A208977 Self-convolution square-root of A005810, where A005810(n) = binomial(4*n,n).
1, 2, 12, 86, 666, 5388, 44832, 380424, 3275172, 28512248, 250413856, 2215112886, 19711078686, 176276723508, 1583186541144, 14271487891512, 129063176166570, 1170480053359908, 10641805703955624, 96970507481607972, 885397365149468076, 8098908925136867112
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 12*x^2 + 86*x^3 + 666*x^4 + 5388*x^5 +... The square of the g.f. equals the g.f. of A005810: A(x)^2 = 1 + 4*x + 28*x^2 + 220*x^3 + 1820*x^4 + 15504*x^5 +... The g.f. of A002293 is G(x) = 1 + x*G(x)^4: G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
Programs
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Maple
a:= proc(n) option remember; `if`(n=0, 1, (binomial(4*n, n)-add(a(j)*a(n-j), j=1..n-1))/2) end: seq(a(n), n=0..21); # Alois P. Heinz, Jun 06 2025 -
Mathematica
nmax = 20; self = ConstantArray[0, nmax + 1]; self[[1]] = 1; self[[2]] = 2; Do[self[[k+1]] = (Binomial[4*k, k] - Sum[self[[j+1]]*self[[k-j+1]], {j, 1, k-1}]) / (2*self[[1]]);, {k, 2, nmax}]; self (* Vaclav Kotesovec, Jun 06 2025 *) -
PARI
{a(n)=polcoeff(sum(k=0,n,binomial(4*k,k)*x^k +x*O(x^n))^(1/2),n)} for(n=0,41,print1(a(n),", "))
Formula
G.f.: A(x) = sqrt( G(x)/(4 - 3*G(x)) ) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293. [From a formula by Mark van Hoeij in A005810]
From Vaclav Kotesovec, Jun 06 2025: (Start)
Recurrence: 81*(n-1)*n*(2*n - 3)*(3*n - 2)*(3*n - 1)*a(n) = 24*(n-1)*(1152*n^4 - 4608*n^3 + 6698*n^2 - 4180*n + 915)*a(n-1) - 16*(2*n - 1)*(8*n - 15)*(8*n - 13)*(8*n - 11)*(8*n - 9)*a(n-2).
a(n) ~ 2^(8*n + 1/4) / (Gamma(1/4) * n^(3/4) * 3^(3*n + 1/4)) * (1 - Gamma(1/4)^2 / (24*Pi*sqrt(3*n))). (End)