A135867 G.f. satisfies A(x) = 1 + x*A(2*x)^2.

1, 1, 4, 36, 640, 21888, 1451008, 188941312, 48768745472, 25069815595008, 25722272102744064, 52730972085034156032, 216091838647321476726784, 1770657164881170759078117376, 29013990909330956353981535748096

Offset: 0

Author

Paul D. Hanna, Dec 02 2007

Keywords

• nonn

Comments

Self-convolution equals A135868 such that 2^n*A135868(n) = a(n+1) for n >= 0.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..81

Crossrefs

Cf. A015083, A135868, A171192.

Cf. A171200, A171202, A171204, A171206, A171208, A171210, A343439.

Programs

Mathematica

nmax = 15; A[] = 0; Do[A[x] = 1 + x*A[2*x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 04 2021 *)

PARI

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,2*x)^2);polcoeff(A,n)}

PARI

a(n)=if(n==0,1,2^(n-1)*sum(k=0,n-1,a(k)*a(n-k-1))) \\ Paul D. Hanna, Feb 09 2010

Formula

a(n) = 2^(n-1)*Sum_{k=0..n-1} a(k)*a(n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Feb 09 2010

a(n) ~ c * 2^(n*(n+1)/2), where c = 0.715337433614869740944075474484711589980951273610257702786245519231799678... - Vaclav Kotesovec, Nov 04 2021