A274961 - OEIS

%I #7 Jul 12 2016 17:58:53

%S 1,2,4,12,32,112,384,1824,6912,34304,154624,852480,4259840,25968640,

%T 143687680,964366336,5515771904,37026332672,230170296320,

%U 1671801339904,10772865351680,80599119298560,557712899309568,4420637088022528,31616746028793856,259184403870121984,1963369608274509824,17005377989510168576,132409252034306375680,1172260103612874620928,9575887243678308106240,89085560504158762565632

%N G.f.: 1 = ...((((1/(1-x) - a(1)*x )^2 - a(2)*x^2 )^2 - a(3)*x^3 )^2 - a(4)*x^4 )^2 -..., an infinite series of nested squares.

%H Paul D. Hanna, <a href="/A274961/b274961.txt">Table of n, a(n) for n = 1..600</a>

%e G.f.: 1 = ... ((((((((1/(1-x) - 1*x)^2 - 2*x^2)^2 - 4*x^3)^2 - 12*x^4)^2 - 32*x^5)^2 - 112*x^6)^2 - 384*x^7)^2 - 1824*x^8)^2 -...- a(n)*x^n)^2 -...

%e ILLUSTRATION OF GENERATING METHOD.

%e Start with G1 = 1/(1-x), and proceed as follows:

%e G2 = (G1 - 1*x)^2 = 1 + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 +...

%e G3 = (G2 - 2*x^2)^2 = 1 + 4*x^3 + 6*x^4 + 8*x^5 + 14*x^6 + 24*x^7 + 39*x^8 +...

%e G4 = (G3 - 4*x^3)^2 = 1 + 12*x^4 + 16*x^5 + 28*x^6 + 48*x^7 + 114*x^8 + 216*x^9 +...

%e G5 = (G4 - 12*x^4)^2 = 1 + 32*x^5 + 56*x^6 + 96*x^7 + 228*x^8 + 432*x^9 +...

%e G6 = (G5 - 32*x^5)^2 = 1 + 112*x^6 + 192*x^7 + 456*x^8 + 864*x^9 + 2144*x^10 +...

%e G7 = (G6 - 112*x^6)^2 = 1 + 384*x^7 + 912*x^8 + 1728*x^9 + 4288*x^10 + 9664*x^11 +...

%e G8 = (G7 - 384*x^7)^2 = 1 + 1824*x^8 + 3456*x^9 + 8576*x^10 + 19328*x^11 +...

%e G9 = (G8 - 1824*x^8)^2 = 1 + 6912*x^9 + 17152*x^10 + 38656*x^11 + 106560*x^12 +...

%e G10 = (G9 - 6912*x^9)^2 = 1 + 34304*x^10 + 77312*x^11 + 213120*x^12 + 532480*x^13 +...

%e G11 = (G10 - 34304*x^10)^2 = 1 + 154624*x^11 + 426240*x^12 + 1064960*x^13 +...

%e ...

%e G_{n+1} = (G_{n} - a(n)*x^n)^2 = 1 + a(n+1)*x^(n+1) + a(n+2)*x^(n+2)/2 + a(n+3)*x^(n+3)/4 + a(n+4)*x^(n+4)/8 +...

%e ...

%o (PARI) {a(n) = my(A=[1],G); for(i=1,n, A=concat(A,0); G = 1/(1-x +x*O(x^#A)); for(m=1,#A, G = (G - A[m]*x^m)^2 ); A[#A] = polcoeff(G,#A)/2 );A[n]}

%o for(n=1,30,print1(a(n),", "))

%o (PARI) /* Informal quick print of initial N terms: */

%o {N=100; A=[1]; G = 1/(1-x +x^2*O(x^N)); for(m=1,N, A=concat(A,0); G = (G - A[m]*x^m)^2; A[m+1] = polcoeff(G,m+1); print1(A[m],", ")); print1(A[N],", ")}

%Y  Cf. A274960.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 12 2016