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G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1+x)^((n+2)^2).
4

%I #3 Aug 06 2012 23:24:14

%S 1,4,26,256,3489,61696,1352518,35566368,1094499820,38670814348,

%T 1545160614694,68970980789472,3404652821768232,184295822142051600,

%U 10861040169788302030,692560292664515634112,47527552597795293035916,3493783983256399634130360,273974326317024551368217200

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

%C Column 2 of triangle A215241.

%e G.f.: 1 = 1/(1+x)^4 + 4*x/(1+x)^9 + 26*x^2/(1+x)^16 + 256*x^3/(1+x)^25 + 3489*x^4/(1+x)^36 + 61696*x^5/(1+x)^49 +...

%e Also forms the final terms in rows of the triangle where row n+1 equals the partial sums of row n with the final term repeated 2*n+1 times, starting with '[1,1,1,1]' in row 1, as illustrated by:

%e 1, 1, 1, 1;

%e 1, 2, 3, 4, 4, 4, 4, 4;

%e 1, 3, 6, 10, 14, 18, 22, 26, 26, 26, 26, 26, 26, 26;

%e 1, 4, 10, 20, 34, 52, 74, 100, 126, 152, 178, 204, 230, 256, 256, 256, 256, 256, 256, 256, 256, 256;

%e 1, 5, 15, 35, 69, 121, 195, 295, 421, 573, 751, 955, 1185, 1441, 1697, 1953, 2209, 2465, 2721, 2977, 3233, 3489, 3489, 3489, 3489, 3489, 3489, 3489, 3489, 3489; ...

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

%o for(n=0,21,print1(a(n)", "))

%Y Cf. A215241, A177447, A215243, A133316.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 06 2012