A282190
E.g.f.: 1/(1 + LambertW(1-exp(x))), where LambertW() is the Lambert W-function.
Original entry on oeis.org
1, 1, 5, 40, 447, 6421, 112726, 2338799, 55990213, 1519122598, 46066158817, 1543974969769, 56677405835276, 2261488166321697, 97455090037460785, 4510770674565054000, 223183550978156866507, 11755122645815049275521, 656670295411196201190366, 38779502115371642484125915, 2413908564514961126280655257
Offset: 0
E.g.f.: A(x) = 1 + x/1! + 5*x^2/2! + 40*x^3/3! + 447*x^4/4! + 6421*x^5/5! + 112726*x^6/6! + ...
- G. C. Greubel, Table of n, a(n) for n = 0..375
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
- Eric Weisstein's MathWorld, Stirling Transform
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b:= proc(n, m) option remember;
`if`(n=0, m^m, m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..23); # Alois P. Heinz, Aug 03 2021
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Range[0, 20]! CoefficientList[Series[1/(1 + ProductLog[1 - Exp[x]]), {x, 0, 20}], x]
Join[{1}, Table[Sum[StirlingS2[n, k] k^k, {k, 1, n}], {n, 1, 20}]]
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x='x+O('x^50); Vec(serlaplace(1/(1 + lambertw(1-exp(x))))) \\ G. C. Greubel, Nov 12 2017
A323280
a(n) = Sum_{k=0..n} binomial(n,k) * k^(2*k).
Original entry on oeis.org
1, 2, 19, 781, 68553, 10100761, 2236373953, 693667946945, 286962262702657, 152652510206521921, 101513694573289791441, 82511051259976074269425, 80480313356721971865934369, 92773167329045961244649105633, 124768226258051318899374299271601
Offset: 0
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Table[1 + Sum[Binomial[n, k]*k^(2*k), {k, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 31 2019 *)
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a(n) = sum(k=0, n, binomial(n, k)*k^(2*k));
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x)^k/(1-x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k^2*x)^k/k!))) \\ Seiichi Manyama, Jul 04 2022
A351182
a(n) = Sum_{k=0..n} k^(2*k) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 15, 683, 61332, 9135004, 2035708760, 634172615600, 263166948202080, 140322186951905736, 93484350581344936344, 76095870609142447018152, 74311960997497053384537408, 85748280952260853814490688656
Offset: 0
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a(n) = sum(k=0, n, k^(2*k)*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^2*log(1+x))^k/k!)))
A308491
a(0) = 1, a(n) = Sum_{k=1..n} stirling2(n,k) * k^(3*k).
Original entry on oeis.org
1, 1, 65, 19876, 16895763, 30685843321, 102018812632786, 560682901512212459, 4738032814084465062121, 58320000513552476843995786, 1002620283226568243192938115197, 23280221638971518379191182864465213, 710336441472841166799952152725333251616
Offset: 0
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Join[{1}, Table[Sum[k^(3*k)*StirlingS2[n, k], {k, 1, n}], {n, 1, 15}]]
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^3*(exp(x)-1))^k/k!))) \\ Seiichi Manyama, Feb 04 2022
A351276
a(n) = Sum_{k=0..n} (2*k)^k * Stirling2(n,k).
Original entry on oeis.org
1, 2, 18, 266, 5506, 146602, 4772162, 183618794, 8152995138, 410307648938, 23079780216386, 1434953808618090, 97716253164212034, 7233006174407149866, 578233606405444793410, 49651123488091636885994, 4557474786380802233761090, 445324385454834015896585386
Offset: 0
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a(n) = sum(k=0, n, (2*k)^k*stirling(n, k, 2));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(2*(1-exp(x))))))
A351181
a(n) = Sum_{k=0..n} k^(k+n) * Stirling2(n,k).
Original entry on oeis.org
1, 1, 17, 826, 79107, 12553011, 2979141058, 988163147091, 436562014218313, 247800100563125728, 175732698005376526429, 152264214647249387402567, 158273183995563848011907696, 194391589002961482387840145341
Offset: 0
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a[0] = 1; a[n_] := Sum[k^(k + n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 04 2022 *)
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a(n) = sum(k=0, n, k^(k+n)*stirling(n, k, 2));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(k*x)-1))^k/k!)))
Showing 1-6 of 6 results.
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