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Self-convolution of A265264.
2

%I #11 Jan 14 2025 15:03:43

%S 1,2,5,12,32,76,192,496,1332,3184,7920,19776,50432,128384,332288,

%T 874368,2324480,5622528,14009856,34833920,87722240,218775040,

%U 551715840,1398160384,3558288384,9005299712,22949810176,58833842176,151438327808,393635946496,1025115512832,2690908487680,7068674293760,17291087904768,43329221885952,108312858722304

%N Self-convolution of A265264.

%H Paul D. Hanna, <a href="/A265265/b265265.txt">Table of n, a(n) for n = 0..3000</a>

%F Terms satisfy:

%F (1) a(n) = A265264(2*n) / A265264(n),

%F (2) a(n) = A265264(2*n+1) / A265264(n+1),

%F (3) a(n) = Sum_{k=0..n} A265264(n-k) * A265264(k),

%F for n>=0, where A(x) = G(x)^2 and G(x) = Sum_{n>=0} A265264(n)*x^n.

%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 12*x^3 + 32*x^4 + 76*x^5 + 192*x^6 + 496*x^7 + 1332*x^8 + 3184*x^9 + 7920*x^10 + 19776*x^11 + 50432*x^12 +...

%e where

%e sqrt(A(x)) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 20*x^5 + 48*x^6 + 120*x^7 + 320*x^8 + 640*x^9 + 1520*x^10 + 3648*x^11 + 9216*x^12 +...+ A265264(n)*x^n +...

%e Illustration of initial terms:

%e a(0) = A265264(1)/A265264(1) = 1/1 = 1;

%e a(1) = A265264(2)/A265264(1) = 2/1 = 2;

%e a(1) = A265264(3)/A265264(2) = 4/2 = 2;

%e a(2) = A265264(4)/A265264(2) = 10/2 = 5;

%e a(2) = A265264(5)/A265264(3) = 20/4 = 5;

%e a(3) = A265264(6)/A265264(3) = 48/4 = 12;

%e a(3) = A265264(7)/A265264(4) = 120/10 = 12;

%e a(4) = A265264(8)/A265264(4) = 320/10 = 32; ...

%o (PARI) {a(n) = my(A=[1, 1]); for(k=2, n, A = concat(A, A[(k+1)\2+1]*Vec(Ser(A)^2)[k\2+1]) ); Vec(Ser(A)^2)[n+1]}

%o for(n=0, 40, print1(a(n), ", "))

%o (PARI) /* Generates N terms rather quickly: */

%o N=300; A=[1, 1]; for(k=2, N, A = concat(A, A[(k+1)\2+1]*Vec(Ser(A)^2)[k\2+1]) ); Vec(Ser(A)^2)

%Y Cf. A265264.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 15 2015