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%I #4 Jun 14 2012 19:37:41
%S 1,1,2,10,30,131,582,3196,13986,70100,336416,1723518,8487202,44468780,
%T 228236112,1241788448,6421700878,34682391148,182473774272,
%U 993091141104,5264377375260,28721435063423,153844326005054,843854383167940
%N a(n) = sum of the squares of the coefficients of x^(2k) in A(x^2)^{2*(n-2k)+1}, as k varies from 0 to floor(n/2), with a(0)=1.
%C This is a variant of the following property of the Catalan sequence:
%C A000108(n) = sum of the squares of the coefficients of x^(2k) in Catalan(x^2)^{n-2k+1}, as k varies from 0 to floor(n/2) where Catalan(x) = g.f. of A000108.
%H Paul D. Hanna, <a href="/A162249/b162249.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = Sum_{k=0..n\2} ( [x^(2*k)] A(x^2)^{2*(n-2*k)+1} )^2 for n>0 with a(0)=1.
%e To illustrate the recurrence, list coefficients of A(x^2)^(2n+1):
%e A^1: . 1,... 1,... 2,... 10,... 30,... 131,.......;
%e A^3: .... 1,... 3,... 9,... 43,... 168,... 735, ...;
%e A^5: ....... 1,... 5,... 20,... 100,... 455,.......;
%e A^7: .......... 1,... 7,... 35,... 189,... 959, ...;
%e A^9: ............. 1,... 9,... 54,... 318,.......;
%e A^11: ............... 1,... 11,... 77,... 495, ...;
%e A^13: .................. 1,... 13,... 104,.......;
%e A^15: ..................... 1,... 15,... 135, ...;
%e A^17: ........................ 1,... 17,.......;
%e A^19: ........................... 1,... 19, ...;
%e A^21: .............................. 1,.......;
%e A^23: ................................. 1, ...;...
%e then sum the squares of the coefficients in each column:
%e a(0) = 1^2 = 1;
%e a(1) = 1^2 = 1;
%e a(2) = 1^2 + 1^2 = 2;
%e a(3) = 3^2 + 1^2 = 10;
%e a(4) = 2^2 + 5^2 + 1^2 = 30;
%e a(5) = 9^2 + 7^2 + 1^2 = 131;
%e a(6) = 10^2 + 20^2 + 8^2 + 1^2 = 582;
%e a(7) = 43^2 + 35^2 + 11^2 + 1^2 = 3196;
%e a(8) = 30^2 + 100^2 + 54^2 + 13^2 + 1^2 = 13986;
%e a(9) = 168^2 + 189^2 + 77^2 + 15^2 + 1^2 = 70100.
%o (PARI) {a(n)=local(A=1+sum(j=1, n\2, a(j)*x^(2*j))+x*O(x^n));if(n==0, 1, sum(k=0, n\2, polcoeff(A^(2*(n-2*k)+1), 2*k)^2))}
%Y Cf. A095892.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 28 2009
%E Comment corrected by _Paul D. Hanna_, Jul 05 2009
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