A265226 Self-convolution of A257889.

1, 2, 5, 14, 34, 96, 261, 692, 1680, 4540, 12540, 34552, 92728, 251572, 662340, 1729628, 4261528, 11130160, 29802200, 80103640, 218398544, 595050400, 1621285648, 4411577744, 11776668772, 31899937136, 85998657296, 231056788736, 607876418544, 1615730650080, 4228062351360, 11047956392096, 27736466241312, 71915999814720, 188591683462784, 495344539985920, 1321221455067520, 3505058052234400

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..2500

Formula

Terms satisfy:

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

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

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

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

Example

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 +...
where
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 +...
Illustration of initial terms:
a(1) = A257889(2)/A257889(1) = 2/1 = 2;
a(2) = A257889(3)/A257889(1) = 5/1 = 5;
a(2) = A257889(4)/A257889(2) = 10/2 = 5;
a(3) = A257889(5)/A257889(2) = 28/2 = 14;
a(3) = A257889(6)/A257889(3) = 70/5 = 14;
a(4) = A257889(7)/A257889(3) = 170/5 = 34;
a(4) = A257889(8)/A257889(4) = 340/10 = 34; ...

Prog

(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)}
for(n=0, 40, print1(a(n), ", "))
(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]}
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\2+1]*Vec(Ser(A)^2)[(k+1)\2+1]) ); Vec(Ser(A)^2)

Crossrefs

Cf. A257889.

Sequence in context: A018015 A080039 A344236 * A357835 A369591 A299164

Adjacent sequences: A265223 A265224 A265225 * A265227 A265228 A265229

Keyword

nonn

Author

Paul D. Hanna, Dec 15 2015

Status

approved