A306147 -id:A306147 - cp's OEIS frontend

A103265 Number of partitions of n in which both even and odd square parts occur in 2 forms c, c* and with multiplicity 1. There is no restriction on parts which are twice squares.

1, 2, 2, 2, 4, 6, 6, 6, 8, 12, 14, 14, 16, 22, 26, 26, 30, 38, 44, 46, 52, 62, 70, 74, 80, 96, 110, 116, 124, 146, 166, 174, 186, 210, 238, 254, 272, 302, 338, 362, 384, 426, 470, 502, 532, 588, 646, 686, 726, 792, 872, 926, 980, 1062

Offset: 0

Noureddine Chair, Feb 27 2005

Convolution of A001156 and A033461. - Vaclav Kotesovec, Aug 18 2015

			E.g. a(8)=8 because 8 can be written as 8, 44*, 422, 4*22, 4211*, 4*211*, 2222, 22211*.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001156, A015128, A033461, A280263, A279227, A306147.

series(product((1+x^(k^2))/(1-x^(k^2)),k=1..100),x=0,100);

nmax = 50; CoefficientList[Series[Product[(1+x^(k^2)) / (1-x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)

G.f.: Product_{k>0}((1+x^k^2)/(1-x^k^2)).

a(n) ~ exp(3 * ((4-sqrt(2))*zeta(3/2))^(2/3) * Pi^(1/3) * n^(1/3) / 4) * ((4-sqrt(2))*zeta(3/2))^(2/3) / (2^(7/2) * sqrt(3) * Pi^(7/6) * n^(7/6)). - Vaclav Kotesovec, Dec 29 2016

A306082 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^(k^2)).

Original entry on oeis.org

1, 1, 3, 13, 99, 901, 8763, 92653, 1125939, 16333141, 274594923, 5041348093, 97841114979, 2007694705381, 44043941312283, 1036207737976333, 25969433606691219, 688418684249653621, 19275116061819888843, 571069469474068377373, 17898523203378840958659

Offset: 0

Vaclav Kotesovec, Jun 20 2018

nonn

Conjecture: for positive integer k, reducing the sequence modulo k produces an eventually periodic sequence with period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 1, 3, 6, 1, 5, 6, 1, 3, 6, 1, 5, 6, 1, 3, 6, 1, 5, 6, ...], with an apparent period of 6 beginning at a(1). - Peter Bala, Feb 22 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Cf. A001156, A294529, A306083, A306147.

a:=series(mul(1/(1-(exp(x)-1)^(k^2)),k=1..100),x=0,21): seq(n!*coeff(a, x, n), n=0..20); # Paolo P. Lava, Mar 26 2019

nmax = 20; CoefficientList[Series[Product[1/(1 - (Exp[x] - 1)^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = Sum_{k=0..n} Stirling2(n,k) * A001156(k) * k!.

a(n) ~ n! * exp(3 * 2^(-5/3) * Zeta(3/2)^(2/3) * (Pi*n/log(2))^(1/3)) * Zeta(3/2)^(2/3) / (2^(13/6) * sqrt(3) * Pi^(7/6) * n^(7/6) * (log(2))^(n - 1/6)).

A306083 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^(k^2)).

Original entry on oeis.org

1, 1, 1, 1, 25, 361, 3361, 25201, 166825, 1383481, 25879921, 651816001, 14450460025, 280347467401, 5253918022081, 107822784560401, 2578135250199625, 69030779356572121, 1953531819704493841, 56903093167217522401, 1689294590583626265625

Offset: 0

Vaclav Kotesovec, Jun 20 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Cf. A033461, A306082, A306147.

a:=series(mul(1+(exp(x)-1)^(k^2),k=1..100),x=0,21): seq(n!*coeff(a, x, n),n=0..20); # Paolo P. Lava, Mar 26 2019

nmax = 20; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = Sum_{k=0..n} Stirling2(n,k) * A033461(k) * k!.

a(n) ~ n! * exp(3 * (Pi/log(2))^(1/3) * ((sqrt(2) - 1) * Zeta(3/2))^(2/3) * n^(1/3) / 4) * ((sqrt(2) - 1) * Zeta(3/2) / Pi)^(1/3) / (2 * sqrt(6) * n^(5/6) * log(2)^(n + 1/6)).

cp's OEIS Frontend

A103265 Number of partitions of n in which both even and odd square parts occur in 2 forms c, c* and with multiplicity 1. There is no restriction on parts which are twice squares.

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A306082 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^(k^2)).

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Author

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Comments

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Maple

Mathematica

Formula

A306083 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^(k^2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula