A249937 G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} x^k * {[x^k] A(x)^(4*n)}.

1, 1, 5, 9, 81, 143, 1209, 2219, 20321, 37286, 342223, 633652, 5909073, 10982266, 103012275, 192218108, 1813703113, 3393964127, 32149901302, 60309293738, 573151325939, 1077320006025, 10264464357888, 19326593165490, 184528265711989, 347951310160204, 3328029193939574, 6283416943044132

Offset

0,3

Links

Table of n, a(n) for n=0..27.

Formula

G.f. satisfies: A(x) = (1 + 4*x^2*G'(x^2)/G(x^2)) / (1 - x*G(x^2)^4), where A(x) = G(x/A(x)^4) and G(x) = A(x*G(x)^4) = ( (1/x)*Series_Reversion(x/A(x)^4) )^(1/4).

Example

 G.f.: A(x) = 1 + x + 5*x^2 + 9*x^3 + 81*x^4 + 143*x^5 + 1209*x^6 +...
Related expansions:
A(x)^4 = 1 + 4*x + 26*x^2 + 100*x^3 + 643*x^4 + 2512*x^5 +...
A(x)^8 = 1 + 8*x + 68*x^2 + 408*x^3 + 2762*x^4 + 15368*x^5 +...
A(x)^12 = 1 + 12*x + 126*x^2 + 988*x^3 + 7605*x^4 + 51480*x^5 +...
A(x)^16 = 1 + 16*x + 200*x^2 + 1904*x^3 + 16676*x^4 + 130416*x^5 +...
GENERATING METHOD.
The initial terms, k=0..n, of the (4*n)-th power of g.f. A(x) begin:
n=0: [1];
n=4: [1, 4];
n=8: [1, 8, 68];
n=12: [1, 12, 126, 988];
n=16: [1, 16, 200, 1904, 16676];
n=20: [1, 20, 290, 3220, 31735, 279424];
n=24: [1, 24, 396, 5000, 54798, 534456, 4818812];
n=28: [1, 28, 518, 7308, 88137, 941192, 9210936, 83847992];
n=32: [1, 32, 656, 10208, 134280, 1556064, 16416112, 160443680, 1474474468]; ...
from which the antidiagonal sums form this sequence:
a(0) = 1;
a(1) = 1;
a(2) = 1 + 4 = 5;
a(3) = 1 + 8 = 9;
a(4) = 1 + 12 + 68 = 81;
a(5) = 1 + 16 + 126 = 143;
a(6) = 1 + 20 + 200 + 988 = 1209;
a(7) = 1 + 24 + 290 + 1904 = 2219; ...
ALTERNATE GENERATING METHOD.
Define G(x) such that G(x) = A(x*G(x)^4) = ( (1/x)*Series_Reversion(x/A(x)^4) )^(1/4):
G(x) = 1 + x + 9*x^2 + 91*x^3 + 1165*x^4 + 15792*x^5 + 228359*x^6 + 3422766*x^7 + 52855277*x^8 + 834274806*x^9 + 13404201980*x^10 +...
then A(x) = (1  + 4*x^2*G'(x^2)/G(x^2)) / (1 - x*G(x^2)^4).
Note that 1  + 4*x^2*G'(x^2)/G(x^2) begins:
1 + 4*x^2 + 68*x^4 + 988*x^6 + 16676*x^8 + 279424*x^10 + 4818812*x^12 +...
where the coefficients form the main diagonal of the above triangle.

Prog

 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, x^k*polcoeff((A+x*O(x^m))^(4*m), k))+x*O(x^n))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* ALTERNATE GENERATING METHOD (faster) */
{a(n)=local(A=1+x, G=1); for(i=0, #binary(n)+1, G=(1/x*serreverse(x/A^4 +x^2*O(x^n)))^(1/4); A=(1+4*x^2*subst(G'/G, x, x^2))/(1-x*subst(G^4, x, x^2))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A222658, A249935, A249936.

Sequence in context: A306123 A365580 A192202 * A344081 A171812 A221741

Adjacent sequences: A249934 A249935 A249936 * A249938 A249939 A249940

Keyword

nonn

Author

Paul D. Hanna, Nov 24 2014

Status

approved