A268302 - OEIS

%I #3 Feb 26 2016 00:11:39

%S 3,8,28,144,736,4024,22912,134784,813476,5010904,31379808,199196320,

%T 1278911808,8290414024,54186864896,356711621984,2362968349568,

%U 15739688709864,105357470567228,708338644347808,4781146692837856,32387329985982176,220104493513881920,1500273861724289984,10253983269166864256,70258772726034956688,482514972838806347776,3320848006096569464080,22900703924095461843008,158216154716853989543080

%N G.f.: Sum_{n>=1} x^(n*(n-1)/2) * (G(x)^n + 1/G(x)^(n-1)), where G(x) is the g.f. of A268300.

%F G.f.: Product_{n>=1} (1-x^n) * (1 + x^n/G(x)) * (1 + x^(n-1)*G(x)), where G(x) is the g.f. of A268300.

%e G.f.: A(x) = 3 + 8*x + 28*x^2 + 144*x^3 + 736*x^4 + 4024*x^5 + 22912*x^6 + 134784*x^7 + 813476*x^8 + 5010904*x^9 + 31379808*x^10 +...

%e such that

%e A(x) = Sum_{n>=1} x^(n*(n-1)/2) * (G(x)^n + 1/G(x)^(n-1)),

%e that is,

%e A(x) = (G(x) + 1) + x*(G(x)^2 + 1/G(x)) + x^3*(G(x)^3 + 1/G(x)^2) + x^6*(G(x)^4 + 1/G(x)^3) + x^10*(G(x)^5 + 1/G(x)^4) + x^15*(G(x)^6 + 1/G(x)^5) +...,

%e where

%e G(x) = 2 + 7*2*x/4 + 119*2*x^2/4^2 + 2118*2*x^3/4^3 + 42523*2*x^4/4^4 + 914922*2*x^5/4^5 + 20745494*2*x^6/4^6 + 487390092*2*x^7/4^7 + 11764545555*2*x^8/4^8 + 289962708802*2*x^9/4^9 +...+ A268300(n)*2*x^n/4^n +...

%e satisfies:

%e -1 = Product_{n>=1} (1-x^n) * (1 - x^n/G(x)) * (1 - x^(n-1)*G(x)).

%Y Cf. A268300, A268301.

%K nonn

%O 0,1

%A _Paul D. Hanna_, Feb 26 2016