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A265265 Self-convolution of A265264. 2
1, 2, 5, 12, 32, 76, 192, 496, 1332, 3184, 7920, 19776, 50432, 128384, 332288, 874368, 2324480, 5622528, 14009856, 34833920, 87722240, 218775040, 551715840, 1398160384, 3558288384, 9005299712, 22949810176, 58833842176, 151438327808, 393635946496, 1025115512832, 2690908487680, 7068674293760, 17291087904768, 43329221885952, 108312858722304 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..3000

FORMULA

Terms satisfy:

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

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

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

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

EXAMPLE

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 +...

where

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 +...

Illustration of initial terms:

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

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

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

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

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

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

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

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

PROG

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

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

(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]}

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

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

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)

CROSSREFS

Cf. A265264.

Sequence in context: A132441 A000840 A232215 * A293868 A162434 A277863

Adjacent sequences:  A265262 A265263 A265264 * A265266 A265267 A265268

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 15 2015

STATUS

approved

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Last modified April 11 03:12 EDT 2021. Contains 342886 sequences. (Running on oeis4.)