The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A355287 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A355287

E.g.f. satisfies A(x) = 1/(1 - x)^(x^2 * A(x)).
(history; published version)

#20 by Alois P. Heinz at Mon Mar 04 11:14:33 EST 2024

STATUS

proposed

approved

#19 by Jean-François Alcover at Mon Mar 04 11:07:38 EST 2024

STATUS

editing

proposed

#18 by Jean-François Alcover at Mon Mar 04 11:07:32 EST 2024

MATHEMATICA

nmax = 21; A[_] = 1;

Do[A[x_] = 1/(1 - x)^(x^2*A[x]) + O[x]^(nmax+1) // Normal, {nmax}];

CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

STATUS

approved

editing

#17 by N. J. A. Sloane at Sun Sep 04 12:33:44 EDT 2022

STATUS

proposed

approved

#16 by Seiichi Manyama at Sun Sep 04 10:47:00 EDT 2022

STATUS

editing

proposed

#15 by Seiichi Manyama at Sun Sep 04 10:44:49 EDT 2022

FORMULA

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

E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-x^2 * log(1-x))^k / k!.

E.g.f.: A(x) = exp( -LambertW(x^2 * log(1-x)) ).

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

#14 by Seiichi Manyama at Sun Sep 04 10:22:05 EDT 2022

LINKS

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

#13 by Seiichi Manyama at Sun Sep 04 10:21:14 EDT 2022

DATA

1, 0, 0, 6, 12, 40, 1260, 8568, 62160, 1473120, 19111680, 232626240, 5403451680, 103176028800, 1822033204992, 45916616592000, 1129459815993600, 26346457488798720, 749439127417466880, 22165051763204582400, 640916967497214643200, 20787453048015928350720

#12 by Seiichi Manyama at Sun Sep 04 10:20:15 EDT 2022

PROG

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-x^2*log(1-x))^k/k!)))

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2*log(1-x)))))

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2*log(1-x))/(x^2*log(1-x))))

#11 by Seiichi Manyama at Sat Sep 03 23:18:17 EDT 2022

CROSSREFS

Cf. A353228, A356910.