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

%I #8 Nov 07 2024 15:21:56

%S 1,3,24,640,61440,22020096,30064771072,158329674399744,

%T 3242591731706757120,259730156557830486753280,

%U 81704042592835098143342198784,101249788741429138756344678419791872,495451126236886402802673428420654515879936

%N G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * 2^(n*(n-1)) * x^n.

%F The convolution cube-root yields A202943.

%e G.f.: A(x) = 1 + 3*x + 24*x^2 + 640*x^3 + 61440*x^4 + 22020096*x^5 +...

%e where

%e A(x) = 1 + 3*x + 6*2^2*x^2 + 10*2^6*x^3 + 15*2^12*x^4 + 21*2^20*x^5 +...

%e Note that the cube root of the g.f. is an integer series:

%e A(x)^(1/3) = 1 + x + 7*x^2 + 199*x^3 + 20026*x^4 + 7296946*x^5 +...+ A202943(n)*x^n +...

%o (PARI) {a(n)=polcoeff(sum(m=0,n,(m+1)*(m+2)/2*2^(m*(m-1))*x^m+x*O(x^n)),n)}

%Y Cf. A202943, A202942.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 26 2011