A002720 -id:A002720 - cp's OEIS frontend

A036969 Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.

1, 1, 1, 1, 5, 1, 1, 21, 14, 1, 1, 85, 147, 30, 1, 1, 341, 1408, 627, 55, 1, 1, 1365, 13013, 11440, 2002, 91, 1, 1, 5461, 118482, 196053, 61490, 5278, 140, 1, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1, 1, 87381, 9668036, 53157079, 46587905

Offset: 1

N. J. A. Sloane

Or, triangle of central factorial numbers T(2n,2k) (in Riordan's notation).
Can be used to calculate the Bernoulli numbers via the formula B_2n = (1/2)*Sum_{k = 1..n} (-1)^(k+1)*(k-1)!*k!*T(n,k)/(2*k+1). E.g., n = 1: B_2 = (1/2)*1/3 = 1/6. n = 2: B_4 = (1/2)*(1/3 - 2/5) = -1/30. n = 3: B_6 = (1/2)*(1/3 - 2*5/5 + 2*6/7) = 1/42. - Philippe Deléham, Nov 13 2003
From Peter Bala, Sep 27 2012: (Start)
Generalized Stirling numbers of the second kind. T(n,k) is equal to the number of partitions of the set {1,1',2,2',...,n,n'} into k disjoint nonempty subsets V1,...,Vk such that, for each 1 <= j <= k, if i is the least integer such that either i or i' belongs to Vj then {i,i'} is a subset of Vj. An example is given below.
Thus T(n,k) may be thought of as a two-colored Stirling number of the second kind. See Matsumoto and Novak, who also give another combinatorial interpretation of these numbers. (End)

			Triangle begins:
  1;
  1,    1;
  1,    5,      1;
  1,   21,     14,      1;
  1,   85,    147,     30,     1;
  1,  341,   1408,    627,    55,    1;
  1, 1365,  13013,  11440,  2002,   91,   1;
  1, 5461, 118482, 196053, 61490, 5278, 140, 1;
  ...
T(3,2) = 5: The five set partitions into two sets are {1,1',2,2'}{3,3'}, {1,1',3,3'}{2,2'}, {1,1'}{2,2',3,3'}, {1,1',3}{2,2',3'} and {1,1',3'}{2,2',3}.

L. Carlitz, A conjecture concerning Genocchi numbers. Norske Vid. Selsk. Skr. (Trondheim) 1971, no. 9, 4 pp. [The triangle appears on page 2.]
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.

Vincenzo Librandi, Rows n = 1..100 of triangle, flattened
Thomas Browning, Counting Parabolic Double Cosets in Symmetric Groups, arXiv:2010.13256 [math.CO], 2020.
P. L. Butzer, M. Schmidt, E. L. Stark and L. Vogt. Central factorial numbers; their main properties and some applications, Num. Funct. Anal. Optim., 10 (1989) 419-488.
José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024. See p. 2.
M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)
Qi Fang, Ya-Nan Feng, and Shi-Mei Ma, Alternating runs of permutations and the central factorial numbers, arXiv:2202.13978 [math.CO], 2022.
F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265. - From _N. J. A. Sloane_, Jan 02 2013
Petro Kolosov, Polynomial identities involving central factorial numbers, GitHub, 2024. See p. 6.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
S. Matsumoto and J. Novak, Jucys-Murphy Elements and Unitary Matrix Integrals arXiv.0905.1992 [math.CO], 2009-2012.
B. K. Miceli, Two q-Analogues of Poly-Stirling Numbers, J. Integer Seq., 14 (2011), 11.9.6.
John Riordan, Letter, Apr 28 1976.
John Riordan, Letter, Jul 06 1978
Richard P. Stanley, Hook Lengths and Contents.

Columns are A002450, A002451.
Diagonals are A000330 and A060493.
Transpose of A008957.
(0,0)-based version: A269945.
Cf. A008955, A008956, A156289, A135920 (row sums), A204579 (inverse), A000290.

a036969 n k = a036969_tabl !! (n-1) (k-1)
a036969_row n = a036969_tabl !! (n-1)
a036969_tabl = iterate f [1] where
   f row = zipWith (+)
     ([0] ++ row) (zipWith (*) (tail a000290_list) (row ++ [0]))
-- Reinhard Zumkeller, Feb 18 2013

A036969 := proc(n,k) local j; 2*add(j^(2*n)*(-1)^(k-j)/((k-j)!*(k+j)!),j=1..k); end;

t[n_, k_] := 2*Sum[j^(2*n)*(-1)^(k-j)/((k-j)!*(k+j)!), {j, 1, k}]; Flatten[ Table[t[n, k], {n, 1, 10}, {k, 1, n}]] (* Jean-François Alcover, Oct 11 2011 *)
t1[n_, k_] := (1/(2 k)!) * Sum[Binomial[2 k, j]*(-1)^j*(k - j)^(2 n), {j, 0, 2 k}]; Column[Table[t1[n, k], {n, 1, 10}, {k, 1, n}]] (* Kolosov Petro ,Jul 26 2023 *)

PARI
```
T(n,k)=if(1M. F. Hasler, Feb 03 2012
```

T(n,k)=2*sum(j=1,k,(-1)^(k-j)*j^(2*n)/(k-j)!/(k+j)!)  \\ M. F. Hasler, Feb 03 2012

def A036969(n,k) : return (2/factorial(2*k))*add((-1)^j*binomial(2*k,j)*(k-j)^(2*n) for j in (0..k))
for n in (1..7) : print([A036969(n,k) for k in (1..n)]) # Peter Luschny, Feb 03 2012

T(n,k) = A156289(n,k)/A001147(k). - Peter Bala, Feb 21 2011
From Peter Bala, Oct 14 2011: (Start)
O.g.f.: Sum_{n >= 1} x^n*t^n/Product_{k = 1..n} (1 - k^2*t^2) = x*t + (x + x^2)*t^2 + (x + 5*x^2 + x^3)*t^3 + ....
Define polynomials x^[2*n] = Product_{k = 0..n-1} (x^2 - k^2). This triangle gives the coefficients in the expansion of the monomials x^(2*n) as a linear combination of x^[2*m], 1 <= m <= n. For example, row 4 gives x^8 = x^[2] + 21*x^[4] + 14*x^[6] + x^[8].
A008955 is a signed version of the inverse.
The n-th row sum = A135920(n). (End)
T(n,k) = (2/(2*k)!)*Sum_{j=0..k-1} (-1)^(j+k+1) * binomial(2*k,j+k+1) * (j+1)^(2*n). This formula is valid for n >= 0 and 0 <= k <= n. - Peter Luschny, Feb 03 2012
From Peter Bala, Sep 27 2012: (Start)
Let E(x) = cosh(sqrt(2*x)) = Sum_{n >= 0} x^n/((2*n)!/2^n). A generating function for the triangle is E(t*(E(x)-1)) = 1 + t*x + t*(1 + t)*x^2/6 + t*(1 + 5*t + t^2)*x^3/90 + ..., where the sequence of denominators [1, 1, 6, 90, ...] is given by (2*n)!/2^n. Cf. A008277 which has generating function exp(t*(exp(x)-1)). An e.g.f. is E(t*(E(x^2/2)-1)) = 1 + t*x^2/2! + t*(1 + t)*x^4/4! + t*(1 + 5*t + t^2)*x^6/6! + ....
Put c(n) := (2*n)!/2^n. The column k generating function is (1/c(k))*(E(x)-1)^k = Sum_{n >= k} T(n,k)*x^n/c(n). The inverse array is A204579.
The production array begins:
  1,  1;
  0,  4,  1;
  0,  0,  9,  1;
  0,  0,  0, 16,  1;
  ... (End)
x^n = Sum_{k=1..n} T(n,k)*Product_{i=0..k-1} (x-i^2), see Stanley link. - Michel Marcus, Nov 19 2014; corrected by Kolosov Petro, Jul 26 2023
From Kolosov Petro, Jul 26 2023: (Start)
T(n,k) = (1/(2*k)!) * Sum_{j=0..2k} binomial(2k, j)*(-1)^j*(k - j)^(2n).
T(n,k) = (1/(k*(2k-1)!)) * Sum_{j=0..k} (-1)^(k-j)*binomial(2k, k-j)*j^(2n). (End)

More terms from Vladeta Jovovic, Apr 16 2000

A289192 A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 7, 6, 1, 4, 14, 34, 24, 1, 5, 23, 86, 209, 120, 1, 6, 34, 168, 648, 1546, 720, 1, 7, 47, 286, 1473, 5752, 13327, 5040, 1, 8, 62, 446, 2840, 14988, 58576, 130922, 40320, 1, 9, 79, 654, 4929, 32344, 173007, 671568, 1441729, 362880

Offset: 0

Alois P. Heinz, Jun 28 2017

			Square array A(n,k) begins:
:   1,    1,    1,     1,     1,     1, ...
:   1,    2,    3,     4,     5,     6, ...
:   2,    7,   14,    23,    34,    47, ...
:   6,   34,   86,   168,   286,   446, ...
:  24,  209,  648,  1473,  2840,  4929, ...
: 120, 1546, 5752, 14988, 32344, 61870, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Columns k=0-10 give: A000142, A002720, A087912, A277382, A289147, A289211, A289212, A289213, A289214, A289215, A289216.
Rows n=0-2 give: A000012, A000027(k+1), A008865(k+2).
Main diagonal gives A277373.
Cf. A253286, A341014.

A:= (n,k)-> n! * add(binomial(n, i)*k^i/i!, i=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := n! * LaguerreL[n, -k];
Table[A[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 05 2019 *)

{T(n, k) = if(n<2, k*n+1, (2*n+k-1)*T(n-1, k)-(n-1)^2*T(n-2, k))} \\ Seiichi Manyama, Feb 03 2021

T(n, k) = n!*pollaguerre(n, 0, -k); \\ Michel Marcus, Feb 05 2021

from sympy import binomial, factorial as f
def A(n, k): return f(n)*sum(binomial(n, i)*k**i/f(i) for i in range(n + 1))
for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Jun 28 2017

A(n,k) = n! * Sum_{i=0..n} k^i/i! * binomial(n,i).
E.g.f. of column k: exp(k*x/(1-x))/(1-x).
A(n, k) = (-1)^n*KummerU(-n, 1, -k). - Peter Luschny, Feb 12 2020
A(n, k) = (2*n+k-1)*A(n-1, k) - (n-1)^2*A(n-2, k) for n > 1. - Seiichi Manyama, Feb 03 2021

A032188 Number of labeled series-reduced mobiles (circular rooted trees) with n leaves (root has degree 0 or >= 2).

Original entry on oeis.org

1, 1, 5, 41, 469, 6889, 123605, 2620169, 64074901, 1775623081, 54989743445, 1882140936521, 70552399533589, 2874543652787689, 126484802362553045, 5977683917752887689, 301983995802099667861, 16239818347465293071401, 926248570498763547197525, 55847464116157184894240201

Offset: 1

Christian G. Bower

With offset 0, a(n) = number of partitions of the multiset {1,1,2,2,...,n,n} into lists of strictly decreasing lists, called blocks, such that the concatenation of all blocks in a list has the Stirling property: all entries between the two occurrences of i exceed i, 1<=i<=n. For example, with slashes separating blocks, a(2) = 5 counts 1/1/2/2; 1/2/2/1; 2/2/1/1; 1/2/2 1; 2/2 1/1, but not, for instance, 2 1/2/1 because it fails the Stirling test for i=2. - David Callan, Nov 21 2011

			D^3(1) = (24*x^2-64*x+41)/(2*x-1)^6. Evaluated at x = 0 this gives a(4) = 41.
a(3) = 5: Denote the colors of the vertices by the letters a,b,c, .... The 5 possible increasing plane trees on 3 vertices with vertices of outdegree k coming in 2^(k-1) colors are
.
   1a       1a        1b        1a        1b
   |       /  \      /  \      /  \      /  \
   2a     2    3    2    3    3    2    3    2
   |
   3

Andrew Howroyd, Table of n, a(n) for n = 1..200
François Bergeron, Philippe Flajolet, and Bruno Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See pp. 6, 16, 30.
Diego Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions arXiv:math/0501052 [math.CA], 2005.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 89
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.
Index entries for sequences related to mobiles

Cf. A006351, A032034.

Order := 20; t1 := solve(series((ln(1-A)+2*A),A)=x,A); A000311 := n->n!*coeff(t1,x,n);
# With offset 0:
a := n -> add(combinat:-eulerian2(n,k)*2^k,k=0..n):
seq(a(n),n=0..19); # Peter Luschny, Jul 09 2015

For[y=x+O[x]^21; oldy=0, y=!=oldy, oldy=y; y=((1-y)Log[1-y]+x*y+y-x)/(2y-1), Null]; Table[n!Coefficient[y, x, n], {n, 1, 20}]
Rest[CoefficientList[InverseSeries[Series[2*x + Log[1-x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

a(n):=sum((n+k-1)!*sum(1/(k-j)!*sum((2^l*(-1)^(n+l+1)*stirling1(n-l+j-1,j-l))/(l!*(n-l+j-1)!),l,0,j),j,0,k),k,0,n-1); /* Vladimir Kruchinin, Feb 06 2012 */

N = 66; x = 'x + O('x^N);
Q(k) = if(k>N, 1,  1 + (k+1)*x - 2*x*(k+1)/Q(k+1) );
gf = 1/Q(0); Vec(gf) \\ Joerg Arndt, May 01 2013

{my(n=20); Vec(serlaplace(serreverse(2*x+log(1-x + O(x*x^n)))))} \\ Andrew Howroyd, Jan 16 2018

Doubles (index 2+) under "CIJ" (necklace, indistinct, labeled) transform.
E.g.f. A(x) satisfies log(1-A(x))+2*A(x)-x = 0. - Vladeta Jovovic, Dec 06 2002
With offset 0, second Eulerian transform of the powers of 2 [A000079]. See A001147 for definition of SET. - Ross La Haye, Feb 14 2005
From Peter Bala, Sep 05 2011: (Start)
The generating function A(x) satisfies the autonomous differential equation A'(x) = (1-A)/(1-2*A) with A(0) = 0. Hence the inverse function A^-1(x) = int {t = 0..x} (1-2*t)/(1-t) = 2*x+log(1-x). The expansion of A(x) can be found by inverting the above integral using the method of [Dominici, Theorem 4.1] to arrive at the result a(n) = D^(n-1)(1) evaluated at x = 0, where D denotes the operator g(x) -> d/dx((1-x)/(1-2*x)*g(x)). Compare with A006351.
Applying [Bergeron et al., Theorem 1] to the result x = int {t = 0..A(x)} 1/phi(t), where phi(t) = (1-t)/(1-2*t) = 1+t+2*t^2+4*t^3+8*t^4+... leads to the following combinatorial interpretation for this sequence: a(n) gives the number of plane increasing trees on n vertices where each vertex of outdegree k >= 1 can be colored in 2^(k-1) ways. An example is given below. (End)
The integral from 0 to infinity w.r.t. w of exp(-2w)(1-z*w)^(-1/z) gives an o.g.f. for the series with offset 0. Consequently, a(n)= sum(j=1 to infinity): St1d(n,j)/(2^(n+j-1)) where St1d(n,j) is the j-th element of the n-th diagonal of A132393 with offset=1; e.g., a(3)= 5 = 0/2^3 + 2/2^4 + 11/2^5 + 35/2^6 + 85/2^7 + ... . - Tom Copeland, Sep 15 2011
A signed o.g.f., with Γ(v,x) the incomplete gamma function (see A111999 with u=1), is g(z) = (2/z)^(-(1/z)-1) exp(2/z) Γ((1/z)+1,2/z)/z. - Tom Copeland, Sep 16 2011
With offset 0, a(n) = Sum[T(n+k,k), k=1..n] where T(n,k) are the associated Stirling numbers of the first kind (A008306). For example, a(3) = 41 = 6 + 20 + 15. - David Callan, Nov 21 2011
a(n) = sum(k=0..n-1, (n+k-1)!*sum(j=0..k, 1/(k-j)!*sum(l=0..j, (2^l*(-1)^(n+l+1)*stirling1(n-l+j-1,j-l))/(l!*(n-l+j-1)!)))), n>0. - Vladimir Kruchinin, Feb 06 2012
G.f.: 1/Q(0), where Q(k)= 1 + (k+1)*x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
a(n) ~ n^(n-1) / (2 * exp(n) * (1-log(2))^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014
a(n) = A032034(n)/2. - Alois P. Heinz, Jul 04 2018
E.g.f: series reversion of 2*x + log(1-x). - Andrew Howroyd, Sep 19 2018

A088699 Array read by antidiagonals of coefficients of generating function exp(x)/(1-y-xy).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 34, 21, 6, 1, 1, 7, 31, 73, 73, 31, 7, 1, 1, 8, 43, 136, 209, 136, 43, 8, 1, 1, 9, 57, 229, 501, 501, 229, 57, 9, 1, 1, 10, 73, 358, 1045, 1546, 1045, 358, 73, 10, 1, 1, 11, 91, 529, 1961, 4051, 4051, 1961

Offset: 0

Michael Somos, Oct 08 2003

A(n,m) is the number of ways to pair the elements of two sets (with respectively n and m elements), where each element of either set may be paired with zero or one elements of the other set; number of n X m matrices of zeros and ones with at most one one in each row and column. E.g., A(2,2)=7 because we can pair {A,B} with {C,D} as {AB,CD}, {AC,BD}, {AC,B,D}, {AD,B,C}, {BC,A,D}, {BD,A,C}, or {A,B,C,D}. - Franklin T. Adams-Watters, Feb 06 2006
Compare with A086885. - Peter Bala, Sep 17 2008
A(n,m) is the number of vertex covers and independent vertex sets in the n X m lattice (rook) graph K_n X K_m. - Andrew Howroyd, May 14 2017

			      1       1       1       1       1       1       1       1       1
      1       2       3       4       5       6       7       8       9
      1       3       7      13      21      31      43      57      73
      1       4      13      34      73     136     229     358     529
      1       5      21      73     209     501    1045    1961    3393
      1       6      31     136     501    1546    4051    9276   19081
      1       7      43     229    1045    4051   13327   37633   93289
      1       8      57     358    1961    9276   37633  130922  394353
      1       9      73     529    3393   19081   93289  394353 1441729

Andrew Howroyd, Table of n, a(n) for n = 0..1274
R. J. Mathar, The number of binary nXm matrices with at most k 1's in each row or column, (2014) Table 1.
Eric Weisstein's World of Mathematics, Rook Graph
Eric Weisstein's World of Mathematics, Vertex Cover
Wikipedia, Rook polynomial

Row sums give A081124.
Main diagonal is A002720.
Cf. A099597, A176120.

A088699 := proc(i,j)
    add(binomial(i,k)*binomial(j,k)*k!,k=0..min(i,j)) ;
end proc: # R. J. Mathar, Feb 28 2015

max = 11; se = Series[E^x/(1 - y - x*y), {x, 0, max}, {y, 0, max}] // Normal // Expand; a[i_, j_] := SeriesCoefficient[se, {x, 0, i}, {y, 0, j}]*i!; Flatten[ Table[ a[i - j, j], {i, 0, max}, {j, 0, i}]] (* Jean-François Alcover, May 15 2012 *)

A(i,j)=if(i<0 || j<0,0,i!*polcoeff(exp(x+x*O(x^i))*(1+x)^j,i))

A(i,j)=if(i<0 || j<0,0,i!*polcoeff(exp(x/(1-x)+x*O(x^i))*(1-x)^(i-j-1),i))

A(i,j)=local(M); if(i<0 || j<0,0,M=matrix(j+1,j+1,n,m,if(n==m,1,if(n==m+1,m))); (M^i)[j+1,]*vectorv(j+1,n,1)) /* Michael Somos, Jul 03 2004 */

E.g.f.: exp(x)/(1-y-xy)=Sum_{i, j} A(i, j) y^j x^i/i!.
A(i, j) = A(i-1, j)+j*A(i-1, j-1)+(i==0) = A(j, i).
T(n, k) = sum{j=0..k, C(n, k-j)*k!/j!} = sum{j=0..k, (k-j)!*C(k, j)C(n, k-j)}. - Paul Barry, Nov 14 2005
A(i,j) = sum_k C(i,k)*C(j,k)*k!. E.g.f.: sum_{i,j} a(i,j)*x^i/i!*y^j/j! = e^{x+y+xy}. - Franklin T. Adams-Watters, Feb 06 2006
The LDU factorization of this array, formatted as a square array, is P * D * transpose(P), where P is Pascal's triangle A007318 and D = diag(0!, 1!, 2!, ... ). Compare with A099597. - Peter Bala, Nov 06 2007
A(i,j) = (-1)^-i HypergeometricU(-i, 1 - i + j, -1). - Eric W. Weisstein, May 10 2017

A277423 a(n) = n!*LaguerreL(n, n).

Original entry on oeis.org

1, 0, -2, 6, 24, -380, 720, 31794, -361088, -2104056, 101548800, -612792290, -25534891008, 593660731404, 2831189530624, -361541172525750, 4481749181890560, 169464194149739536, -6805365045197340672, -9663483091971306186, 6883830206467440640000

Offset: 0

Vaclav Kotesovec, Oct 14 2016

sign

G. C. Greubel, Table of n, a(n) for n = 0..415
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials

Cf. A277373, A277391, A277392, A277418, A277419, A277420, A277421, A277422.

Cf. A002720, A087912, A277382.

[Factorial(n)*(&+[Binomial(n,k)*(-1)^k*n^k/Factorial(k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, May 16 2018

Table[n!*LaguerreL[n, n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * (-1)^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]
Table[n! * Hypergeometric1F1[-n, 1, n], {n, 0, 20}] (* Vaclav Kotesovec, Feb 20 2020 *)

a(n) = n!*sum(k=0,n, binomial(n,k)*(-1)^k*n^k/k!); \\ G. C. Greubel, May 16 2018

a(n) = n! * Sum_{k=0..n} binomial(n, k) * (-1)^k * n^k / k!.
a(n) = n! * [x^n] exp(-n*x/(1 - x))/(1 - x). - Ilya Gutkovskiy, Nov 21 2017
a(n) = Sum_{k=0..n} (-n)^(n-k)*k!*binomial(n,k)^2. - Ridouane Oudra, Jul 08 2025

A123202 Triangle of coefficients of n!(1 - x)^nL_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.

Original entry on oeis.org

1, 1, -2, 2, -8, 7, 6, -36, 63, -34, 24, -192, 504, -544, 209, 120, -1200, 4200, -6800, 5225, -1546, 720, -8640, 37800, -81600, 94050, -55656, 13327, 5040, -70560, 370440, -999600, 1536150, -1363572, 653023, -130922, 40320, -645120, 3951360, -12794880

Offset: 0

Roger L. Bagula, Oct 04 2006

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A021009(n,j)*x^j*(1 - x)^(n - j).

			Triangle begins:
       1;
       1,    -2;
       2,    -8,     7;
       6,   -36,    63,    -34;
      24,  -192,   504,   -544,   209;
     120, -1200,  4200,  -6800,  5225,  -1546;
     720, -8640, 37800, -81600, 94050, -55656, 13327;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 13 2018

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, 1972, p. 782.
Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, p. 164, figure 26.1.

G. C. Greubel, Table of n, a(n) for n = 0..5150 (Rows n=0..100 of triangle, flattened; offset corrected by _Georg Fischer_, Jan 31 2019)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Laguerre Polynomial

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221.

M := (n,x) -> n!*subs(x=(x/(1-x)),orthopoly[L](n,x))*(1-x)^n:
seq(print(seq(coeff(simplify(M(n,x)),x,k),k=0..n)),n=0..6); # Peter Luschny, Jan 05 2015

w = Table[n!*CoefficientList[LaguerreL[n, x], x], {n, 0, 10}];
v = Table[CoefficientList[Sum[w[[n + 1]][[m + 1]]*x^ m*(1 - x)^(n - m), {m, 0, n}], x], {n, 0, 10}]; Flatten[v]

create_list(ratcoef(n!*(1 - x)^n*laguerre(n, x/(1 - x)), x, k), n, 0, 10, k, 0, n); /* Franck Maminirina Ramaharo, Oct 13 2018 */

row(n) = Vecrev(n!*(1-x)^n*pollaguerre(n, 0, x/(1 - x))); \\ Michel Marcus, Feb 06 2021

T(n, k) = [x^k] (n!*L_n(x)*(1 - x)^n) with L_n(x) the Laguerre polynomial after substituting x by x/(1 - x). - Peter Luschny, Jan 05 2015
From Franck Maminirina Ramaharo, Oct 13 2018: (Start)
G.f.: exp(-x*y/(1 - (1 - x)*y))/(1 - (1 - x)*y).
T(n,1) = A000142(n).
T(n,2) = -A052582(n).
T(n,n) = A002720(n). (End)

Edited by N. J. A. Sloane, Jun 12 2007

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 13 2018

A130191 Square of the Stirling2 matrix A048993.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 5, 6, 1, 0, 15, 32, 12, 1, 0, 52, 175, 110, 20, 1, 0, 203, 1012, 945, 280, 30, 1, 0, 877, 6230, 8092, 3465, 595, 42, 1, 0, 4140, 40819, 70756, 40992, 10010, 1120, 56, 1, 0, 21147, 283944, 638423, 479976, 156072, 24570, 1932, 72, 1

Offset: 0

Wolfdieter Lang, Jun 01 2007

Without row n=0 and column k=0 this is triangle A039810.
This is an associated Sheffer matrix with e.g.f. of the m-th column ((exp(f(x))-1)^m)/m! with f(x)=:exp(x)-1.
The triangle is also called the exponential Riordan array [1, exp(exp(x)-1)]. - Peter Luschny, Apr 19 2015
Also the Bell transform of shifted Bell numbers A000110(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle starts:
  1;
  0,   1;
  0,   2,    1;
  0,   5,    6,    1;
  0,  15,   32,   12,    1;
  0,  52,  175,  110,   20,   1;
  0, 203, 1012,  945,  280,  30,  1;
  0, 877, 6230, 8092, 3465, 595, 42, 1;

G. C. Greubel, Rows n=0..100 of triangle, flattened
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Wolfdieter Lang, First 10 rows and more
John Riordan, Letter, Apr 28 1976. (See third page)

Columns k=0..3 give A000007, A000110 (for n > 0), A000558, A000559.
Row sums: A000258.
Alternating row sums: A130410.
T(2n,n) gives A321712.
Cf. A039810 (another version), A048993.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> combinat:-bell(n+1), 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[BellB[# + 1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
a[n_, m_]:= Sum[StirlingS2[n, k]*StirlingS2[k, m], {k,m,n}]; Table[a[n, m], {n, 0, 100}, {m, 0, n}]//Flatten (* G. C. Greubel, Jul 10 2018 *)

for(n=0, 9, for(k=0, n, print1(sum(j=k, n, stirling(n, j, 2)*stirling(j, k, 2)), ", "))) \\ G. C. Greubel, Jul 10 2018

# uses[riordan_array from A256893]
riordan_array(1, exp(exp(x) - 1), 8, exp=true) # Peter Luschny, Apr 19 2015

a(n,k) = Sum_{j=k..n} S2(n,j) * S2(j,k), n>=k>=0.
E.g.f. row polynomials with argument x: exp(x*f(f(z))).
E.g.f. column k: ((exp(exp(x) - 1) - 1)^k)/k!.

A002740 Number of tree-rooted bridgeless planar maps with two vertices and n faces.

Original entry on oeis.org

0, 0, 0, 2, 15, 84, 420, 1980, 9009, 40040, 175032, 755820, 3233230, 13728792, 57946200, 243374040, 1017958725, 4242920400, 17631691440, 73078721100, 302202005490, 1247182879800, 5137916074200, 21132472200840, 86794082253450, 356013544661424, 1458583920435600, 5969389748449400

Offset: 0

N. J. A. Sloane

a(n) is the sum of the major indices of all Dyck words of length 2n-2. The major index of a Dyck word is the sum of the positions of those 1's that are followed by a 0. Example: a(4)=15 because the Dyck words of length 6 are 010101, 010011, 001101, 001011 and 000111 having major indices 6,2,4,3 and 0, respectively. a(n) = Sum_{k=0..n(n-1)} k*A129175(n,k). - Emeric Deutsch, Apr 20 2007

J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933, p. 97.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Mireille Bousquet-Mélou, New enumerative results on two-dimensional directed animals, Discr. Math., Vol. 180, No. 1-3 (1998), pp. 73-106. See Cor. 6.6.
Luca Ferrari and Emanuele Munarini, Enumeration of saturated chains in Dyck lattices, arXiv preprint arXiv:1203.6807 [math.CO], 2012.
J. Fürlinger and J. Hofbauer, q-Catalan numbers, J. Comb. Theory, Series A, Vol. 40, No. 2 (1985), pp. 248-264.
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933. (Annotated scans of some selected pages)
Mark Shattuck, Parity theorems for statistics on permutations and Catalan words, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5 (2005), Paper A07.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus III: Nonseparable maps, J. Combinatorial Theory Ser. B, Vol. 18, No. 3 (1975), pp. 222-259. See Table VIIIb.

Cf. A129175.

A diagonal of A253180.

[(n-2)*Binomial(2*n-2,n-2)/2 : n in [0..30]]; // Wesley Ivan Hurt, Sep 24 2014

with(combinat):for n from 0 to 22 do printf(`%d, `,n*sum(binomial(2*n, n)/(n+1)/2, k=2..n)) od: # Zerinvary Lajos, Mar 13 2007
a:=n->sum(sum(binomial(2*n,n)/(n+1)/2, j=1..n),k=2..n): seq(a(n), n=0..25); # Zerinvary Lajos, May 09 2007
A002740:=n->(n-2)*binomial(2*n-2,n-2)/2+0^n: seq(A002740(n), n=0..30); # Wesley Ivan Hurt, Sep 24 2014

a[n_] := (n-1)(n-2)Binomial[2(n-1), n-1]/(2n); a[0] = 0; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Nov 16 2011 *)

combinat::catalan(n) *binomial(n,2) $ n = 0..22 // Zerinvary Lajos, Feb 15 2007

a(n)=if(n<3,0,(2*(n-1))!/(2*n!*(n-3)!)); /* Joerg Arndt, Sep 28 2012 */

G.f.: (1/2)*(1-(1 - 6*t + 6*t^2)/(1-4*t)^(3/2)).
a(n+3) = (2*(n+2))!/(2*n!*(n+3)!). - Wolfdieter Lang
a(n+2) = Sum_{k=0..n} k*binomial(k+n, k). - Benoit Cloitre, Oct 25 2003
a(n) = Sum_{k=2..n} Sum_{j=1..n} binomial(2*n,n)/(2*(n+1)), n >= 0. - Zerinvary Lajos, May 09 2007
a(n) = (n-2)*binomial(2n-2, n-2)/2 + 0^n. - Wesley Ivan Hurt, Sep 24 2014
E.g.f.: (1 + exp(2*x) * ((2*x - 1) * BesselI(0,2*x) - x * BesselI(1,2*x))) / 2. - Ilya Gutkovskiy, Nov 03 2021
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=3} 1/a(n) = 3 - 4*Pi/(3*sqrt(3)).
Sum_{n>=3} (-1)^(n+1)/a(n) = 16*log(phi)/sqrt(5) - 3, where phi is the golden ratio (A001622). (End)

Name clarified by Noam Zeilberger, Aug 18 2017

A046089 Triangle read by rows, the Bell transform of (n+2)!/2 without column 0.

Original entry on oeis.org

1, 3, 1, 12, 9, 1, 60, 75, 18, 1, 360, 660, 255, 30, 1, 2520, 6300, 3465, 645, 45, 1, 20160, 65520, 47880, 12495, 1365, 63, 1, 181440, 740880, 687960, 235305, 35700, 2562, 84, 1, 1814400, 9072000, 10372320, 4452840, 877905, 86940, 4410, 108, 1

Offset: 1

Wolfdieter Lang

Previous name was: A triangle of numbers related to triangle A030523.
a(n,1)= A001710(n+1). a(n,m)=: S1p(3; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n,m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n,m) (unsigned Lah numbers).
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A035342(n,m) := S2(3; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+2 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
a(4,2)=75=4*(3*4)+3*(3*3) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*3*4)=12 colored versions, e.g. ((1c1),(2c1,3c3,4c2)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 3 colors, c1, c2 and c3, can be chosen and the vertex labeled 4 with j=2 can come in 4 colors, e.g. c1, c2, c3 and c4. Therefore there are 4*(1)*(1*3*4)=48 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*3)*(1*3))=27 such forests, e.g. ((1c1,3c2)(2c1,4c1)) or ((1c1,3c2)(2c1,4c2)), etc. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of A001710(n+2) (adding 1,0,0,.. as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins:
  [1],
  [3, 1],
  [12, 9, 1],
  [60, 75, 18, 1],
  [360, 660, 255, 30, 1],
  [2520, 6300, 3465, 645, 45, 1],
  ...

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First ten rows.
E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's board, Discr. Maths. 239 (2001) 33-51.
John Riordan, Letter, Apr 28 1976.

Cf. A001710, A049376, A105752.

Alternating row sums A134138.

a[n_, m_] /; n >= m >= 1 := a[n, m] = (2m + n - 1)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* _Jean-François Alcover, Jul 22 2011 *)
a[n_, k_] := -(-1/2)^k*(n+1)!*HypergeometricPFQ[{1-k, n/2+1, (n+3)/2}, {3/2, 2}, 1]/(k-1)!; Table[a[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 28 2013, after Vladimir Kruchinin *)
a[0] = 0; a[n_] := (n + 1)!/2;
T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, a[0]^n], Sum[Binomial[n - 1, j - 1] a[j] T[n - j, k - 1], {j, 0, n - k + 1}]];
Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 19 2016, after Peter Luschny, updated Jan 01 2021 *)
rows = 9;
a[n_, m_] := BellY[n, m, Table[(k+2)!/2, {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

a(n,k):=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+2*j-1,2*j-1),j,1,k))/(2^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: factorial(n+2)//2, 9) # Peter Luschny, Jan 19 2016

a(n, m) = n!*A030523(n, m)/(m!*2^(n-m)); a(n, m) = (2*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

a(n, m) = sum(|S1(n, j)|* A075497(j, m), j=m..n) (matrix product), with S1(n, j) := A008275(n, j) (signed Stirling1 triangle). Priv. comm. to Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference.

a(n, k) = (n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+2*j-1,2*j-1)))/(2^k*k!) - Vladimir Kruchinin, Apr 01 2011

New name from Peter Luschny, Jan 19 2016

A197458 Number of n X n binary matrices with at most two 1's in each row and column, other entries 0.

Original entry on oeis.org

1, 2, 16, 265, 7343, 304186, 17525812, 1336221251, 129980132305, 15686404067098, 2297230134084416, 400977650310256537, 82188611938415464231, 19536244019455339261970, 5328019975275896220786388, 1651867356348327784988233291, 577522171260292028520919811777

Offset: 0

Felix A. Pahl, Oct 15 2011

nonn

Filling out an n x n square grid with 0's and 1's at math.stackexchange
R. J. Mathar, The number of binary n X m matrices with at most k 1's in each row or column, (2014) Table 2.

Cf. A001499, A002720. Column of A283500.

import java.math.BigInteger;
public class AtMostTwoOnes {
    public static void main (String [] args) {
        for (int n = 0;n <= 40;n++) {
            BigInteger [] [] [] counts = new BigInteger [n + 1] [n + 1] [n + 1]; // counts [m] [k] [l] : number of mxn matrices with k column sums 0, l column sums 1
            for (int k = 0;k <= n;k++)
                for (int l = 0;l <= n;l++)
                    counts [0] [k] [l] = BigInteger.ZERO;
            counts [0] [n] [0] = BigInteger.ONE; // only one 0xn matrix, with all n column sums 0
            for (int m = 1;m <= n;m++) {
                BigInteger [] [] old = counts [m - 1];
                for (int k = 0;k <= n;k++)
                    for (int l = 0;l <= n;l++) {
                        BigInteger sum = BigInteger.ZERO;
                        // new row contains no 1s
                        sum = sum.add (old [k] [l]);
                        // new row contains one 1
                        //   added to column sum 0
                        if (k < n && l > 0)
                            sum = sum.add (old [k + 1] [l - 1].multiply (BigInteger.valueOf (k + 1)));
                        //   added to column sum 1
                        if (l < n)
                            sum = sum.add (old [k] [l + 1].multiply (BigInteger.valueOf (l + 1)));
                        // new row contains two 1s
                        //   added to two column sums 0
                        if (k < n - 1 && l > 1)
                            sum = sum.add (old [k + 2] [l - 2].multiply (BigInteger.valueOf (((k + 2) * (k + 1)) / 2)));
                        //   added to one column sum 0, one column sum 1
                        if (k < n)
                            sum = sum.add (old [k + 1] [l].multiply (BigInteger.valueOf ((k + 1) * l)));
                        //   added to two column sums 1
                        if (l < n - 1)
                            sum = sum.add (old [k] [l + 2].multiply (BigInteger.valueOf (((l + 2) * (l + 1)) / 2)));
                        counts [m] [k] [l] = sum;
                    }
            }
            BigInteger sum = BigInteger.ZERO;
            for (int k = 0;k <= n;k++)
                for (int l = 0;l <= n;l++)
                    sum = sum.add (counts [n] [k] [l]);
            System.out.println (n + " : " + sum);
        }
    }
}

a(n) = Sum_{k=0}^n Sum_{l=0}^{n-k} a(k,l,n,n) where a(k,l,m,n) is the number of binary m X n matrices with at most two 1's per row, k columns with sum 0, l columns with sum 1 and the remaining n - k - l columns with sum 2.

a(k,l,m,n) = a(k,l,m-1,n) +(k+1)*a(k+1,l-1,m-1,n) +(l+1)*a(k,l+1,m-1,n) +(k+1)*2*a(k+2,l-2,m-1,n)/(k+2) +(k+1)*l*a(k+1,l,m-1,n) +(l+1)*2*a(k,l+2,m-1,n)/(l+2).

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A036969 Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.

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A289192 A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A032188 Number of labeled series-reduced mobiles (circular rooted trees) with n leaves (root has degree 0 or >= 2).

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A088699 Array read by antidiagonals of coefficients of generating function exp(x)/(1-y-xy).

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A277423 a(n) = n!*LaguerreL(n, n).

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A123202 Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.

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A123202 Triangle of coefficients of n!(1 - x)^nL_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.