A264927 Terms satisfy: a(2*n) = b(n)^2 and a(2*n+1) = b(n)*b(n+1) for n>=0 with a(0)=1, where A(x)^2 = Sum_{n>=0} b(n)*x^n and g.f. A(x) = Sum_{n>=0} a(n)*x^n.

1, 1, 4, 18, 81, 396, 1936, 9416, 45796, 234972, 1205604, 6188328, 31764496, 162249168, 828748944, 4224034452, 21529399441, 113085801048, 593996986944, 3126735424128, 16458794619136, 86717178000000, 456890625000000, 2407551579000000, 12686416153856064, 66648950832253248, 350144800010358336, 1838080630013808960, 9648980656951105600, 50559912359694495680

Offset

0,3

Comments

What is the limit a(n)^(1/n) = ? The limit is at least 5.3...

Links

Table of n, a(n) for n=0..29.

Example

 G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 81*x^4 + 396*x^5 + 1936*x^6 + 9416*x^7 + 45796*x^8 + 234972*x^9 + 1205604*x^10 +...
where
A(x)^2 = 1 + 2*x + 9*x^2 + 44*x^3 + 214*x^4 + 1098*x^5 + 5636*x^6 + 28788*x^7 + 146729*x^8 + 770712*x^9 + 4056944*x^10 + 21375000*x^11 + 112633992*x^12 + 591730344*x^13 + 3106280840*x^14 + 16276671352*x^15 + 85215198522*x^16 +...
Illustration of initial terms, both a(n) of A(x) and b(n) of A(x)^2:
a(0) = 1; b(0) = 1;
a(1) = 1; b(1) = 2 = 1*1 + 1*1;
a(2) = b(1)*b(1) = 4; b(2) = 9 = 1*4 + 1*1 + 4*1;
a(3) = b(1)*b(2) = 18; b(3) = 44 = 1*18 + 1*4 + 4*1 + 18*1;
a(4) = b(2)*b(2) = 81; b(4) = 214 = 1*81 + 1*18 + 4*4 + 18*1 + 81*1;
a(5) = b(2)*b(3) = 396; b(5) = 1098;
a(6) = b(3)*b(3) = 1936; b(6) = 5636;
a(7) = b(3)*b(4) = 9416; b(7) = 28788;
a(8) = b(4)*b(4) = 45796; b(8) = 146729; ...

Prog

(PARI) {a(n) = my(A=1+x); for(k=2, n, B = A^2; A = A + polcoeff(B, k\2) * polcoeff(B, (k+1)\2) * x^k +x*O(x^n) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1, 1]); for(k=2, n, B = Vec(Ser(A)^2); A = concat(A, B[k\2+1]*B[(k+1)\2+1]) ); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* Generates N terms rather quickly: */
N=500; A=[1, 1]; for(k=2, N, B = Vec(Ser(A)^2); A = concat(A, B[k\2+1]*B[(k+1)\2+1]) ); A

Crossrefs

Cf. A257889.

Sequence in context: A264191 A257060 A181610 * A257059 A194460 A356289

Adjacent sequences: A264924 A264925 A264926 * A264928 A264929 A264930

Keyword

nonn

Author

Paul D. Hanna, Dec 13 2015

Status

approved