A138294 G.f.: A(x) = 1 + x*A_1(x)^2; A_1(x) = (1+x) + x*A_2(x)^2; A_2(x) = (1+x)^2 + x*A_3(x)^2; ...; A_{n}(x) = (1+x)^n + x*A_{n+1}(x)^2 for n>=0 with A(x) = A_0(x).

1, 1, 4, 16, 78, 420, 2454, 15297, 100660, 694022, 4986128, 37171749, 286619290, 2279866324, 18668221560, 157080129914, 1356186583276, 11999018622158, 108672944038356, 1006528378511868, 9525454067974148, 92037443236217412

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

G.f.: A(x) = 1 + x*A_1(x)^2 where A_1(x) = g.f. of A138295.

Example

G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 78*x^4 + 420*x^5 + 2454*x^6 +...
Given A_{n}(x) = (1+x)^n + x*A_{n+1}(x)^2 for n>=0,
the initial coefficients of the functions A_{n} for n=0..8 are:
A_0 = [1, 1, 4, 16, 78, 420, 2454, 15297, 100660, 694022, ...];
A_1 = [1, 2, 6, 27, 138, 789, 4878, 32114, 222690, 1614412,...];
A_2 = [1, 3, 9, 42, 228, 1377, 8992, 62400, 455252, 3465728,...];
A_3 = [1, 4, 13, 62, 356, 2266, 15586, 113752, 871378, 6953751,...];
A_4 = [1, 5, 18, 88, 531, 3554, 25676, 196609, 1577930, 13174337,...];
A_5 = [1, 6, 24, 121, 763, 5356, 40536, 324882, 2725852, 23763583,...];
A_6 = [1, 7, 31, 162, 1063, 7805, 61731, 516648, 4522200, 41085199,...];
A_7 = [1, 8, 39, 212, 1443, 11053, 91151, 794909, 7244078, 68460164,...];
A_8 = [1, 9, 48, 272, 1916, 15272, 131046, 1188417, 11254609, 110444000,..].

Prog

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

Crossrefs

Cf. A138295 (A_1).

Sequence in context: A020006 A207653 A204208 * A014514 A000780 A002713

Adjacent sequences: A138291 A138292 A138293 * A138295 A138296 A138297

Keyword

nonn

Author

Paul D. Hanna, Mar 13 2008

Status

approved