A304632 Expansion of (1/(1 - x))* Product_{k>=1} (1 + x^k)/(1 + x^(3*k)).

1, 2, 3, 4, 5, 7, 9, 12, 15, 18, 22, 27, 33, 40, 48, 57, 67, 79, 93, 109, 127, 147, 170, 196, 226, 260, 298, 340, 387, 440, 500, 567, 641, 723, 814, 916, 1030, 1156, 1295, 1448, 1617, 1804, 2011, 2239, 2489, 2763, 3064, 3395, 3759, 4158, 4594, 5070, 5590, 6159, 6781, 7460, 8199, 9003

Offset

0,2

Comments

Partial sums of A003105.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Eric Weisstein's World of Mathematics, Schur's Partition Theorem

Index entries for sequences related to partitions

Formula

G.f.: (1/(1 - x))*Product_{k>=0} 1/((1 - x^(6*k+1))*(1 - x^(6*k+5))).

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

a(n) ~ exp(sqrt(2*n)*Pi/3) * sqrt(3) / (Pi * 2^(3/4) * n^(1/4)). - Vaclav Kotesovec, May 19 2018

Mathematica

nmax = 57; CoefficientList[Series[1/(1 - x) Product[(1 + x^k)/(1 + x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 57; CoefficientList[Series[1/(1 - x) Product[1/((1 - x^(6 k + 1)) (1 - x^(6 k + 5))), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 57; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k + x^(2 k)), {k, 0, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000070, A003105, A036469, A304630, A304631.

Sequence in context: A033100 A030741 A190269 * A306385 A039853 A062188

Adjacent sequences: A304629 A304630 A304631 * A304633 A304634 A304635

Keyword

nonn

Author

Ilya Gutkovskiy, May 15 2018

Status

approved