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
Keywords
Examples
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 +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..430
Crossrefs
Cf. A227845.
Programs
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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), ", "))
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