3, 6, 6, 5, 5, 4, 5, 7, 8, 8, 7, 6, 6, 6, 8, 9, 9, 9, 7, 8, 9, 9, 11, 10, 11, 11, 10, 12, 10, 14, 15, 14, 14, 11, 13, 13, 17, 17, 14, 15, 14, 17, 20, 19, 20, 20, 20, 21, 20, 21, 21, 25, 26, 23, 22, 21, 24, 27

Offset: 0

Ilya Gutkovskiy, Aug 14 2017

nonn

			a(15) = 3 because we have [15], [12, 2, 1] and [7, 5, 2, 1].

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

Cf. A000009, A001318, A095699, A218379, A218380, A279221, A280952, A281083.

nmax = 90; CoefficientList[Series[Product[(1 + x^(k (3 k - 1)/2)) (1 + x^(k (3 k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

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

A294621 Number of partitions of n into generalized heptagonal numbers (A085787).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 18, 20, 21, 23, 26, 29, 32, 35, 38, 41, 45, 49, 53, 59, 64, 69, 73, 80, 87, 94, 101, 109, 117, 125, 134, 145, 156, 167, 178, 190, 202, 217, 232, 249, 265, 282, 299, 318, 339, 361, 384, 408, 432, 457, 484, 514, 545, 578, 610, 646

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

			a(8) = 4 because we have [7, 1], [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

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

Cf. A007294, A085787, A095699, A279012, A294622, A294623.

nmax = 70; CoefficientList[Series[Product[1/((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/((1 - x^(k*(5*k-3)/2))*(1 - x^(k*(5*k+3)/2))).

A294622 Number of partitions of n into generalized octagonal numbers (A001082).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 8, 8, 9, 9, 10, 13, 13, 14, 16, 17, 20, 20, 21, 24, 25, 28, 31, 33, 36, 37, 40, 45, 47, 50, 55, 59, 65, 67, 70, 77, 81, 87, 94, 99, 107, 111, 117, 127, 133, 141, 152, 160, 172, 178, 186, 201, 210, 223, 237, 249, 267, 276, 289, 308, 322, 341, 360, 378, 401

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

			a(8) = 3 because we have [8], [5, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

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

Cf. A001082, A007294, A095699, A279041, A294621, A294624.

nmax = 74; CoefficientList[Series[Product[1/((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/((1 - x^(k*(3*k-2)))*(1 - x^(k*(3*k+2)))).

A296237 Expansion of Product_{k>0} 1/(1 - x^(k(3k+1)/2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 3, 5, 4, 6, 4, 6, 5, 6, 6, 7, 7, 7, 7, 9, 8, 10, 9, 11, 10, 11, 12, 12, 13, 13, 14, 15, 14, 17, 16, 19, 18, 20, 20, 21, 22, 23, 24, 25, 25, 28, 27, 30, 29, 33, 32, 35, 35, 37, 38, 39

Offset: 0

Seiichi Manyama, Dec 09 2017

nonn

Integer partitions into second or "negative" pentagonal numbers (A005449) .

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A005449, A095699, A218379, A296238.

nmax = 100; CoefficientList[Series[Product[1/(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(1/prod(k=1, N, (1-x^(k*(3*k+1)/2))))

A025781 Expansion of 1/((1-x)(1-x^5)(1-x^12)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 10, 10, 11, 11, 12, 13, 13, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts 1, 5, and 12. - Joerg Arndt, Mar 18 2013

Up to and including a(21) this is the same as the expansion of Product_{k>=1} 1/(1-x^(k*(3*k-1)/2)), which appears as a convolution factor in A095699. - R. J. Mathar, Mar 18 2013

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1,0,0,0,0,0,1,-1,0,0,0,-1,1).

CoefficientList[Series[1/((1-x)(1-x^5)(1-x^12)),{x,0,70}],x] (* or *) LinearRecurrence[{1,0,0,0,1,-1,0,0,0,0,0,1,-1,0,0,0,-1,1},{1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,5,5,6},70] (* Harvey P. Dale, May 11 2014 *)

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(5)=2, a(6)=2, a(7)=2, a(8)=2, a(9)=2, a(10)=3, a(11)=3, a(12)=4, a(13)=4, a(14)=4, a(15)=5, a(16)=5, a(17)=6, a(n)=a(n-1)+a(n-5)-a(n-6)+a(n-12)-a(n-13)-a(n-17)+a(n-18). - Harvey P. Dale, May 11 2014

cp's OEIS Frontend

A218379 Number of partitions of n into pentagonal parts.

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A294591 Expansion of Product_{k>=1} 1/((1 - x^(2*k-1))^(k*(3*k-1)/2)*(1 - x^(2*k))^(k*(3*k+1)/2)).

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A290942 Number of partitions of n into distinct generalized pentagonal numbers (A001318).

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A294621 Number of partitions of n into generalized heptagonal numbers (A085787).

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A294622 Number of partitions of n into generalized octagonal numbers (A001082).

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A296237 Expansion of Product_{k>0} 1/(1 - x^(k*(3*k+1)/2)).

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Programs

Mathematica

PARI

A025781 Expansion of 1/((1-x)(1-x^5)(1-x^12)).

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Mathematica

Formula

A294591 Expansion of Product_{k>=1} 1/((1 - x^(2k-1))^(k(3k-1)/2)(1 - x^(2k))^(k(3*k+1)/2)).

A296237 Expansion of Product_{k>0} 1/(1 - x^(k(3k+1)/2)).