A392953 E.g.f. A(x) satisfies A(x) = 1 + (1/x) * (exp(x^2*A(x)) - 1).

1, 1, 2, 9, 60, 500, 5160, 64050, 925680, 15274224, 283681440, 5858178480, 133168432320, 3305096308800, 88940036465280, 2579622274429200, 80228312766201600, 2663566170795129600, 94025979044975577600, 3516890931415484663040, 138943364103192391142400

Offset

0,3

Links

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} 1/(k+1)! * Stirling2(n-k,n-2*k).

a(n) ~ sqrt(1 - r + 2*r^2) * n^(n-1) / (exp(n) * r^(n+2)), where r = 0.47199626373441347265516... is the root of the equation 1 - r + r^2 = -log(r). - Vaclav Kotesovec, Jan 28 2026

Mathematica

Table[n! * Sum[1/(k+1)! * StirlingS2[n-k, n-2*k], {k, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 28 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, 1/(k+1)!*stirling(n-k, n-2*k, 2));
(Magma) [Factorial(n)*&+[1/Factorial(k+1)* StirlingSecond(n-k, n-2*k): k in [0..Floor(n/2)] ] : n in [0..24] ]; // Vincenzo Librandi, Feb 02 2026

Crossrefs

Cf. A000272, A392959.

Cf. A392929.

Sequence in context: A354314 A354496 A357683 * A120970 A339360 A111558

Adjacent sequences: A392950 A392951 A392952 * A392954 A392955 A392956

Keyword

nonn

Author

Seiichi Manyama, Jan 28 2026

Status

approved