A002720 -id:A002720 - cp's OEIS frontend

A261767 Triangle read by rows: T(n,k) is the number of subpermutations of an n-set, whose orbits are each of size at most k with at least one orbit of size exactly k.

1, 1, 1, 1, 3, 3, 1, 7, 18, 8, 1, 15, 99, 64, 30, 1, 31, 510, 560, 300, 144, 1, 63, 2745, 4800, 3150, 1728, 840

Offset: 0

Samira Stitou, Sep 21 2015

			T(3, 2) = 18 because there are 18 subpermutations on {1,2,3} whose orbits are each of size at most 2 with at least one orbit of size exactly 2, namely: (1 2 --> 2 1), (1 3 --> 3 1), (2 3 --> 3 2), (123 --> 213), (123 --> 321), (123 --> 132); (1-->2), (1-->3), (2-->1), (2-->3), (3-->1), (3-->2); (13-->23), (12-->32), (23-->13), (32-->33), (23-->21), (13-->12).
Triangle starts:
1;
1, 1;
1, 3, 3;
1, 7, 18, 8;
1, 15, 99, 64, 30;
1, 31, 510, 560, 300, 144;
...

A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.

Cf. A261762, A261763, A261764, A261765, A261766, A261767.

Row sums give A002720.

T(n, k) = A261763(n, k) - A261763(n, k-1), T(n, n) = A261766(n) for all n not equal to 1 and T(1, 1) = 1.

A289147 Number of (n+1) X (n+1) binary matrices M with at most one 1 in each of the first n rows and each of the first n columns and M[n+1,n+1] = 0.

Original entry on oeis.org

1, 5, 34, 286, 2840, 32344, 414160, 5876336, 91356544, 1542401920, 28075364096, 547643910400, 11389266525184, 251428006132736, 5869482147358720, 144413021660821504, 3733822274973040640, 101181690628832198656, 2867011297057247002624, 84764595415605494743040

Offset: 0

Alois P. Heinz, Jun 26 2017

nonn

Number of marriage patterns between a labeled set X of n women and a labeled set Y of n men (all heterosexual): some couples can be formed where one partner is from X and the other from Y, some members of X and Y marry external (unlabeled) partners, and some do not marry.

			a(1) = 5:
[0 0]  [1 0]  [0 1]  [0 0]  [0 1]
[0 0]  [0 0]  [0 0]  [1 0]  [1 0] .
.
a(2) = 34:
[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]
[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 0 1]
[0 0 0]  [0 1 0]  [1 0 0]  [1 1 0]  [0 0 0]  [0 1 0]  [1 0 0]
.
[0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]
[0 0 1]  [0 1 0]  [0 1 0]  [1 0 0]  [1 0 0]  [0 0 0]  [0 0 0]
[1 1 0]  [0 0 0]  [1 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [0 1 0]
.
[0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]
[0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]  [0 1 0]
[1 0 0]  [1 1 0]  [0 0 0]  [0 1 0]  [1 0 0]  [1 1 0]  [0 0 0]
.
[0 0 1]  [0 0 1]  [0 0 1]  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 0]
[0 1 0]  [1 0 0]  [1 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]
[1 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [1 0 0]  [0 0 0]  [1 0 0]
.
[0 1 0]  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 0]
[1 0 0]  [0 0 0]  [0 0 0]  [0 0 1]  [0 0 1]  [0 1 0]
[0 0 0]  [0 0 0]  [0 1 0]  [0 0 0]  [0 1 0]  [0 0 0]  .

Alois P. Heinz, Table of n, a(n) for n = 0..437
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Column k=4 of A289192.

Cf.: A000142, A000165, A000302, A002720, A025167, A084771, A087912, A102773, A160611, A160612, A277382.

a:= proc(n) option remember; `if`(n<2, 4*n+1,
      (2*n+3)*a(n-1)-(n-1)^2*a(n-2))
    end:
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n-> n! * add(binomial(n, i)*4^i/i!, i=0..n):
seq(a(n), n=0..25);
# third Maple program:
a:= n-> n!* simplify(LaguerreL(n, -4), 'LaguerreL'):
seq(a(n), n=0..25);

Table[n! LaguerreL[n, -4], {n, 0, 30}] (* Indranil Ghosh, Jul 06 2017 *)

from mpmath import *
mp.dps=150
l=chop(taylor(lambda x:exp(4*x/(1-x))/(1-x), 0, 31))
print([int(fac(i)*l[i]) for i in range(len(l))]) # Indranil Ghosh, Jul 06 2017
# or #
from mpmath import *
mp.dps=100
def a(n): return int(fac(n)*laguerre(n, 0, -4))
print([a(n) for n in range(31)]) # Indranil Ghosh, Jul 06 2017

E.g.f.: exp(4*x/(1-x))/(1-x).
a(n) = Sum_{i=0..n} i! * (2^(n-i)*binomial(n,i))^2.
a(n) = Sum_{i=0..n} (n-i)! * 4^i * binomial(n,i)^2.
a(n) = n! * Sum_{i=0..n} 4^i/i! * binomial(n,i).
a(n) = (2*n+3)*a(n-1)-(n-1)^2*a(n-2) for n>=2, a(n) = 4*n+1 for n<2.
a(n) = n! * Laguerre(n,-4) = n! * A160611(n)/A160612(n).
a(n) ~ exp(-2 + 4*sqrt(n) - n) * n^(n + 1/4) / 2 * (1 + 163/(96*sqrt(n))). - Vaclav Kotesovec, Nov 13 2017
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 4^n * x^n / (n!)^2. - Ilya Gutkovskiy, Jul 17 2020

A326252 Number of digraphs with vertices {1..n} whose increasing edges are crossing.

Original entry on oeis.org

0, 0, 0, 0, 16384, 22020096, 62679678976, 556181084962816

Offset: 0

Gus Wiseman, Jun 30 2019

A directed edge (a,b) is increasing if a < b. Two edges (a,b), (c,d) are crossing if a < c < b < d or c < a < d < b.

Simple graphs whose edges are crossing are A326210.
Digraphs whose increasing edges are not crossing are A326251.
Digraphs whose edges are not crossing are A326237.
Cf. A002416, A002720, A229865, A324170, A326209, A326252, A326258.

croXQ[eds_]:=MatchQ[eds,{_,{x_,y_},_,{z_,t_},_}/;x

a(n) = 2^(n * (n + 1)/2) * A326210(n).

A331432 Triangle T(n,k) (n >= k >= 0) read by rows: T(n,0) = (1+(-1)^n)/2; for k>=1, set T(0,k) = 0, S(n,k) = binomial(n,k)*binomial(n+k+1,k), and for n>=1, T(n,k) = S(n,k)-T(n-1,k).

Original entry on oeis.org

1, 0, 3, 1, 5, 10, 0, 10, 35, 35, 1, 14, 91, 189, 126, 0, 21, 189, 651, 924, 462, 1, 27, 351, 1749, 4026, 4290, 1716, 0, 36, 594, 4026, 13299, 22737, 19305, 6435, 1, 44, 946, 8294, 36751, 89375, 120835, 85085, 24310, 0, 55, 1430, 15730, 89375, 289003, 551837, 615043, 369512, 92378, 1, 65, 2080, 27950, 197275, 811733, 2047123, 3203837, 3031678, 1587222, 352716

Offset: 0

N. J. A. Sloane, Jan 17 2020

The scanned pages of Ser are essentially illegible, and the book is out of print and hard to locate.

For Table IV on page 93, it is simplest to ignore the minus signs. The present triangle then matches all the given terms in that triangle, so it seems best to define the triangle by the recurrences given here, and to conjecture (strongly) that this is the same as Ser's triangle.

			Triangle begins:
  1;
  0,  3;
  1,  5,   10;
  0, 10,   35,    35;
  1, 14,   91,   189,    126;
  0, 21,  189,   651,    924,    462;
  1, 27,  351,  1749,   4026,   4290,    1716;
  0, 36,  594,  4026,  13299,  22737,   19305,    6435;
  1, 44,  946,  8294,  36751,  89375,  120835,   85085,   24310;
  0, 55, 1430, 15730,  89375, 289003,  551837,  615043,  369512,   92378;
  1, 65, 2080, 27950, 197275, 811733, 2047123, 3203837, 3031678, 1587222, 352716;

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)

Columns 1 and 2 are A176222 and A331429; the last three diagonals are A002739, A002737, A001700.
Taking the component-wise sums of the rows by pairs give the triangle in A178303.
Ser's tables I and III are A331430 and A331431 (both are still mysterious).

SS := (n,k)->binomial(n,k)*binomial(n+k+1,k);
T4:=proc(n,k) local i; global SS; option remember;
if k=0 then return((1+(-1)^n)/2); fi;
if n=0 then 0 else SS(n,k)-T4(n-1,k); fi; end;
rho:=n->[seq(T4(n,k),k=0..n)];
for n from 0 to 14 do lprint(rho(n)); od:

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0, (1 + (-1)^n)/2, Binomial[n, k]*Binomial[n+k+1, k] - T[n-1, k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)

def T(n,k): # A331432
    if (n<0): return 0
    elif (k==0): return ((n+1)%2)
    else: return binomial(n,k)*binomial(n+k+1,k) - T(n-1,k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

T(n, k) = binomial(n,k)*binomial(n+k+1,k) - T(n-1, k), with T(n, 0) = (1 + (-1)^n)/2.
T(n, 0) = A000035(n+1).
T(n, 1) = A176222(n).
T(n, 2) = A331429(n).
T(n, n-2) = A002739(n).
T(n, n-1) = A002737(n).
T(n, n) = A001700(n).

A119400 a(n) = Sum_{k=0..n} (n!/k!)^2*binomial(n,k).

Original entry on oeis.org

1, 2, 13, 172, 3809, 126526, 5874517, 362848088, 28744087297, 2839192902874, 341922922464701, 49297062811573732, 8380916229314577313, 1658770724530766046422, 378056469777362366873989, 98286603829297813268996176, 28907477297195536067142301697

Offset: 0

Vladeta Jovovic, Jul 25 2006

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

Cf. A000522, A002720, A216831.

Table[Sum[(n!/k!)^2*Binomial[n, k], {k, 0, n}], {n, 0, 16}] (* Stefan Steinerberger, Jun 17 2007 *)

a(n)=n!^3*polcoeff(exp(x+x*O(x^n))*sum(m=0, n, x^m/m!^3), n)
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 27 2012

Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0,2*sqrt(x/(1-x)))/(1-x).
Recurrence: a(n) = (3*n^2-3*n+2)*a(n-1)-3*(n-1)^4*a(n-2)+(n-2)^3*(n-1)^3*a(n-3). - Vaclav Kotesovec, Sep 30 2012
a(n) ~ 1/sqrt(3)*n^(2*n+2/3)/exp(2*n-3*n^(1/3)). - Vaclav Kotesovec, Sep 30 2012
E.g.f.: exp(x) * Sum_{n>=0} x^n/n!^3  =  Sum_{n>=0} a(n)*x^n/n!^3. - Paul D. Hanna, Nov 27 2012
a(n) = Sum_{k=0..n} k!^2*binomial(n,k)^3. - Ridouane Oudra, Jun 14 2025

More terms from Stefan Steinerberger, Jun 17 2007

A121629 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^2(a+a), where a+ and a are boson creation and annihilation operators, respectively.

Original entry on oeis.org

1, 3, 16, 121, 1179, 14026, 196783, 3177861, 58019356, 1181098459, 26515026561, 650572403218, 17316566815441, 496889918749251, 15288155067806104, 502024850361876481, 17522822345606176083, 647790109599863145106, 25283238154309049107231

Offset: 0

Karol A. Penson, Aug 12 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..350
K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. vol. 50, 083512 (2009)

Cf. A002720, A121630, A121631, A239301.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(((1-2*x)^(-1/2))-1)/(1-2*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 17 2018

CoefficientList[Series[E^(((1-2*x)^(-1/2))-1)/(1-2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 29 2013 *)

x='x+O('x^30); Vec(serlaplace(exp(((1-2*x)^(-1/2))-1)/(1-2*x))) \\ G. C. Greubel, May 17 2018

a(n) = Sum_{p=1..n+1} abs(stirling1(n+1,p))*2^(n-p+1)*bell(p-1), n=0,1...
E.g.f.: exp(((1-2*x)^(-1/2))-1)/(1-2*x). - Vladeta Jovovic, Aug 13 2006
Recurrence: a(n) = (6*n-5)*a(n-1) - (2*n-3)*(6*n-7)*a(n-2) + 4*(2*n-3)*(n-2)^2*a(n-3). - Vaclav Kotesovec, Jun 29 2013
a(n) ~ 2^(n+5/6)*exp(3/2*(2*n)^(1/3)-1-n)*n^(n+1/3)/sqrt(3). - Vaclav Kotesovec, Jun 29 2013

Terms a(17) onward added by G. C. Greubel, May 17 2018

A176120 Triangle read by rows: Sum_{j=0..k} binomial(n, j)binomial(k, j)j!.

Original entry on oeis.org

1, 1, 2, 1, 3, 7, 1, 4, 13, 34, 1, 5, 21, 73, 209, 1, 6, 31, 136, 501, 1546, 1, 7, 43, 229, 1045, 4051, 13327, 1, 8, 57, 358, 1961, 9276, 37633, 130922, 1, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 1, 10, 91, 748, 5509, 36046, 207775, 1047376, 4596553, 17572114

Offset: 0

Roger L. Bagula, Apr 09 2010

The number of ways of placing any number k = 0, 1, ..., min(n,m) of non-attacking rooks on an n X m chessboard. - R. J. Mathar, Dec 19 2014

			Triangle begins
  1;
  1,  2;
  1,  3,   7;
  1,  4,  13,   34;
  1,  5,  21,   73,  209;
  1,  6,  31,  136,  501,  1546;
  1,  7,  43,  229, 1045,  4051,  13327;
  1,  8,  57,  358, 1961,  9276,  37633,  130922;
  1,  9,  73,  529, 3393, 19081,  93289,  394353,  1441729;
  1, 10,  91,  748, 5509, 36046, 207775, 1047376,  4596553, 17572114;
  1, 11, 111, 1021, 8501, 63591, 424051, 2501801, 12975561, 58941091, 234662231;

O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 46.

Seiichi Manyama, Rows n = 0..139, flattened
Wikipedia, Rook polynomial

Cf. A086885 (table without column 0), A129833 (row sums).

Cf. A000262, A002061, A002720, A082545, A135859, A343832.

A176120:=func< n,k| (&+[Factorial(j)*Binomial(n,j)*Binomial(k,j): j in [0..k]]) >;
[A176120(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 11 2022

A176120 := proc(i,j)
        add(binomial(i,k)*binomial(j,k)*k!,k=0..j) ;
end proc: # R. J. Mathar, Jul 28 2016

T[n_, m_]:= T[n,m]= Sum[Binomial[n, k]*Binomial[m, k]*k!, {k, 0, m}];
Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten

def A176120(n,k): return sum(factorial(j)*binomial(n,j)*binomial(k,j) for j in (0..k))
flatten([[A176120(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 11 2022

Sum_{k=0..n} T(n, k) = A129833(n).
T(n,m) = A088699(n, m). - Peter Bala, Aug 26 2013
T(n,m) = A086885(n, m). - R. J. Mathar, Dec 19 2014
From G. C. Greubel, Aug 11 2022: (Start)
T(n, k) = Hypergeometric2F1([-n, -k], [], 1).
T(2*n, n) = A082545(n).
T(2*n+1, n) = A343832(n).
T(n, n) = A002720(n).
T(n, n-1) = A000262(n), n >= 1.
T(n, 1) = A000027(n+1).
T(n, 2) = A002061(n+1).
T(n, 3) = A135859(n+1). (End)

A182924 Generalized vertical Bell numbers of order 4.

Original entry on oeis.org

1, 52, 43833, 149670844, 1346634725665, 25571928251231076, 893591647147188285577, 52327970757667659912764908, 4796836032234830356783078467969, 653510798275634770675047022800897940, 127014654376520087360456517007106313763801

Offset: 0

Peter Luschny, Mar 28 2011

nonn

The name "generalized 'vertical' Bell numbers" is used to distinguish them from the generalized (horizontal) Bell numbers with reference to the square array representation of the generalized Bell numbers as given in A090210. a(n) is column 5 in this representation. The order is the parameter M in Penson et al., p. 6, eq. 29.

Apparently a(n) = A157280(n+1) for 0 <= n <= 8. - Georg Fischer, Oct 24 2018 (and true considering the hypergeometric comment in A157280, R. J. Mathar, Apr 23 2024).

G. C. Greubel, Table of n, a(n) for n = 0..129
P. Blasiak and P. Flajolet, Combinatorial models of creation-annihilation, arXiv:1010.0354 [math.CO], 2010-2011.
K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. 50, 083512 (2009).

Cf. A090210, A002720, A069948,A157280, A182925.

A182924 := proc(n) exp(-x)*GAMMA(n+1)^4*hypergeom([n+1,n+1,n+1,n+1],[1,1,1,1],x): simplify(subs(x=1, %)) end;
seq(A182924(i),i=0..10);

fallfac[n_, k_] := Pochhammer[n-k+1, k]; f[m_][n_, k_] := (-1)^k/k!* Sum[(-1)^p*Binomial[k, p]*fallfac[p, m]^n, {p, m, k}]; a[n_] := Sum[f[n][5, k], {k, n, 5*n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Sep 05 2012 *)

a(n) = exp(-1)*Gamma(n+1)^4*[4F4]([n+1,n+1,n+1,n+1], [1,1,1,1] | 1); here [4F4] is the generalized hypergeometric function of type 4F4.

Let B_{n}(x) = sum_{j>=0}(exp(j!/(j-n)!*x-1)/j!) then a(n) = 5! [x^5] taylor(B_{n}(x)), where [x^5] denotes the coefficient of x^5 in the Taylor series for B_{n}(x).

A182933 Generalized Bell numbers based on the rising factorial powers; square array read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 5, 27, 13, 1, 1, 15, 409, 778, 73, 1, 1, 52, 9089, 104149, 37553, 501, 1, 1, 203, 272947, 25053583, 57184313, 2688546, 4051, 1, 1, 877, 10515147, 9566642254, 192052025697, 56410245661, 265141267, 37633, 1

Offset: 0

Peter Luschny, Mar 29 2011

These numbers are related to the generalized Bell numbers based on the falling factorial powers (A090210).
The square array starts for n>=0, k>=0:
n\k=0,1,..  A000012,A000262,A182934,...
A000012: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
A000110: 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
A094577: 1, 3, 27, 409, 9089, 272947, 10515147, ...
A182932: 1, 13, 778, 104149, 25053583, 9566642254, ...
       : 1, 73, 37553, 57184313, 192052025697, ...
       : 1, 501, 2688546, 56410245661, ...
 ....  : 1, 4051, 265141267, 89501806774945, ...

Cf. A000110, A020556, A069223, A071379, A090209, A002720, A069948, A182925, A182924, A182933.

A182933_AsSquareArray := proc(n,k) local r,s,i;
r := [seq(n+1,i=1..k)]; s := [seq(1,i=1..k-1),2];
exp(-x)*n!^k*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) end:
seq(lprint(seq(A182933_AsSquareArray(n,k),k=0..6)),n=0..6);

a[n_, k_] := Exp[-1]*n!^k*HypergeometricPFQ[ Table[n+1, {k}], Append[ Table[1, {k-1}], 2], 1.]; Table[ a[n-k, k] // Round , {n, 0, 8}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

Let r_k = [n+1,...,n+1] (k occurrences of n+1), s_k = [1,...,1,2] (k-1 occurrences of 1) and F_k the generalized hypergeometric function of type k_F_k, then a(n,k) = exp(-1)*n!^k*F_k(r_k, s_k | 1).

Let B_{n}(x) = sum_{j>=0}(exp((j+n-1)!/(j-1)!*x-1)/j!) then a(n,k) = k! [x^k] series(B_{n}(x)), where [x^k] denotes the coefficient of x^k in the Taylor series for B_{n}(x).

A216831 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k!.

Original entry on oeis.org

1, 2, 11, 88, 905, 11246, 162607, 2668436, 48830273, 983353690, 21570885011, 511212091952, 13001401709881, 352856328962918, 10170853073795975, 310093415465876716, 9964607161173899777, 336439048405066012466, 11902368222382731461083, 440122520333417057761160

Offset: 0

Vaclav Kotesovec, Sep 17 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000522, A002720, A000172, A119400, A206178, A021009.

Table[Sum[Binomial[n, k]^3*k!, {k, 0, n}], {n, 0, 25}]
Table[HypergeometricPFQ[{-n, -n, -n}, {1}, -1], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 28 2016 *)

a(n) = sum(k=0, n, binomial(n,k)^3 * k!); \\ Michel Marcus, May 04 2021

Recurrence: (8*n^2+31*n+21)*a(n+3) - (24*n^3+157*n^2+308*n+162)*a(n+2) + (24*n^4+117*n^3+178*n^2+71*n-18)*a(n+1) - (8*n^2+31*n+30)*(n+1)^3*a(n) = 0.
a(n) ~ n^(n-1/6)/(sqrt(6*Pi)*exp(n+n^(1/3)-3*n^(2/3)-1/3)). - Vaclav Kotesovec, Sep 30 2012
a(n) = hypergeom([-n, -n, -n], [1], -1). - Vladimir Reshetnikov, Sep 28 2016
a(n) = Sum_{k=0..n} binomial(n, k)*|A021009(n, k)|. - Peter Luschny, May 04 2021
Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} x^n / n!^3. - Ilya Gutkovskiy, Jun 19 2022

cp's OEIS Frontend

A261767 Triangle read by rows: T(n,k) is the number of subpermutations of an n-set, whose orbits are each of size at most k with at least one orbit of size exactly k.

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A289147 Number of (n+1) X (n+1) binary matrices M with at most one 1 in each of the first n rows and each of the first n columns and M[n+1,n+1] = 0.

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A326252 Number of digraphs with vertices {1..n} whose increasing edges are crossing.

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A331432 Triangle T(n,k) (n >= k >= 0) read by rows: T(n,0) = (1+(-1)^n)/2; for k>=1, set T(0,k) = 0, S(n,k) = binomial(n,k)*binomial(n+k+1,k), and for n>=1, T(n,k) = S(n,k)-T(n-1,k).

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A119400 a(n) = Sum_{k=0..n} (n!/k!)^2*binomial(n,k).

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A121629 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^2*(a+*a), where a+ and a are boson creation and annihilation operators, respectively.

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A176120 Triangle read by rows: Sum_{j=0..k} binomial(n, j)*binomial(k, j)*j!.

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A121629 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^2(a+a), where a+ and a are boson creation and annihilation operators, respectively.

A176120 Triangle read by rows: Sum_{j=0..k} binomial(n, j)binomial(k, j)j!.