A278428 - OEIS

%I #15 Oct 03 2023 03:07:18

%S 1,1,1,2,6,17,46,128,373,1119,3405,10464,32478,101781,321642,1023512,

%T 3276326,10543100,34088806,110690682,360810160,1180195810,3872588051,

%U 12743937024,42049240694,139082885503,461072582522,1531697761470,5098246648103,17000237006441

%N Series reversion of g.f. (1/2)*x*(-1; -x)_inf, where (a; q)_inf is the q-Pochhammer symbol.

%C (1/2)*x*(-1; -x)_inf is the g.f. for A081360 shifted right.

%H Vaclav Kotesovec, <a href="/A278428/b278428.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.

%F a(n) ~ c * d^n / n^(3/2), where c = 0.1211369424750398272226454930396... and d = A318204 = 3.509754327949703340437273523375193698454789733931739911... - _Vaclav Kotesovec_, Nov 23 2016

%t InverseSeries[x QPochhammer[-1, -x]/2 + O[x]^35][[3]]

%t (* Calculation of constant c: *) 1/Sqrt[(4/s^2 - s*Derivative[0, 2][QPochhammer][-1, -s]/r) * Pi] /. FindRoot[{2*r == s*QPochhammer[-1, -s], 2*r == s^2*Derivative[0, 1][QPochhammer][-1, -s]}, {r, 1/3}, {s, 1/2}, WorkingPrecision -> 120] (* _Vaclav Kotesovec_, Oct 03 2023 *)

%Y Cf. A081360, A109085, A171805, A181315, A255526.

%K nonn

%O 1,4

%A _Vladimir Reshetnikov_, Nov 21 2016