A227846 Self-convolution equals A227845.

1, 1, 3, 11, 47, 215, 1029, 5077, 25615, 131455, 683749, 3595341, 19075913, 101978321, 548683499, 2968535115, 16138308655, 88107960847, 482839660509, 2654879900085, 14641704563577, 80968432526961, 448856443832643, 2493858308981331, 13884356040349161, 77445573778294041

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..430

Formula

G.f.: 1 / sqrt( AGM((1+x)^2, (1+x)^2 - 8*x) ), where AGM denotes the arithmetic-geometric mean. - Paul D. Hanna, Jul 31 2014

G.f.: sqrt( Sum_{n>=0} A227845(n)*x^n ), where A227845(n) = Sum_{k=0..[n/2]} Sum_{j=k..n-k} binomial(n-k,j)^2*binomial(j,k)^2.

a(n) ~ (1+sqrt(2))^(2*n+1) / (4*n*sqrt(Pi*log(n))) * (1 - (2*gamma + 5*log(2))/(4*log(n)) + (12*gamma^2 + 60*gamma*log(2) + 75*log(2)^2 - 2*Pi^2) / (32*log(n)^2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

Example

G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 47*x^4 + 215*x^5 + 1029*x^6 +...
where A(x)^2 equals the g.f. of A227845:
A(x)^2 = 1/(1-x) + x/(1-x)^3*(1+x)^2 + x^2/(1-x)^5*(1 + 2^2*x + x^2)^2
+ x^3/(1-x)^7*(1 + 3^2*x + 3^2*x^2 + x^3)^2
+ x^4/(1-x)^9*(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2
+ x^5/(1-x)^11*(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2
+ x^6/(1-x)^13*(1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)^2 +...
and also:
A(x)^2 = 1 + x*(1 + (1+x)) + x^2*(1 + 2^2*(1+x) + (1+2^2*x+x^2)) + x^3*(1 + 3^2*(1+x) + 3^2*(1+2^2*x+x^2) + (1+3^2*x+3^2*x^2+x^3)) + x^4*(1 + 4^2*(1+x) + 6^2*(1+2^2*x+x^2) + 4^2*(1+3^2*x+3^2*x^2+x^3) + (1+4^2*x+6^2*x^2+4^2*x^3+x^4)) + ...
Explicitly,
A(x)^2 = 1 + 2*x + 7*x^2 + 28*x^3 + 125*x^4 + 590*x^5 + 2891*x^6 +...+ A227845(n)*x^n +...

Prog

(PARI) {a(n)=polcoeff(sqrt(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*sum(j=0, k, binomial(k, j)^2*x^j)+x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From g.f. 1/sqrt( AGM((1+x)^2, (1+x)^2 - 8*x) ): */
{a(n)=local(A); A = 1/sqrt( agm((1+x)^2, (1+x)^2 - 8*x +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A227845.

Sequence in context: A151159 A151160 A151161 * A124890 A301770 A308227

Adjacent sequences: A227843 A227844 A227845 * A227847 A227848 A227849

Keyword

nonn

Author

Paul D. Hanna, Aug 01 2013

Status

approved