The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A354288 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A354288

Expansion of e.g.f. (1 + x)^(2/(1 - 2 * log(1+x))).
(history; published version)

#15 by Harvey P. Dale at Thu Oct 13 14:29:33 EDT 2022

STATUS

editing

approved

#14 by Harvey P. Dale at Thu Oct 13 14:29:30 EDT 2022

MATHEMATICA

With[{nn=20}, CoefficientList[Series[(1+x)^(2/(1-2Log[1+x])), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Oct 13 2022 *)

STATUS

approved

editing

#13 by Vaclav Kotesovec at Mon May 23 09:39:45 EDT 2022

STATUS

editing

approved

#12 by Vaclav Kotesovec at Mon May 23 09:39:40 EDT 2022

FORMULA

a(n) ~ exp(-7/8 + 1/(4*(exp(1/2) - 1)) + sqrt((2*n)/(exp(1/2) - 1))*exp(1/4) - n) * n^(n - 1/4) / (2^(3/4) * (exp(1/2) - 1)^(n + 1/4)). - Vaclav Kotesovec, May 23 2022

STATUS

approved

editing

#11 by Michael De Vlieger at Mon May 23 09:11:25 EDT 2022

STATUS

proposed

approved

#10 by Seiichi Manyama at Mon May 23 08:19:42 EDT 2022

STATUS

editing

proposed

#9 by Seiichi Manyama at Mon May 23 07:45:35 EDT 2022

DATA

1, 2, 10, 72, 664, 7440, 97712, 1468768, 24825184, 465516672, 9582002688, 214642099584, 5195322070656, 135064965744384, 3752151488840448, 110892824334154752, 3473236656134243328, 114893633354895538176, 4002000861023966189568, 146388324613230926979072

#8 by Seiichi Manyama at Mon May 23 06:51:06 EDT 2022

PROG

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((1+x)^(2/(1-2*log(1+x)))))

#7 by Seiichi Manyama at Mon May 23 03:40:01 EDT 2022

FORMULA

a(0) = 1; a(n) = Sum_{k=1..n} A088501(k) * binomial(n-1,k-1) * a(n-k).

CROSSREFS

Cf. A000262, A088501, A354286, A354290.

#6 by Seiichi Manyama at Mon May 23 03:23:57 EDT 2022

PROG

(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 2^k*k!*stirling(j, k, 1))*binomial(i-1, j-1)*v[i-j+1])); v;