A296164 - OEIS

%I #15 Jan 17 2024 07:23:29

%S 1,1,3,10,35,131,498,1919,7459,29170,114653,452552,1792754,7124040,

%T 28386081,113372690,453743907,1819317153,7306575042,29386858821,

%U 118348662525,477188876405,1926137365804,7782398551661,31472648050930,127384123318906,515978637418884

%N a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^n.

%H G. C. Greubel, <a href="/A296164/b296164.txt">Table of n, a(n) for n = 0..500</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SchursPartitionTheorem.html">Schur's Partition Theorem</a>

%F a(n) = [x^n] Product_{k>=1} 1/((1 - x^(6*k-1))*(1 - x^(6*k-5)))^n.

%F a(n) ~ c * d^n / sqrt(n), where d = 4.129321588075726742506... and c = 0.25764349816429874321... - _Vaclav Kotesovec_, May 18 2018

%t Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(3 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]

%t Table[SeriesCoefficient[Product[1/((1 - x^(6 k - 1)) (1 - x^(6 k - 5)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]

%t (* Calculation of constants {d,c}: *) With[{k = 3}, {1/r, Sqrt[QPochhammer[-1, (r*s)^k] / (2*Pi*(r^2*s*Derivative[0, 2][QPochhammer][-1, r*s] - k^2*(r*s)^(2*k) * Derivative[0, 2][QPochhammer][-1, (r*s)^k] - k*(1 + k)*(r*s)^k * Derivative[0, 1][QPochhammer][-1, (r*s)^k]))]} /. FindRoot[{s == QPochhammer[-1, r*s]/QPochhammer[-1, (r*s)^k], QPochhammer[-1, (r*s)^k] + k*(r*s)^k*Derivative[0, 1][QPochhammer][-1, (r*s)^k] == r*Derivative[0, 1][QPochhammer][-1, r*s]}, {r, 1/4}, {s, 2}, WorkingPrecision -> 70]] (* _Vaclav Kotesovec_, Jan 17 2024 *)

%Y Cf. A003105, A058484, A058539, A103262, A255526, A296163.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Dec 06 2017