A049020 -id:A049020 - cp's OEIS frontend

A059099 Expansion of exp(exp(x)-1)/(2-exp(x)).

1, 2, 7, 33, 198, 1453, 12669, 128320, 1482721, 19260421, 277913552, 4410640919, 76360030701, 1432144732762, 28926138244883, 625974400305541, 14449445989893990, 354384475357492593, 9202837263156670345, 252260867710562944224, 7278710072406887897461

Offset: 0

Vladeta Jovovic, Jan 02 2001

Row sums of A227343. - Peter Bala, Jul 11 2013

The sequence gives the number of barred preferential arrangements of an n-set having one bar, where one fixed section is a free section and elements which are to go into the other section are partitioned into unordered nonempty subsets. - Sithembele Nkonkobe, Jul 02 2015

			exp(exp(x)-1)/(2-exp(x)) = 1 + 2*x + 7/2*x^2 + 11/2*x^3 + 33/4*x^4 + 1453/120*x^5 + 4223/240*x^6 + 1604/63*x^7 + ...

Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 12, 19, 29.
S. Nkonkobe and V. Murali, A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements, arXiv:1503.06172 [math.CO], 2015.

Row sums of A059098.

Cf. A001861, A000110, A005493, A049020, A227343.

s := series(exp(exp(x)-1)/(2-exp(x)), x, 60): for i from 0 to 50 do printf(`%d,`,i!*coeff(s,x,i)) od:

CoefficientList[Series[E^(E^x-1)/(2-E^x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jul 02 2015 *)

a(n) = Sum_{m=0..n} Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..m} (i-j+1).
Stirling transform of A000522. - Vladeta Jovovic, May 10 2004
a(n) ~ n!*exp(1)/(2*(log(2))^(n+1)). - Vaclav Kotesovec, Jul 02 2015

More terms from James Sellers, Jan 03 2001

A059115 Expansion of e.g.f.: ((1-x)/(1-2x))exp(x/(1-x)).

Original entry on oeis.org

1, 2, 9, 58, 485, 4986, 60877, 861554, 13878153, 250854130, 5030058161, 110837000682, 2662669300909, 69270266115818, 1940260799150325, 58220372514830626, 1863293173842259217, 63356877145370671074

Offset: 0

Vladeta Jovovic, Jan 06 2001

L'(n,i) are unsigned Lah numbers (Cf. A008297): L'(n,i) = (n!/i!)*binomial(n-1,i-1) for i >= 1, L'(0,0) = 1, L'(n,0) = 0 for n > 0.

			(1-x)/(1-2*x)*exp(x/(1-x)) = 1 + 2*x + 9/2*x^2 + 29/3*x^3 + 485/24*x^4 + 831/20*x^5 + ...

G. C. Greubel, Table of n, a(n) for n = 0..250
Index entries for sequences related to Laguerre polynomials

Cf. A001861, A049020, A052897, A059099, A059110.

Row sums of A059114.

[Factorial(n)*(&+[Evaluate(LaguerrePolynomial(n-k, k-1), -1) : k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 23 2021

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

With[{nn=20},CoefficientList[Series[(1-x)/(1-2x) Exp[x/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 18 2020 *)
Table[n!*Sum[LaguerreL[n-k, k-1, -1], {k,0,n}], {n,0,30}] (* G. C. Greubel, Feb 23 2021 *)

{a(n)=if(n<0, 0, n!*polcoeff( (1-x)/(1-2*x)*exp(x/(1-x)+x*O(x^n)), n))} /* Michael Somos, Aug 03 2006 */

{a(n)=local(A); if(n<0,0, n++; A=vector(n); A[n]=1; for(k=1,n-1, A[n-k]=1; if(k>1, A[n-k+1]=A[n-k+2]); for(i=n-k+1,n, A[i]=A[i-1]+k*A[i])); A[n])} /* Michael Somos, Aug 03 2006 */

a(n) = n!*sum(k=0, n, pollaguerre(n-k, k-1, -1)); \\ Michel Marcus, Feb 23 2021

[factorial(n)*sum( gen_laguerre(n-k, k-1, -1) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Feb 23 2021

Sum_{m=0..n} Sum_{i=0..n} L'(n, i)*Product_{j=1..m} (i-j+1).
Given g.f. A(x), then g.f. A000522 = A(x/(1+x)). - Michael Somos, Aug 03 2006
a(n) = n!*Sum_{k=0..n} LaguerreL(n-k, k-1, -1). - G. C. Greubel, Feb 23 2021
a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n + 1/2) / exp(n-1). - Vaclav Kotesovec, Feb 23 2021

Definition clarified by Harvey P. Dale, Jul 18 2020

A245109 G.f.: Sum_{n>=0} exp(-(1 + n^2x)) (1 + n^2*x)^n / n!.

Original entry on oeis.org

1, 3, 31, 520, 11991, 350889, 12428746, 516450792, 24619176153, 1323971052261, 79280864647205, 5231080689880500, 377062508515478306, 29479066783583059530, 2484534527715953700780, 224559818606249783480400, 21666961097367611148157815, 2222844864226101120054773295

Offset: 0

Paul D. Hanna, Jul 12 2014

nonn

Compare the g.f. to:
(1) Sum_{n>=0} exp(-(1+n*x)) * (1+n*x)^n / n! = 1/(1-x).
(2) Sum_{n>=1} exp(-n^2*x) * n^(2*n) * x^n/n! = Sum_{n>=1} S2(2*n,n)*x^n (A007820).

			G.f.: A(x) = 1 + 3*x + 31*x^2 + 520*x^3 + 11991*x^4 + 350889*x^5 +...
where
A(x) = exp(-1) + exp(-(1+x))*(1+x) + exp(-(1+2^2*x))*(1+2^2*x)^2/2!
+ exp(-(1+3^2*x))*(1+3^2*x)^3/3! + exp(-(1+4^2*x))*(1+4^2*x)^4/4!
+ exp(-(1+5^2*x))*(1+5^2*x)^5/5! + exp(-(1+6^2*x))*(1+6^2*x)^6/6!
+ exp(-(1+7^2*x))*(1+7^2*x)^7/7! + exp(-(1+8^2*x))*(1+8^2*x)^8/8! +...
simplifies to a power series in x with integer coefficients.

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (first 100 terms from Paul D. Hanna)

Cf. A049020, A245110.

Cf. A187655, A217899, A217900.

Table[SeriesCoefficient[Sum[E^(-(1+k^2*x))*(1+k^2*x)^k/k!,{k,0,Infinity}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 12 2014 *)

/* Must first set suitable precision */ \p300
{a(n)=local(A=1+x); A=suminf(k=0, exp(-(1+k^2*x)+x*O(x^n))*(1+k^2*x)^k/k!); round(polcoeff(A, n))}
for(n=0, 30, print1(a(n), ", "))

a(n) ~ c * d^n * (n-1)!, where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.17655460948348035823168..., and c = 10.427337127699040838035... . - Vaclav Kotesovec, Jul 12 2014

a(n) = A049020(2n,n). - Alois P. Heinz, Aug 23 2017

A059098 Triangle read by rows. T(n, k) = Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..k} (i - j + 1) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 5, 10, 12, 6, 15, 37, 62, 60, 24, 52, 151, 320, 450, 360, 120, 203, 674, 1712, 3120, 3720, 2520, 720, 877, 3263, 9604, 21336, 33600, 34440, 20160, 5040, 4140, 17007, 56674, 147756, 287784, 394800, 352800, 181440, 40320, 21147, 94828

Offset: 0

Vladeta Jovovic, Jan 02 2001

The transpose of this lower unitriangular array is the U factor in the LU decomposition of the Hankel matrix (Bell(i+j-2))A000110(n).%20The%20L%20factor%20is%20A049020%20(see%20Chamberland,%20p.%201672).%20-%20_Peter%20Bala">i,j >= 1, where Bell(n) = A000110(n). The L factor is A049020 (see Chamberland, p. 1672). - _Peter Bala, Oct 15 2023

			Triangle begins:
  [0] [ 1]
  [1] [ 1,    1]
  [2] [ 2,    3,    2]
  [3] [ 5,   10,   12,    6]
  [4] [15,   37,   62,   60,   24]
  [5] [52,  151,  320,  450,  360,  120]
  [6] [203, 674, 1712, 3120, 3720, 2520, 720]
  ...;
E.g.f. for T(n, 2) = (exp(x)-1)^2*(exp(exp(x)-1)) = x^2 + 2*x^3 + 31/12*x^4 + 8/3*x^5 + 107/45*x^6 + 343/180*x^7 + 28337/20160*x^8 + 349/360*x^9 + ...;
E.g.f. for T(n, 3) = (exp(x)-1)^3*(exp(exp(x)-1)) = x^3 + 5/2*x^4 + 15/4*x^5 + 13/3*x^6 + 127/30*x^7 + 1759/480*x^8 + 34961/12096*x^9 + ...

Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.

Cf. A000110(n) = T(n,0), A005493(n) = T(n,1), A059099 (row sums).

Cf. A049020, A001861, A046716.

T := proc(n, k) option remember; `if`(k < 0 or k > n, 0,
      `if`(n = 0, 1, k*T(n-1, k-1) + (k+1)*T(n-1, k) + T(n-1, k+1)))
    end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..15); # Peter Bala, Oct 15 2023

E.g.f. for T(n, k): (exp(x)-1)^k*(exp(exp(x)-1)).
n-th row is M^n*[1,0,0,0,...], where M is a tridiagonal matrix with all 1's in the superdiagonal, (1,2,3,...) in the main and subdiagonals; and the rest zeros. - Gary W. Adamson, Jun 23 2011
T(n, k) = k!*A049020(n, k). - R. J. Mathar, May 17 2016
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j). - Peter Luschny, Dec 06 2023

A346842 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^3 / 3!.

Original entry on oeis.org

1, 10, 75, 520, 3556, 24626, 174805, 1279240, 9677151, 75750752, 613656836, 5142797660, 44557627661, 398786697398, 3683575764083, 35084121263136, 344242894197456, 3476490965903174, 36104281709286841, 385257741260565844, 4220537246457019687, 47432055430482106880

Offset: 3

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000110, A000292, A000392, A003128, A005493, A049020, A189924, A346843, A346844.

b:= proc(n, m) option remember;
     `if`(n=0, binomial(m, 3), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=3..24);  # Alois P. Heinz, Aug 05 2021

nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^3/3!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
Table[Sum[StirlingS2[n, k] Binomial[k, 3], {k, 0, n}], {n, 3, 24}]
Table[Sum[Binomial[n, k] StirlingS2[k, 3] BellB[n - k], {k, 0, n}], {n, 3, 24}]
Table[(BellB[n+3] - 6*BellB[n+2] + 8*BellB[n+1] - BellB[n])/6, {n, 3, 24}] (* Vaclav Kotesovec, Aug 06 2021 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^3/3!)) \\ Michel Marcus, Aug 06 2021

a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,3).
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,3) * Bell(n-k).
a(n) = (Bell(n+3) - 6*Bell(n+2) + 8*Bell(n+1) - Bell(n))/6. - Vaclav Kotesovec, Aug 06 2021
a(n) ~ exp(-1 - n + n/LambertW(n)) * (n - LambertW(n))^3 * n^n / (6 * sqrt(1 + LambertW(n)) * LambertW(n)^(n+3)). - Vaclav Kotesovec, Jun 28 2022

A346843 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^4 / 4!.

Original entry on oeis.org

1, 15, 155, 1400, 11991, 101031, 853315, 7300260, 63641006, 567304452, 5181338526, 48538121450, 466611951261, 4603782469653, 46613101232933, 484188586821376, 5157850655391981, 56321812548867229, 630125374420189131, 7219368394888423554, 84658119388335562972

Offset: 4

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000110, A000332, A000453, A003128, A005493, A049020, A346842, A346844.

b:= proc(n, m) option remember;
     `if`(n=0, binomial(m, 4), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=4..24);  # Alois P. Heinz, Aug 05 2021

nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^4/4!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
Table[Sum[StirlingS2[n, k] Binomial[k, 4], {k, 0, n}], {n, 4, 24}]
Table[Sum[Binomial[n, k] StirlingS2[k, 4] BellB[n - k], {k, 0, n}], {n, 4, 24}]
Table[(BellB[n] - 24*BellB[n+1] + 29*BellB[n+2] - 10*BellB[n+3] + BellB[n+4])/24, {n, 4, 24}] (* Vaclav Kotesovec, Aug 06 2021 *)
With[{nn=30},Drop[CoefficientList[Series[(Exp[Exp[x]-1](Exp[x]-1)^4)/4!,{x,0,nn}],x] Range[0,nn]!,4]] (* Harvey P. Dale, Oct 03 2024 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^4/4!)) \\ Michel Marcus, Aug 06 2021

a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,4).
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,4) * Bell(n-k).
a(n) = (Bell(n) - 24*Bell(n+1) + 29*Bell(n+2) - 10*Bell(n+3) + Bell(n+4))/24. - Vaclav Kotesovec, Aug 06 2021

A346844 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^5 / 5!.

Original entry on oeis.org

1, 21, 287, 3290, 34671, 350889, 3492511, 34669734, 346231886, 3497726232, 35872743270, 374387203190, 3982122624117, 43207791878715, 478532965417529, 5411213661200830, 62482405993313229, 736696756305382411, 8868148033487285103, 108969560832001750716

Offset: 5

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000110, A000389, A000481, A003128, A005493, A049020, A094816, A346842, A346843.

b:= proc(n, m) option remember;
     `if`(n=0, binomial(m, 5), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=5..24);  # Alois P. Heinz, Aug 05 2021

nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^5/5!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
Table[Sum[StirlingS2[n, k] Binomial[k, 5], {k, 0, n}], {n, 5, 24}]
Table[Sum[Binomial[n, k] StirlingS2[k, 5] BellB[n - k], {k, 0, n}], {n, 5, 24}]
Table[(-BellB[n] + 89*BellB[n+1] - 145*BellB[n+2] + 75*BellB[n+3] - 15*BellB[n+4] + BellB[n+5])/120, {n, 5, 24}] (* Vaclav Kotesovec, Aug 06 2021 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^5/5!)) \\ Michel Marcus, Aug 06 2021

a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,5).
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,5) * Bell(n-k).
a(n) = (-Bell(n) + 89*Bell(n+1) - 145*Bell(n+2) + 75*Bell(n+3) - 15*Bell(n+4) + Bell(n+5))/120. - Vaclav Kotesovec, Aug 06 2021

A090809 Coefficient of the irreducible character of S_m indexed by (m-2n+2,2n-2) in the n-th Kronecker power of the representation indexed by (m-2,2).

Original entry on oeis.org

0, 0, 2, 10, 31, 75, 155, 287, 490, 786, 1200, 1760, 2497, 3445, 4641, 6125, 7940, 10132, 12750, 15846, 19475, 23695, 28567, 34155, 40526, 47750, 55900, 65052, 75285, 86681, 99325, 113305, 128712, 145640, 164186, 184450, 206535, 230547

Offset: 0

Alain Goupil, Feb 10 2004

For n > 0, the terms of this sequence are related to A000124 by a(n) = Sum_{i=0..n-1} i*A000124(i). - Bruno Berselli, Dec 20 2013

A. Goupil, Combinatorics of the Kronecker products of irreducible representations of Sn, in preparation.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000124, A049020.

f := proc(k) 2*binomial(k,2)+4*binomial(k,3)+3*binomial(k,4); end;
seq (f(n), n=0..50);

f[n_] := 2Binomial[n, 2] + 4Binomial[n, 3] + 3Binomial[n, 4]; Table[ f[n], {n, 0, 40}] (* Robert G. Wilson v, Feb 13 2004 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 2, 10, 31}, 38] (* Jean-François Alcover, Sep 25 2017 *)

a(n) = 2*binomial(n, 2) + 4*binomial(n, 3) + 3*binomial(n, 4) = (n-1)*n*(3*n^2 + n + 10)/24.
a(n) = A049020(n, n-2), for n >= 2. - Philippe Deléham, Mar 06 2004
G.f.: x^2*(2 + x^2) / (1-x)^5. - Colin Barker, Nov 21 2012

More terms from Robert G. Wilson v, Feb 13 2004

A174516 Partial sums of A002896.

Original entry on oeis.org

1, 7, 97, 1957, 46687, 1219243, 33715399, 970085119, 28740443449, 870830918389, 26860099935529, 840549807424369, 26620996978712269, 851664885506669269, 27482469263443730269, 893460843597349019629, 29235859228655427097639

Offset: 0

Jonathan Vos Post, Mar 20 2010

nonn

			a(4) = 1 + 6 + 90 + 1860 + 44730 = 46687.

Cf. A002896, A049020, A049037, A084261, A138540, A140476.

b[n_] := b[n] = (* A002896 *) Binomial[2*n, n]*HypergeometricPFQ[{1/2, -n, -n}, {1, 1}, 4]; a[n_] := Sum[b[k], {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Dec 20 2011 *)

a(n) = Sum_{i=0..n} A002896(i).
G.f.: g/(1-x) where g is the o.g.f. of A002896. - Mark van Hoeij, Nov 12 2011
a(n) ~ 2^(2*n) * 3^(2*n + 7/2) / (35 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Feb 17 2024

A244489 Triangle read by rows: T(n,k) = Sum_{j=k..n} binomial(n,j)Stirling_2(j,k)Bell(n-j), where Bell(n) = A000110(n), for n >= 1, 0 <= k <= n-1.

Original entry on oeis.org

1, 2, 3, 5, 10, 6, 15, 37, 31, 10, 52, 151, 160, 75, 15, 203, 674, 856, 520, 155, 21, 877, 3263, 4802, 3556, 1400, 287, 28, 4140, 17007, 28337, 24626, 11991, 3290, 490, 36, 21147, 94828, 175896, 174805, 101031, 34671, 6972, 786, 45, 115975, 562595, 1146931, 1279240, 853315, 350889, 88977, 13620, 1200, 55

Offset: 1

N. J. A. Sloane, Jul 04 2014

			Triangle begins:
1
2 3
5 10 6
15 37 31 10
52 151 160 75 15
203 674 856 520 155 21
877 3263 4802 3556 1400 287 28
4140 17007 28337 24626 11991 3290 490 36
...

J. East, R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

Same as A049020 (which is the main entry for this triangle) except the present sequence has an extra 1 at the end of each row. - R. J. Mathar and N. J. A. Sloane, May 17 2016

T[n_, k_] := Sum[Binomial[n, j] StirlingS2[j, k] BellB[n-j], {j, k, n}];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Oct 09 2018 *)

cp's OEIS Frontend

A059099 Expansion of exp(exp(x)-1)/(2-exp(x)).

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A059115 Expansion of e.g.f.: ((1-x)/(1-2*x))*exp(x/(1-x)).

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A245109 G.f.: Sum_{n>=0} exp(-(1 + n^2*x)) * (1 + n^2*x)^n / n!.

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A059098 Triangle read by rows. T(n, k) = Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..k} (i - j + 1) for 0 <= k <= n.

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A346842 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^3 / 3!.

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A346843 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^4 / 4!.

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A346844 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^5 / 5!.

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A059115 Expansion of e.g.f.: ((1-x)/(1-2x))exp(x/(1-x)).

A245109 G.f.: Sum_{n>=0} exp(-(1 + n^2x)) (1 + n^2*x)^n / n!.

A244489 Triangle read by rows: T(n,k) = Sum_{j=k..n} binomial(n,j)Stirling_2(j,k)Bell(n-j), where Bell(n) = A000110(n), for n >= 1, 0 <= k <= n-1.