A072034 -id:A072034 - cp's OEIS frontend

A003725 E.g.f.: exp( x * exp(-x) ).

1, 1, -1, -2, 9, -4, -95, 414, 49, -10088, 55521, -13870, -2024759, 15787188, -28612415, -616876274, 7476967905, -32522642896, -209513308607, 4924388011050, -38993940088199, 11731860520780, 3807154270837281

Offset: 0

R. H. Hardin

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..557
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3.

Cf. A000248, A069856, A072034. A215652.

Cf. this sequence (k=1), A292909 (k=2), A292910 (k=3), A292912 (k=4).

With[{nn=30},CoefficientList[Series[Exp[x Exp[-x]],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Oct 20 2011 *)

Vec(serlaplace(exp(exp(-x) * x))) \\ Charles R Greathouse IV, Sep 26 2017

a(n) = Sum_{k=0..n} (-k)^(n-k)*binomial(n, k). - Vladeta Jovovic, Mar 15 2003
First column of A215652. - Peter Bala, Sep 14 2012
G.f.: Sum_{k>=0} x^k/(1 + k*x)^(k+1). - Ilya Gutkovskiy, Jun 25 2018

A256016 a(n) = n! * Sum_{k=0..n} k^n/k!.

Original entry on oeis.org

1, 1, 6, 57, 796, 15145, 374526, 11669665, 447595800, 20733553809, 1141067915290, 73552752257281, 5484203261135028, 467864288815609465, 45236104846954021014, 4915818294874879570305, 596044703812665607374256, 80118478395137652912476449, 11870487496575403846760198322

Offset: 0

Vaclav Kotesovec, Jun 01 2015

Seiichi Manyama, Table of n, a(n) for n = 0..283
Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials

Cf. A000110, A031971, A072034, A242446, A337001, A337002, A354436.

Main diagonal of A337085.

Join[{1}, Table[n!*Sum[k^n/k!,{k,0,n}],{n,1,20}]]

a(n) = n!*sum(k=0, n, k^n/k!); \\ Michel Marcus, Aug 15 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x)^k/(k!*(1-k*x))))) \\ Seiichi Manyama, Aug 23 2022

a(n) ~ e*Bell(n)*n!, for the Bell numbers see A000110.
a(n) ~ sqrt(2*Pi) * n^(2*n+1/2) * exp(n/LambertW(n)-2*n) / (sqrt(1+LambertW(n)) * LambertW(n)^n).
E.g.f.: Sum_{k>=0} (k * x)^k / (k! * (1 - k * x)). - Seiichi Manyama, Aug 23 2022
a(n) = n! * [x^n] B_n(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - Seiichi Manyama, Jan 04 2024
a(n) = Sum_{k=0..n} k^n*(n-k)!*binomial(n,k). - Ridouane Oudra, Jun 16 2025

a(0)=1 prepended by Seiichi Manyama, Aug 14 2020

A216857 Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} that have a fixed point summed over all subsets.

Original entry on oeis.org

0, 1, 4, 24, 224, 2880, 47232, 942592, 22171648, 600698880, 18422374400, 630897721344, 23864653578240, 988197253808128, 44460603225407488, 2159714024218951680, 112652924603290615808, 6280048587936003784704, 372616014329572403183616, 23445082059018189741752320, 1559275240299007139066675200

Offset: 0

Geoffrey Critzer, Sep 17 2012

nonn

Essentially the same as A038049.
Also the number of rooted trees whose nodes are labeled with the blocks of a set partition of {1,2,...,n} each having a distinguished element. (See A000248.)
The bijection is straightforward. The trees correspond to functional digraphs mapping the distinguished elements towards the root. All the elements within each block are mapped to the distinguished element of that block. The distinguished element in the root node is the fixed point.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Cf. A048802, A072034.

With[{nmax = 20}, CoefficientList[Series[-LambertW[-x*Exp[x]], {x, 0, nmax}], x]*Range[0, nmax]!] (* modified by G. C. Greubel, Nov 15 2017 *)

for(n=0,30, print1(sum(k=1,n, binomial(n,k)*k^(n-1)), ", ")) \\ G. C. Greubel, Nov 15 2017

my(x='x+O('x^50)); concat([0], Vec(serlaplace(-lambertw(-x*exp(x))))) \\ G. C. Greubel, Nov 15 2017

E.g.f.: T(x*exp(x)) where T(x) is the e.g.f. for A000169.
a(n) = Sum_{k=1..n} binomial(n,k)*k^(n-1).
a(n) ~ sqrt(1+LambertW(exp(-1))) * n^(n-1) / (exp(n)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Jul 09 2013
O.g.f.: Sum_{n>=0} n^(n-1)* x^n / (1 - n*x)^(n+1). - Paul D. Hanna, May 22 2018
E.g.f.: the compositional inverse of A(x) is -A(-x). - Alexander Burstein, Aug 11 2018

A096131 Sum of the terms of the n-th row of triangle pertaining to A096130.

Original entry on oeis.org

1, 7, 105, 2386, 71890, 2695652, 120907185, 6312179764, 375971507406, 25160695768715, 1869031937691061, 152603843369288819, 13584174777196666630, 1309317592648179024666, 135850890740575408906465

Offset: 1

Amarnath Murthy, Jul 04 2004

nonn

The product of the terms of the n-th row is given by A034841.

Collection of partial binary matrices: 1 to n rows of length n and a total of n entries set to one in each partial matrix. - Olivier Gérard, Aug 08 2016

			From _Seiichi Manyama_, Aug 18 2018: (Start)
a(1) = (1/1!) * (1) = 1.
a(2) = (1/2!) * (1*2 + 3*4) = 7.
a(3) = (1/3!) * (1*2*3 + 4*5*6 + 7*8*9) = 105.
a(4) = (1/4!) * (1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16) = 2386. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..338

Cf. A014062, A096130, A034841, A007318, A226391, A167009, A167008, A167010, A072034, A086331, A349470.

List(List([1..20],n->List([1..n],k->Binomial(k*n,n))),Sum); # Muniru A Asiru, Aug 12 2018

A096130 := proc(n,k) binomial(k*n,n) ; end: A096131 := proc(n) local k; add( A096130(n,k),k=1..n) ; end: for n from 1 to 18 do printf("%d, ",A096131(n)) ; od ; # R. J. Mathar, Apr 30 2007
seq(add((binomial(n*k,n)), k=0..n), n=1..15); # Zerinvary Lajos, Sep 16 2007

Table[Sum[Binomial[k*n, n], {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jun 06 2013 *)

a(n) = sum(k=1, n, binomial(k*n, n)); \\ Michel Marcus, Aug 20 2018

a(n) = Sum_{k=1..n} binomial(k*n, n). - Reinhard Zumkeller, Jan 09 2005
a(n) = (1/n!) * Sum_{j=1..n} Product_{i=n*(j-1)+1..n*j} i. - Reinhard Zumkeller, Jan 09 2005 [corrected by Seiichi Manyama, Aug 17 2018]
a(n) ~ exp(1)/(exp(1)-1) * binomial(n^2,n). - Vaclav Kotesovec, Jun 06 2013

More terms from R. J. Mathar, Apr 30 2007

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

A226391 a(n) = Sum_{k=0..n} binomial(k*n, k).

Original entry on oeis.org

1, 2, 9, 103, 2073, 58481, 2101813, 91492906, 4671050401, 273437232283, 18046800575211, 1325445408799007, 107200425419863009, 9466283137384124247, 906151826270369213655, 93459630239922214535911, 10331984296666203358431361, 1218745075041575200343722415

Offset: 0

Vaclav Kotesovec, Jun 06 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A072034, A086331, A096131, A167008, A167010, A349471.

[(&+[Binomial(n*j,j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Aug 31 2022

Table[Sum[Binomial[k*n, k], {k, 0, n}], {n, 0, 20}]

A226391(n):=sum(binomial(k*n,k), k,0,n); makelist(A226391(n),n,0,30); /* Martin Ettl, Jun 06 2013 */

@CachedFunction
def A226391(n): return sum(binomial(n*j, j) for j in (0..n))
[A226391(n) for n in (0..30)] # G. C. Greubel, Aug 31 2022

a(n) ~ binomial(n^2, n).

A242446 a(n) = Sum_{k=1..n} C(n,k) * k^(2*n).

Original entry on oeis.org

1, 18, 924, 93320, 15609240, 3903974592, 1364509038592, 635177480713344, 379867490829555840, 283825251434680651520, 259092157573229145859584, 283735986144895532781391872, 367138254141051794797009309696, 554136240038549806366753446051840

Offset: 1

Vaclav Kotesovec, May 14 2014

Generally, for p>=1, a(n) = Sum_{k=1..n} C(n,k) * k^(p*n) is asymptotic to sqrt(r/(p+r-p*r)) * r^(p*n) * n^(p*n) / (exp(p*n) * (1-r)^n), where r = p/(p+LambertW(p*exp(-p))).

Sum_{k=1..n} (-1)^(n-k) * C(n,k) * k^(p*n) = n! * stirling2(p*n,n).

Cf. A007820, A072034, A242449, A256016.

Table[Sum[Binomial[n,k]*k^(2*n),{k,1,n}],{n,1,20}]

a(n) ~ sqrt(r/(2-r)) * r^(2*n) * n^(2*n) / (exp(2*n) * (1-r)^n), where r = 2/(2+LambertW(2*exp(-2))).

A242449 a(n) = Sum_{k=0..n} C(n,k) * (2k+1)^(2n+1).

Original entry on oeis.org

1, 28, 3612, 1064480, 560632400, 462479403072, 550095467201728, 891290348282967040, 1887146395301619304704, 5058811707344107766328320, 16746136671945501439084657664, 67088193422344140016282100785152, 319900900946743851959321101768511488

Offset: 0

Vaclav Kotesovec, May 14 2014

Generally, for p>=1, a(n) = Sum_{k=0..n} C(n,k) * (p*k+1)^(p*n+1) is asymptotic to n^(p*n+1) * p^(p*n+1) * r^(p*n+3/2+1/p) / (sqrt(p+r-p*r) * exp(p*n) * (1-r)^(n+1/p)), where r = p/(p+LambertW(p*exp(-p))).

G. C. Greubel, Table of n, a(n) for n = 0..190

Cf. A242373, A220955, A242446, A072034, A007820, A221214, A213193, A195242.

Table[Sum[Binomial[n,k]*(2*k+1)^(2*n+1),{k,0,n}],{n,0,20}]

for(n=0,30, print1(sum(k=0,n, binomial(n,k)*(2*k+1)^(2*n+1)), ", ")) \\ G. C. Greubel, Nov 16 2017

a(n) ~ n^(2*n+1) * 2^(2*n+1) * r^(2*n+2) / (sqrt(2-r) * exp(2*n) * (1-r)^(n+1/2)), where r = 2/(2+LambertW(2*exp(-2))) = 0.901829091937052...

A360618 Expansion of Sum_{k>=0} (k * x * (1 + k*x))^k.

Original entry on oeis.org

1, 1, 5, 43, 515, 7950, 150086, 3349945, 86296849, 2519907605, 82249222661, 2967449372028, 117266100841668, 5037282382077353, 233701540415817409, 11645959855678136519, 620389246928233860127, 35181554115178393462386, 2116059351692554708911298

Offset: 0

Seiichi Manyama, Feb 14 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..373

Cf. A072034, A360592, A360611.

Flatten[{1, Table[Sum[Binomial[n-k, k] * (n-k)^n, {k, 0, n/2}], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 14 2023 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+k*x))^k))

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

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

a(n) ~ c * d^n * n^n, where d = (1-r)^(2-r) / (r^r * (1-2*r)^(1-2*r)) where r = 0.163662210494891118101893756356803907477984542... is the root of the equation (1-2*r)^2 = r*(1-r) * exp(1/(1-r)) and c = 0.78619174295244329885973980954744130517052330684023764340463604028671858569... - Vaclav Kotesovec, Feb 14 2023

A258773 Triangle read by rows, T(n,k) = (-1)^(n-k)C(n,k)k^n, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -2, 4, 0, 3, -24, 27, 0, -4, 96, -324, 256, 0, 5, -320, 2430, -5120, 3125, 0, -6, 960, -14580, 61440, -93750, 46656, 0, 7, -2688, 76545, -573440, 1640625, -1959552, 823543, 0, -8, 7168, -367416, 4587520, -21875000, 47029248, -46118408, 16777216

Offset: 0

Peter Luschny, Jun 09 2015

The row polynomials are p(0, x) = 1, and p(n, x) = Eu(x)^n (x-1)^n, for n >= 1, where Eu(x) := x*d/dx is the Euler derivative with respect to x. See A075513. - Wolfdieter Lang, Oct 12 2022

Coefficients of the Sidi polynomials (-1)^n*x*D_{n,0,n}(x), for n >= 0, where D_{k,n,m}(z) is given in Theorem 4.2., p. 862, of Sidi [1980]. - Wolfdieter Lang, Apr 10 2023

			Triangle begins:
  [1]
  [0,  1]
  [0, -2,     4]
  [0,  3,   -24,     27]
  [0, -4,    96,   -324,     256]
  [0,  5,  -320,   2430,   -5120,    3125]
  [0, -6,   960, -14580,   61440,  -93750,    46656]
  [0,  7, -2688,  76545, -573440, 1640625, -1959552, 823543]

G. C. Greubel, Table of n, a(n) for n = 0..1326
Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874.

Cf. A000142, A000312, A072034, A075513, A154715.

seq(seq((-1)^(n-k)*binomial(n, k)*k^n, k=0..n), n=0..8);
T_row := proc(n) (-1)^n*(1-exp(x))^n/n!; diff(%,[x$n]); subs(exp(x)=t, n!*expand(%,x)); CoefficientList(%,t) end: seq(print(T_row(n)), n=0..7);

Flatten@Table[(-1)^(n - k) Binomial[n, k] k^n, {n, 0 , 10}, {k, 0, n}] (* G. C. Greubel, Dec 17 2015 *)

Sum_{k=0..n} T(n,k) = n!.
Sum_{k=0..n} |T(n,k)| = A072034(n).
Sum_{n>=0} Sum_{k=0..n} T(n,k) x^k y^n/n! = 1/(1 + W(-x*y*exp(-y))) where W is the Lambert W function. - Robert Israel, Dec 16 2015
T(n,n) = A000312(n). - Peter Luschny, Dec 17 2015
T(n, k+1) = n * A075513(n, k) if n > 0. - Michael Somos, May 13 2018

A332408 a(n) = Sum_{k=0..n} binomial(n,k) * k! * k^n.

Original entry on oeis.org

1, 1, 10, 213, 8284, 513105, 46406286, 5772636373, 945492503320, 197253667623681, 51069324556151290, 16067283861476491941, 6037615013420387657844, 2670812587802323522405393, 1373842484756310928089102022, 813119045938378747809030359445

Offset: 0

Ilya Gutkovskiy, Apr 23 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A000522, A006153, A031971, A063170, A072034, A256016, A277452, A332627.

Join[{1}, Table[Sum[Binomial[n, k] k! k^n, {k, 0, n}], {n, 1, 15}]]

a(n) = sum(k=0, n, binomial(n,k) * k! * k^n); \\ Michel Marcus, Apr 24 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x*exp(x))^k))) \\ Seiichi Manyama, Feb 19 2022

G.f.: Sum_{k>=0} k! * k^k * x^k / (1 - k*x)^(k+1).
a(n) = n! * Sum_{k=0..n} k^n / (n-k)!.
a(n) ~ c * n! * n^n, where c = A073229 = exp(exp(-1)). - Vaclav Kotesovec, Feb 20 2021
E.g.f.: Sum_{k>=0} (k*x*exp(x))^k. - Seiichi Manyama, Feb 19 2022

cp's OEIS Frontend

A003725 E.g.f.: exp( x * exp(-x) ).

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Mathematica

PARI

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A256016 a(n) = n! * Sum_{k=0..n} k^n/k!.

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A216857 Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} that have a fixed point summed over all subsets.

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PARI

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A096131 Sum of the terms of the n-th row of triangle pertaining to A096130.

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GAP

Maple

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A226391 a(n) = Sum_{k=0..n} binomial(k*n, k).

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Magma

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Maxima

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A242446 a(n) = Sum_{k=1..n} C(n,k) * k^(2*n).

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A242449 a(n) = Sum_{k=0..n} C(n,k) * (2*k+1)^(2*n+1).

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A242449 a(n) = Sum_{k=0..n} C(n,k) * (2k+1)^(2n+1).

A258773 Triangle read by rows, T(n,k) = (-1)^(n-k)C(n,k)k^n, for n >= 0 and 0 <= k <= n.