A265226 - OEIS

%I #19 Nov 22 2019 13:18:48

%S 1,2,5,14,34,96,261,692,1680,4540,12540,34552,92728,251572,662340,

%T 1729628,4261528,11130160,29802200,80103640,218398544,595050400,

%U 1621285648,4411577744,11776668772,31899937136,85998657296,231056788736,607876418544,1615730650080,4228062351360,11047956392096,27736466241312,71915999814720,188591683462784,495344539985920,1321221455067520,3505058052234400

%N Self-convolution of A257889.

%H Paul D. Hanna, <a href="/A265226/b265226.txt">Table of n, a(n) for n = 0..2500</a>

%F Terms satisfy:

%F (1) a(n) = A257889(2*n) / A257889(n),

%F (2) a(n+1) = A257889(2*n+1) / A257889(n),

%F (3) a(n) = Sum_{k=0..n} A257889(n-k) * A257889(k),

%F for n>=0, where A(x) = G(x)^2 and G(x) = Sum_{n>=0} A257889(n)*x^n.

%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 34*x^4 + 96*x^5 + 261*x^6 + 692*x^7 + 1680*x^8 + 4540*x^9 + 12540*x^10 + 34552*x^11 + 92728*x^12 +...

%e where

%e sqrt(A(x)) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 28*x^5 + 70*x^6 + 170*x^7 + 340*x^8 + 960*x^9 + 2688*x^10 + 7308*x^11 + 18270*x^12 +...+ A257889(n)*x^n +...

%e Illustration of initial terms:

%e a(1) = A257889(2)/A257889(1) = 2/1 = 2;

%e a(2) = A257889(3)/A257889(1) = 5/1 = 5;

%e a(2) = A257889(4)/A257889(2) = 10/2 = 5;

%e a(3) = A257889(5)/A257889(2) = 28/2 = 14;

%e a(3) = A257889(6)/A257889(3) = 70/5 = 14;

%e a(4) = A257889(7)/A257889(3) = 170/5 = 34;

%e a(4) = A257889(8)/A257889(4) = 340/10 = 34; ...

%o (PARI) {a(n) = my(A=1+x); for(k=2,n, A = A + a(k\2) * polcoeff(A^2, (k+1)\2) * x^k +x*O(x^n) ); polcoeff(A^2,n)}

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

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

%Y Cf. A257889.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 15 2015