A094557 a(2*n) equals coefficient of x^n in A(x)^(n+1) and a(2*n+1) equals coefficient of x^n in A(x)^(n+2), for n>=0.

1, 1, 2, 3, 9, 14, 40, 65, 210, 339, 1080, 1764, 5775, 9448, 30992, 50931, 168849, 277920, 925240, 1525887, 5106288, 8431260, 28309440, 46796334, 157627548, 260788843, 880639004, 1458096900, 4934715105, 8175734400, 27721876064

Offset: 0

Author

Paul D. Hanna, May 11 2004

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 9*x^4 + 14*x^5 + 40*x^6 + 65*x^7 +...
Terms are produced by main and secondary diagonals in the table of successive self-convolutions of this sequence:
  [(1), 1, 2, 3, 9, 14, 40, 65, 210, 339, 1080, ...];
  [(1),(2), 5, 10, 28, 58, 153, 320, 875, 1850, ...];
  [1, (3),(9), 22, 63, 153, 410, 978, 2607, 6222, ...];
  [1, 4, (14),(40), 121, 328, 918, 2392, 6504, 16708, ...];
  [1, 5, 20, (65),(210), 621, 1830, 5110, 14395, 39085, ...];
  [1, 6, 27, 98, (339),(1080), 3356, 9942, 29163, 83008, ...];
  [1, 7, 35, 140, 518, (1764),(5775), 18040, 55160, 163863, ...];
  [1, 8, 44, 192, 758, 2744, (9448),(30992), 98729, 305240, ...];
  [1, 9, 54, 255, 1071, 4104, 14832, (50931),(168849), 542164, ...];
  [1, 10, 65, 330, 1470, 5942, 22495, 80660, (277920),(925240), ...]; ...
from which A094558 may be formed from the main diagonal:
  [1/1, 2/2, 9/3, 40/4, 210/5, 1080/6, 5775/7, 30992/8, 168849/9, 925240/10,...].
Let G(x) be the g.f. of A094558:
G(x) = 1 + x + 3*x^2 + 10*x^3 + 42*x^4 + 180*x^5 + 825*x^6 + 3874*x^7 +...
then the coefficients of G(x)^2 generates the secondary diagonal:
[1*2/2, 3*2/3, 14*2/4, 65*2/5, 339*2/6, 1764*2/7, 9448*2/8, 50931*2/9,...]
and may be derived from the odd-indexed terms of this sequence.
		

Crossrefs

Cf. A094558, A094600.

Programs

PARI

{a(n)=local(A=1+x,G);for(i=1,n,G=serreverse(x/A+x*O(x^n));A=subst(deriv(G),x,x^2)+subst(deriv(G^2/2),x,x^2)/x);polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

a(2*n) = (n+1)*A094558(n).

G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A094558.

G.f. satisfies: A(x) = F'(x^2)*(1 + F(x^2)/x) where F(x) = Series_Reversion(x/A(x)) and F(x)/x is the g.f. of A094558.

Extensions

Entry revised by Paul D. Hanna, Apr 17 2013