A392932 E.g.f. A(x) satisfies A(x) = 1 - (A(x)/x) * log(1 - x^2*A(x)^2).

1, 1, 6, 75, 1416, 36100, 1162500, 45298260, 2073120000, 109025236656, 6479480327040, 429490443634560, 31416341604500160, 2513890936457746560, 218444719852713594240, 20484930511629760176000, 2062040084613184977100800, 221771483779991107676313600, 25379627910164829624284881920

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

E.g.f.: (1/x) * Series_Reversion( x * (1 + sqrt(1 + 4*log(1-x^2)/x))/2 ).

a(n) = n! * Sum_{k=0..floor(n/2)} (3*n-4*k)!/(2*n-2*k+1)! * |Stirling1(n-k,n-2*k)|/(n-k)!.

Mathematica

Table[n! * Sum[(3*n-4*k)!/(2*n-2*k+1)!*Abs[StirlingS1[n-k, n-2*k]/(n -k)!], {k, 0, Floor[n/2]}], {n, 0, 18}] (* Vincenzo Librandi, Jan 27 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, (3*n-4*k)!/(2*n-2*k+1)!*abs(stirling(n-k, n-2*k, 1))/(n-k)!);
(Magma) [ Factorial(n) * &+[ Factorial(3*n-4*k)/Factorial(2*n-2*k+1) * Abs(StirlingFirst(n-k, n-2*k))/Factorial(n-k) : k in [0..Floor(n/2)] ]: n in [0..23] ]; // Vincenzo Librandi, Jan 27 2026

Crossrefs

Cf. A392934, A392935.

Sequence in context: A381445 A381172 A392956 * A381477 A162863 A216136

Adjacent sequences: A392929 A392930 A392931 * A392933 A392934 A392935

Keyword

nonn

Author

Seiichi Manyama, Jan 27 2026

Status

approved