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A278296 Expansion of ((sqrt(2)-1)*(-sqrt(2);x)_inf - (sqrt(2)+1)*(sqrt(2);x)_inf)/2, where (a;q)_inf is the q-Pochhammer symbol.

%I A278296 #15 Feb 16 2025 08:33:37
%S A278296 1,0,0,2,2,4,4,6,6,8,12,14,18,24,32,38,50,60,76,90,110,136,164,194,
%T A278296 234,280,336,402,474,564,668,790,926,1096,1276,1494,1754,2040,2368,
%U A278296 2758,3186,3692,4268,4922,5670,6528,7492,8594,9858,11272,12888,14722,16786
%N A278296 Expansion of ((sqrt(2)-1)*(-sqrt(2);x)_inf - (sqrt(2)+1)*(sqrt(2);x)_inf)/2, where (a;q)_inf is the q-Pochhammer symbol.
%C A278296 The q-Pochhammer symbol (a;q)_inf = Product_{k>=0} (1 - a*q^k).
%C A278296 a(n) agrees with A238132(n) for 0 < n < 21.
%H A278296 G. C. Greubel, <a href="/A278296/b278296.txt">Table of n, a(n) for n = 0..1000</a>
%H A278296 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.
%p A278296 qP := (x,y) -> (y-1)*QDifferenceEquations:-QPochhammer(-y,x,99):
%p A278296 dP := x -> (qP(x,sqrt(2)) + qP(x,-sqrt(2)))/2:
%p A278296 simplify(expand(dP(x),x)): seq(coeff(%,x,n), n=0..52); # _Peter Luschny_, Nov 17 2016
%t A278296 Simplify@(((Sqrt[2] - 1) QPochhammer[-Sqrt[2], x] - (Sqrt[2] + 1) QPochhammer[Sqrt[2], x])/2 + O[x]^53)[[3]]
%Y A278296 Cf. A238132.
%K A278296 nonn
%O A278296 0,4
%A A278296 _Vladimir Reshetnikov_, Nov 16 2016