0,3

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

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,..].

(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)}

Cf. A138295 (A_1).

Sequence in context: A159926 A020051 A020006 this_sequence A014514 A000780 A002713

Adjacent sequences: A138291 A138292 A138293 this_sequence A138295 A138296 A138297

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Mar 13 2008