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Revision History for A367920

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A367920

Expansion of e.g.f. exp(4*(exp(x) - 1) - 2*x).
(history; published version)

#6 by Hugo Pfoertner at Tue Dec 05 04:28:57 EST 2023

STATUS

reviewed

approved

#5 by Joerg Arndt at Tue Dec 05 00:11:20 EST 2023

STATUS

proposed

reviewed

#4 by Ilya Gutkovskiy at Mon Dec 04 20:27:10 EST 2023

STATUS

editing

proposed

#3 by Ilya Gutkovskiy at Mon Dec 04 19:36:46 EST 2023

CROSSREFS

Cf. A000296, A078944, A126617, A194689, A367888, A367891, A367919, A367921.

#2 by Ilya Gutkovskiy at Mon Dec 04 19:32:19 EST 2023

NAME

allocated for Ilya Gutkovskiy

Expansion of e.g.f. exp(4*(exp(x) - 1) - 2*x).

DATA

1, 2, 8, 36, 196, 1196, 8116, 60108, 481140, 4126540, 37671540, 364068172, 3707910772, 39645022540, 443540780660, 5177560304972, 62903920321140, 793654042136908, 10378403752717940, 140413475790402892, 1962339063781284468, 28287778534523140428, 420059992540347885172

OFFSET

0,2

FORMULA

G.f. A(x) satisfies: A(x) = 1 - 2 * x * ( A(x) - 2 * A(x/(1 - x)) / (1 - x) ).

a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-2)^n / k!.

a(0) = 1; a(n) = -2 * a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).

MATHEMATICA

nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1) - 2 x], {x, 0, nmax}], x] Range[0, nmax]!

a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

CROSSREFS

Cf. A000296, A078944, A126617, A194689, A367888, A367891.

KEYWORD

allocated

nonn

AUTHOR

Ilya Gutkovskiy, Dec 04 2023

STATUS

approved

editing

#1 by Ilya Gutkovskiy at Mon Dec 04 19:32:19 EST 2023

NAME

allocated for Ilya Gutkovskiy

KEYWORD

allocated

STATUS

approved