A009153 -id:A009153

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A357649

Expansion of e.g.f. cosh( (exp(3*x) - 1)/3 ).

+10
4

1, 0, 1, 9, 64, 435, 3097, 24822, 232759, 2517345, 30070954, 382827225, 5110770205, 71421582024, 1049487311485, 16286699945853, 267145966335088, 4616924929100535, 83622792656855125, 1578916985654901366, 30957723637379211115, 628927539690331202661

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..21.

FORMULA

a(n) = Sum_{k=0..floor(n/2)} 3^(n-2*k) * Stirling2(n,2*k).

a(n) ~ 3^n * exp(n/LambertW(3*n) - n - 1/3) * n^n / (LambertW(3*n)^n * 2*sqrt(1 + LambertW(3*n))). - Vaclav Kotesovec, Oct 07 2022

MATHEMATICA

With[{m = 20}, Range[0, m]! * CoefficientList[Series[Cosh[(Exp[3*x] - 1)/3], {x, 0, m}], x]] (* Amiram Eldar, Oct 07 2022 *)

PROG

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(cosh((exp(3*x)-1)/3)))

(PARI) a(n) = sum(k=0, n\2, 3^(n-2*k)*stirling(n, 2*k, 2));

CROSSREFS

Cf. A009153, A024430, A357650.

Cf. A356572.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 07 2022

STATUS

approved

A357650

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

+10
3

1, 0, 1, 12, 113, 1000, 8977, 86996, 959905, 12303888, 179038689, 2840696540, 47684181393, 835731314808, 15277172343409, 292597596283684, 5900038421042753, 125488177929542944, 2809541905807203009, 65903118624174027436, 1610968753088423886257

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..20.

FORMULA

a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * Stirling2(n,2*k).

a(n) ~ 2^(2*n-1) * exp(n/LambertW(4*n) - n - 1/4) * n^n / (LambertW(4*n)^n * sqrt(1 + LambertW(4*n))). - Vaclav Kotesovec, Oct 07 2022

MATHEMATICA

With[{m = 20}, Range[0, m]! * CoefficientList[Series[Cosh[(Exp[4*x] - 1)/4], {x, 0, m}], x]] (* Amiram Eldar, Oct 07 2022 *)

PROG

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(cosh((exp(4*x)-1)/4)))

(PARI) a(n) = sum(k=0, n\2, 4^(n-2*k)*stirling(n, 2*k, 2));

CROSSREFS

Cf. A009153, A024430, A357649.

Cf. A357617.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 07 2022

STATUS

approved

A357661

Expansion of e.g.f. cosh( (exp(2*x) - 1)/sqrt(2) ).

+10
3

1, 0, 2, 12, 60, 320, 2040, 15568, 133648, 1230336, 11962400, 123144384, 1349008320, 15731096576, 194349866880, 2527082917120, 34392647418112, 488243791183872, 7216792525799936, 110936087161801728, 1771199461131500544, 29324602146652307456

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..21.

FORMULA

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

PROG

(PARI) my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cosh((exp(2*x)-1)/sqrt(2)))))

(PARI) a(n) = sum(k=0, n\2, 2^(n-k)*stirling(n, 2*k, 2));

CROSSREFS

Cf. A024430, A357662, A357663.

Cf. A009153, A264036, A357664.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 07 2022

STATUS

approved

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