A059605 -id:A059605 - cp's OEIS frontend

A059606 Expansion of (1/2)(exp(2x)-1)*exp(exp(x)-1).

0, 1, 4, 16, 68, 311, 1530, 8065, 45344, 270724, 1709526, 11376135, 79520644, 582207393, 4453142140, 35500884556, 294365897104, 2533900264547, 22604669612078, 208656457858161, 1990060882027600

Offset: 0

Vladeta Jovovic, Jan 29 2001

Starting (1, 4, 16, 68, 311, ...), = A008277 * A000217, i.e., the product of the Stirling2 triangle and triangular series. - Gary W. Adamson, Jan 31 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vladeta Jovovic, More information.

Cf. A000110, A005493, A059604, A059605.
Cf. A035098.
Cf. A008277, A000217.

s := series(1/2*(exp(2*x)-1)*exp(exp(x)-1), x, 21): for i from 0 to 20 do printf(`%d,`,i!*coeff(s,x,i)) od:

With[{nn=20},CoefficientList[Series[((Exp[2x]-1)Exp[Exp[x]-1])/2,{x,0,nn}] ,x] Range[0,nn]!] (* Harvey P. Dale, Nov 10 2011 *)

a(n) = Sum_{i=0..n} Stirling2(n, i)*binomial(i+1, 2).
a(n) = (1/2)*(Bell(n+2)-Bell(n+1)-Bell(n)). - Vladeta Jovovic, Sep 23 2003
G.f.: Sum_{k>=1} (k*(k + 1)/2)*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, Jun 19 2018
a(n) ~ n^2 * Bell(n) / (2*LambertW(n)^2) * (1 - LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021

A059604 Coefficients of polynomials (n-1)!P(n,k), P(n,k) = Sum_{i=0..n} Stirling2(n,i)binomial(k+i-1,k).

Original entry on oeis.org

1, 1, 2, 1, 9, 10, 1, 24, 107, 90, 1, 50, 575, 1750, 1248, 1, 90, 2135, 16050, 38244, 24360, 1, 147, 6265, 95445, 537334, 1078728, 631440, 1, 224, 15610, 424340, 4734289, 21569996, 38105220, 20865600, 1, 324, 34482, 1529640, 30128049

Offset: 1

Vladeta Jovovic, Jan 29 2001

			[1],
[1, 2],
[1, 9, 10],
[1, 24, 107, 90],
[1, 50, 575, 1750, 1248],
[1, 90, 2135, 16050, 38244, 24360],
[1, 147, 6265, 95445, 537334, 1078728, 631440],
...
P(2,k) = k + 2,
P(3,k) = (1/2!)*(k^2 + 9*k + 10),
P(4,k) = (1/3!)*(k^3 + 24*k^2 + 107*k + 90).

Vladeta Jovovic, More information

Cf. A059605, A059606, A320962.

P := (n, k) -> (n-1)!*add(Stirling2(n,i)*binomial(k+i-1,k), i=0..n):
for n from 1 to 8 do seq(coeff(expand(P(n,x)),x,n-k), k=1..n) od; # Peter Luschny, Nov 07 2018

row[n_] := (n-1)! CoefficientList[Sum[StirlingS2[n,i] Binomial[k+i-1,k] // FunctionExpand, {i,0,n}], k] // Reverse;
Array[row,10] // Flatten (* Jean-François Alcover, Jun 03 2019 *)

row(n)={Vec((n-1)!*sum(i=0, n, stirling(n,i,2)*binomial(x+i-1,i-1)))}
for(n=1, 10, print(row(n))) \\ Andrew Howroyd, Nov 07 2018

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A059606 Expansion of (1/2)(exp(2x)-1)*exp(exp(x)-1).

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A059604 Coefficients of polynomials (n-1)!P(n,k), P(n,k) = Sum_{i=0..n} Stirling2(n,i)binomial(k+i-1,k).

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cp's OEIS Frontend

A059606 Expansion of (1/2)*(exp(2*x)-1)*exp(exp(x)-1).

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A059604 Coefficients of polynomials (n-1)!*P(n,k), P(n,k) = Sum_{i=0..n} Stirling2(n,i)*binomial(k+i-1,k).

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A059606 Expansion of (1/2)(exp(2x)-1)*exp(exp(x)-1).

A059604 Coefficients of polynomials (n-1)!P(n,k), P(n,k) = Sum_{i=0..n} Stirling2(n,i)binomial(k+i-1,k).