A009963 -id:A009963 - cp's OEIS frontend

A060854 Array T(m,n) read by antidiagonals: T(m,n) (m >= 1, n >= 1) = number of ways to arrange the numbers 1,2,...,m*n in an m X n matrix so that each row and each column is increasing.

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 14, 42, 14, 1, 1, 42, 462, 462, 42, 1, 1, 132, 6006, 24024, 6006, 132, 1, 1, 429, 87516, 1662804, 1662804, 87516, 429, 1, 1, 1430, 1385670, 140229804, 701149020, 140229804, 1385670, 1430, 1, 1, 4862, 23371634, 13672405890, 396499770810, 396499770810, 13672405890, 23371634, 4862, 1

Offset: 1

R. H. Hardin, May 03 2001

Multidimensional Catalan numbers; a special case of the "hook-number formula".
Number of paths from (0,0,...,0) to (n,n,...,n) in m dimensions, all coordinates increasing: if (x_1,x_2,...,x_m) is on the path, then x_1 <= x_2 <= ... <= x_m. Number of ways to label an n by m array with all the values 1..n*m such that each row and column is strictly increasing. Number of rectangular Young Tableaux. Number of linear extensions of the n X m lattice (the divisor lattice of a number having exactly two prime divisors). - Mitch Harris, Dec 27 2005
Given m*n lines in a {(m + 1)(n - 1)}-dimensional space, T(m, n) is the number of {n*(m-1)-1}-dimensional spaces cutting these lines in points (see Fontanari and Castelnuovo). - Stefano Spezia, Jun 19 2022

			Array begins:
  1,   1,     1,         1,            1,                1, ...
  1,   2,     5,        14,           42,              132, ...
  1,   5,    42,       462,         6006,            87516, ...
  1,  14,   462,     24024,      1662804,        140229804, ...
  1,  42,  6006,   1662804,    701149020,     396499770810, ...
  1, 132, 87516, 140229804, 396499770810, 1671643033734960, ...

Guido Castelnuovo, Numero degli spazi che segano più rette in uno spazio ad n dimensioni, Rendiconti della R. Accademia dei Lincei, s. IV, vol. V, 4 agosto 1889. In Guido Castelnuovo, Memorie scelte, Zanichelli, Bologna 1937, pp. 55-64 (in Italian).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 7.23.19(b).

Alois P. Heinz, Antidiagonals n = 1..36
Albrecht Böttcher, Wiener-Hopf Determinants with Rational Symbols, Math. Nachr. 144 (1989), 39-64.
Freddy Cachazo and Nick Early, Minimal Kinematics: An all k and n peek into Trop^+G(k,n), arXiv:2003.07958 [hep-th], 2020.
Freddy Cachazo and Nick Early, Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes, arXiv:2010.09708 [math.CO], 2020.
Freddy Cachazo and Nick Early, Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes, arXiv:2204.01743 [hep-th], 2022.
Andrzej Dudek, Jarosław Grytczuk, Jakub Przybyło, and Andrzej Ruciński, Homogeneous substructures in random ordered hyper-matchings, arXiv:2507.20374 [math.CO], 2025. See p. 19.
Nick Early, Planarity in Generalized Scattering Amplitudes: PK Polytope, Generalized Root Systems and Worldsheet Associahedra, arXiv:2106.07142 [math.CO], 2021, see p. 14.
Ömer Eğecioğlu, On Böttcher's mysterious identity, Australasian Journal of Combinatorics, Volume 43 (2009), 307-316.
Paul Drube, Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers, arXiv:1606.04869 [math.CO], 2016.
Claudio Fontanari, Guido Castelnuovo and his heritage: geometry, combinatorics, teaching, arXiv:2206.06709 [math.HO], 2022. See pp. 2-3.
J. S. Frame, G. de B. Robinson and R. M. Thrall, The hook graphs of a symmetric group, Canad. J. Math. 6 (1954), pp. 316-324.
Alexander Garver and Thomas McConville, Chapoton triangles for nonkissing complexes, Algebraic Combinatorics, 3 (2020), pp. 1331-1363.
Katarzyna Górska and Karol A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008 [math.CO], 2013.
Owen John Levens, Joel Brewster Lewis, and Bridget Eileen Tenner, Global patterns in signed permutations, arXiv:2504.13108 [math.CO], 2025. See p. 18.
Dimana Miroslavova Pramatarova, Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions, MIT Res. Sci. Instit. (2025). See p. 5.
Francisco Santos, Christian Stump, and Volkmar Welker, Noncrossing sets and a Graßmannian associahedron, in FPSAC 2014, Chicago, USA; Discrete Mathematics and Theoretical Computer Science (DMTCS) Proceedings, 2014, 609-620.
Wikipedia, Hook length formula

Rows give A000108 (Catalan numbers), A005789, A005790, A005791, A321975, A321976, A321977, A321978.
Diagonals give A039622, A060855, A060856.
Cf. A227578. - Alois P. Heinz, Jul 18 2013
Cf. A321716.

T:= (m, n)-> (m*n)! * mul(i!/(m+i)!, i=0..n-1):
seq(seq(T(n, 1+d-n), n=1..d), d=1..10);

maxm = 10; t[m_, n_] := Product[k!, {k, 0, n - 1}]*(m*n)! / Product[k!, {k, m, m + n - 1}]; Flatten[ Table[t[m + 1 - n, n], {m, 1, maxm}, {n, 1, m}]] (* Jean-François Alcover, Sep 21 2011 *)
Table[ BarnesG[n+1]*(n*(m-n+1))!*BarnesG[m-n+2] / BarnesG[m+2], {m, 1, 10}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jan 30 2016 *)

{A(i, j) = if( i<0 || j<0, 0, (i*j)! / prod(k=1, i+j-1, k^vecmin([k, i, j, i+j-k])))}; /* Michael Somos, Jan 28 2004 */

T(m, n) = 0!*1!*..*(n-1)! *(m*n)! / ( m!*(m+1)!*..*(m+n-1)! ).

T(m, n) = A000142(m*n)*A000178(m-1)*A000178(n-1)/A000178(m+n-1) = A000142(A004247(m, n)) * A007318(m+n, n)/A009963(m+n, n). - Henry Bottomley, May 22 2002

More terms from Frank Ellermann, May 21 2001

A079478 Coefficient of x^0 in P(n,x) = (Product_{i=0..n-1} i!^2)/matdet(M(n)) of degree n^2 where M(n) is the n X n matrix m(i,j) = 1/(i+j+x).

Original entry on oeis.org

1, 2, 72, 172800, 60963840000, 5574884681318400000, 205619158526859285626880000000, 4394314874750658447092552646524928000000000, 73955304765761130113502867875624106401967636480000000000000

Offset: 0

Benoit Cloitre, Jan 15 2003

nonn

Product of all matrix elements of n X n matrix M(i,j) = i+j (i,j=1..n). - Alexander Adamchuk, Apr 12 2006

			Determinant of M(2) is 1/(x^4 + 12*x^3 + 53*x^2 + 102*x + 72) hence a(2)=72.

Alois P. Heinz, Table of n, a(n) for n = 0..20

Cf. A011379.

Central column in triangle A009963.

seq(mul(mul(k+j,j=1..n), k=1..n), n=0..8); # Zerinvary Lajos, Jun 01 2007

Table[Product[Product[(i+j),{i,1,n}],{j,1,n}],{n,0,10}] (* Alexander Adamchuk, Apr 12 2006 *)
Table[BarnesG[2*n+2] / BarnesG[n+2]^2, {n,0,10}] (* Vaclav Kotesovec, Feb 28 2019 *)

a(n)=(n+1)*prod(i=0,n,(n+i)!)/prod(i=1,n+1,i!)

a(n) = prod(i=1, n, prod(j=1, n, i+j)); \\ Michel Marcus, Feb 27 2019

from math import prod, factorial
def A079478(n): return prod(i+j for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)<Chai Wah Wu, Nov 26 2023

a(n) = (n+1)*(Product_{i=0..n} (n+i)!)/Product_{i=1..n+1} i!.
a(n) = A000178(2n)/A000178(n)^2, i.e., "central supercombinations" by analogy with A000984. - Henry Bottomley, May 14 2005
a(n) = Product_{j=1..n} Product_{i=1..n} (i + j). - Alexander Adamchuk, Apr 12 2006
Asymptotic: a(n) ~ (2*n+1)^(2*n^2 + 2*n + 5/12)*(n+1)^(-n^2 - 2*n - 5/6) * exp(-zeta'(-1) - (3/2)*n^2 + 3/4)/(sqrt(2*Pi)). - Peter Luschny, Nov 26 2012
a(n) = BarnesG(2*n+2) / BarnesG(n+2)^2. - Vaclav Kotesovec, Feb 28 2019
a(n) ~ A * 2^(2*n*(n+1) - 1/12) * n^(n^2 - 5/12) / (sqrt(Pi) * exp(3*n^2/2 + 1/12)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Dec 04 2023

A010796 a(n) = n!*(n+1)!/2.

Original entry on oeis.org

1, 6, 72, 1440, 43200, 1814400, 101606400, 7315660800, 658409472000, 72425041920000, 9560105533440000, 1491376463216640000, 271430516305428480000, 57000408424139980800000, 13680098021793595392000000, 3720986661927857946624000000

Offset: 1

N. J. A. Sloane

Column 2 in triangle A009963.
a(n) = A078740(n, 2), first column of (3, 2)-Stirling2 array.
Also the number of undirected Hamiltonian paths in the complete bipartite graph K_{n,n+1}. - Eric W. Weisstein, Sep 03 2017
Also, the number of undirected Hamiltonian cycles in the complete bipartite graph K_{n+1,n+1}. - Pontus von Brömssen, Sep 06 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Index entries for sequences related to factorial numbers.

Cf. A006472, A009963, A078740, A010790.

Main diagonal of A291909.

[Factorial(n)* Factorial(n+1) / 2: n in [1..20]]; // Vincenzo Librandi, Jun 11 2013

Table[n! (n + 1)! / 2, {n, 1, 20}] (* Vincenzo Librandi, Jun 11 2013 *)
Times@@@Partition[Range[20]!,2,1]/2 (* Harvey P. Dale, Jul 04 2017 *)

for(n=1,30, print1(n!*(n+1)!/2, ", ")) \\ G. C. Greubel, Feb 07 2018

a(n) = 2^(n-1) * A006472(n+1).
a(n) = A010790(n)/2.
E.g.f.: (hypergeom([1, 2], [], x)-1)/2.
a(n) = Product_{k=1..n-1} (k^2+3*k+2). - Gerry Martens, May 09 2016
E.g.f.: x*hypergeom([1, 3], [], x). - Robert Israel, May 09 2016
From Amiram Eldar, Jun 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*(BesselI(1, 2) - 1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(1 - BesselJ(1, 2)). (End)

A090441 Symmetric triangle of certain normalized products of decreasing factorials.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 12, 6, 1, 1, 24, 144, 144, 24, 1, 1, 120, 2880, 8640, 2880, 120, 1, 1, 720, 86400, 1036800, 1036800, 86400, 720, 1, 1, 5040, 3628800, 217728000, 870912000, 217728000, 3628800, 5040, 1, 1, 40320, 203212800, 73156608000

Offset: 0

Wolfdieter Lang, Dec 23 2003

Similar to, but different from, superfactorial Pascal triangle A009963.
A009963(n,m) = (Product_{p=0..m-1} (n-p)!)/superfac(m) with n >= m >= 0, otherwise 0.
From Natalia L. Skirrow, Apr 13 2025 (Start)
Denoting this sequence as the superbinomial sb(n,k), the hook length formula for a j X k rectangular Young tableau states the number of configurations of j*k distinct numbers such that each row and column is strictly increasing is (j*k)!/sb(j+k,j), ie. 1/sb(j+k,j) is the probability that a random permutation is a Young tableau.
Meanwhile, if the numbers are placed into the array with repetition, but the columns are still strictly increasing, there are c(n,j,k) = sb(n+1,j+k)/(sb(n+1-j,k)*sb(n+1-k,j)) configurations.
If the strict criterion is relaxed to monotonic, this becomes C(n,j,k) = sb(n-1+j+k,j+k)/(sb(n-1+j,j)*sb(n-1+k,k)).
By proposition 13.2(i) of Stanley's PhD thesis, for fixed j,k, c(n,j,k) and C(n,j,k) are polynomials in n of degree j*k, and c(n,j,k) = (-1)^(j*k)*C(-n,j,k).
For example, c(n,1,k)=(n choose k) and C(n,1,k)=(n+k-1 choose k), while c(n,2,k) = N(n,k+1) and C(n,2,k) = N(n+k,k+1), so the binomial coefficients and Narayana numbers N=A001263 obey the dualities (under continuation as polynomials) (n choose k) = (-1)^k*(k-1-n choose k) and N(n,k) = N(k-1-n,k).
(End)

			Rows for n = 0, 1, 2, 3, ...:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  2,  1;
  1,  6, 12,  6,  1;
  ...

Donald Knuth, Two Notes on Notation, arXiv:math/9205211 [math.HO], 1992. (Page 16-17 explain and give examples; the case with Narayana numbers come from tying together the poset P_k's 'shoelaces' with inequalities, into a 2 X k rectangle.)
Wolfdieter Lang, First 9 rows
Richard P. Stanley, Ordered Structures and Partitions, 1971.

Column sequences give: A000012 (powers of 1), A000142 (factorials), A010790, A090443-4, etc.

Cf. A090445 (row sums), A090446 (alternating row sums).

spf(n) = prod(k=2, n, k!);
T(n,m) = spf(n-1)/spf(m-1)/spf(n-m-1);
row(n) = vector(n+1, k, T(n, k-1)); \\ Michel Marcus, Apr 13 2025

a(n, m) = 0 if n < m;
a(n, m) = 1 if m = 0 or m = n;
a(n, m) = (Product_{p=1..m} (n-p)!)/superfac(m-1) if n >= 0, 1 <= m <= n+1, where superfac(n) := A000178(n), n >= 0, (superfactorials).
a(n, m) = superfac(n-1)/superfac(m-1)/superfac(n-m-1)
With offset 1, equals ConvOffsStoT transform of the factorials, A000142: (1, 1, 2, 6, 24, ...); e.g., ConvOffs transform of (1, 1, 2, 6) = (1, 6, 12, 6, 1). - Gary W. Adamson, Apr 21 2008

OFFSET changed from -1 to 0 by Natalia L. Skirrow, Apr 13 2025

A193520 a(n) = Sum_{k=0..n} G(n)/(G(k)*G(n-k)) where G(n) = Product_{k=0..n} k!.

Original entry on oeis.org

1, 2, 4, 14, 122, 3122, 260642, 76214882, 85552669442, 381014246511362, 7442029915221081602, 632869669701185574873602, 264542347321693265938488883202, 517169258108069965039831739271321602, 5495073385198979486456081260457854269542402

Offset: 0

Paul D. Hanna, Jul 29 2011

nonn

			Let F(x) = 1 + x + x^2/(1!*2!) + x^3/(1!*2!*3!) + x^4/(1!*2!*3!*4!) +...+ x^n/G(n) +...
then
F(x)^2 = 1 + 2*x + 4*x^2/(1!*2!) + 14*x^3/(1!*2!*3!) + 122*x^4/(1!*2!*3!*4!) + 3122*x^5/(1!*2!*3!*4!*5!) +...+ a(n)*x^n/G(n) +...
Illustration of initial terms:
a(3) = 1 + 3! + 3! + 1 = 14;
a(4) = 1 + 4! + 4!*3!/2! + 4! + 1 = 122;
a(5) = 1 + 5! + 5!*4!/2! + 5!*4!/2! + 5! + 1 = 3122;
a(6) = 1 + 6! + 6!*5!/2! + 6!*5!*4!/(3!*2!) + 6!*5!/2! + 6! + 1 = 260642; ...

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A000178, A193521.

Row sums of A009963.

A009963:= func< n,k | (1/Factorial(n+1))*(&*[ Factorial(n-j+1)/Factorial(j): j in [0..k]]) >;
A193520:= func< n | (&+[A009963(n,j): j in [0..n]]) >;
[A193520(n): n in [0..20]]; // G. C. Greubel, Jan 05 2022

Table[Sum[BarnesG[n+2] / (BarnesG[k+2] * BarnesG[n-k+2]), {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Mar 04 2019 *)

{a(n)=sum(k=0,n,prod(j=0,n,j!)/(prod(j=0,k,j!)*prod(j=0,n-k,j!)))}

{a(n)=prod(k=1,n,k!)*polcoeff((sum(m=0,n+1,x^m/prod(k=0,m,k!)+x*O(x^n))^2),n)}

from mpmath import mp
mp.dps = 98; mp.pretty = True
def superbinomial(n,k):
    return mp.superfac(n)/(mp.superfac(k)*mp.superfac(n-k))
def A193520(n): return add(superbinomial(n,k) for k in (0..n))
[int(A193520(n)) for n in (0..14)]  # Peter Luschny, Nov 28 2012

G.f.: A(x) = ( Sum_{n>=0} x^n/G(n) )^2 where A(x) = Sum_{n>=0} a(n)*x^n/G(n), and G(n) = Product_{k=0..n} k!.

a(n) ~ 2^(n^2/4 + n - 5*(-1)^n/8 + 23/24) * n^(n^2/4 + (-1)^n/8 - 13/24) / (sqrt(Pi) * exp(3*n^2/8 + Zeta'(-1))). - Vaclav Kotesovec, Mar 04 2019

A010797 a(n) = n! * (n+1)! * (n+2)! / (2! * 3!).

Original entry on oeis.org

1, 24, 1440, 172800, 36288000, 12192768000, 6145155072000, 4424511651840000, 4380266535321600000, 5781951826624512000000, 9921829334487662592000000, 21669275266521055100928000000

Offset: 1

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 1..180
Index entries for sequences related to factorial numbers

Column 3 in triangle A009963.

Times@@@Partition[Range[15]!,3,1]/(2!3!) (* Harvey P. Dale, Aug 06 2022 *)

Offset changed to 1 by Seiichi Manyama, Aug 14 2023

A010798 a(n) = n! * (n+1)! * (n+2)! * (n+3)! / (2! * 3! * 4!).

Original entry on oeis.org

1, 120, 43200, 36288000, 60963840000, 184354652160000, 929147446886400000, 7358847779340288000000, 87423111618562621440000000, 1500180595374534583910400000000, 36040338623277818843863449600000000

Offset: 1

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 1..142
Index entries for sequences related to factorial numbers

Column 4 in triangle A009963.

#/(2!3!4!)&/@Times@@@Partition[Range[20]!,4,1] (* Harvey P. Dale, Jul 29 2022 *)

Offset changed to 1 by Seiichi Manyama, Aug 14 2023

A010799 a(n) = n!(n+1)!(n+2)!(n+3)!(n+4)! / ( 2!3!4!*5! ).

Original entry on oeis.org

1, 720, 1814400, 12192768000, 184354652160000, 5574884681318400000, 309071606732292096000000, 29374165503837040803840000000, 4536546120412592581745049600000000

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..96
Index entries for sequences related to factorial numbers

Column 5 in triangle A009963.

With[{c=Times@@(Range[2,5]!),l=Times@@@Partition[Range[20]!,5,1]}, #/c&/@l] (* Harvey P. Dale, Nov 10 2011 *)

Offset changed to 1 by Seiichi Manyama, Aug 14 2023

A010800 a(n) = n! * (n+1)! * (n+2)! * (n+3)! * (n+4)! * (n+5)! / (2! * 3! * 4! * 5! * 6!).

Original entry on oeis.org

1, 5040, 101606400, 6145155072000, 929147446886400000, 309071606732292096000000, 205619158526859285626880000000, 254046582743105184577722777600000000, 549289359343832305886569080815616000000000, 1979419135331434097492840339627153817600000000000

Offset: 1

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 1..102
Index entries for sequences related to factorial numbers

Column 6 in triangle A009963.

Module[{nn=20,c=Times@@(Range[2,6]!)},(Times@@@Partition[ Range[ nn]!, 6,1])/c] (* Harvey P. Dale, Jul 29 2015 *)

More terms from Harvey P. Dale, Jul 29 2015

Offset changed to 1 by Seiichi Manyama, Aug 14 2023

A107252 a(n) = Product_{k=0..n-1} (n+k)!/(k+1)!.

Original entry on oeis.org

1, 1, 6, 1440, 36288000, 184354652160000, 309071606732292096000000, 254046582743105184577722777600000000, 142518177743863255019484504453155074867200000000000

Offset: 0

Henry Bottomley, May 14 2005

nonn

			a(3) = 1!*2!*3!*4!*5!/(1!*2!*3!*1!*2!) = 34560/24 = 1440.

Cf. A000178, A001700, A005130, A009963, A110131, A112332, A107254, A000142.

[1] cat [(&*[Factorial(n+k)/Factorial(k+1): k in [0..n-1]]): n in [1..10]]; // G. C. Greubel, May 21 2019

Table[Product[(n+k)!/(k+1)!,{k,0,n-1}],{n,0,10}] (* Alexander Adamchuk, Jul 10 2006 *)
a[n_] := (BarnesG[1 + 2 n] n! ((BarnesG[2 + n] Gamma[2 + n])/ BarnesG[3 + n])^(-1 + n))/BarnesG[2 + n]^2; Table[a[n], {n, 0, 10}] (* Peter Luschny, May 20 2019 *)

{a(n) = prod(k=0,n-1, (n+k)!/(k+1)!)}; \\ G. C. Greubel, May 21 2019

[product(factorial(n+k)/factorial(k+1) for k in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 21 2019

a(n) = (n+1)!*(n+2)!*...*(2n-1)!/(1!*2!*...*(n-1)!).
a(n) = A000178(2n-1)/(A000178(n)*A000178(n-1)).
a(n) = A079478(n)/A001813(n).
a(n) = A079478(n-1)*A006963(n+1).
a(n) = A107251(n)/A000108(n).
a(n) = A107251(n-1)*A009445(n-1).
a(n) = A107254(n)/A000142(n).
a(n) = A009963(2n-1, n-1).
a(n) = A009963(2n-1, n).
a(n) = (G(1+2*n)*n!*((G(2+n)*Gamma(2+n))/G(3+n))^(n-1))/G(2+n)^2, where G(x) is the Barnes G function. - Peter Luschny, May 20 2019
a(n) ~ A * 2^(2*n^2 - 7/12) * n^(n^2 - n - 5/12) / (sqrt(Pi) * exp(3*n^2/2 - n + 1/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 21 2019

cp's OEIS Frontend

A060854 Array T(m,n) read by antidiagonals: T(m,n) (m >= 1, n >= 1) = number of ways to arrange the numbers 1,2,...,m*n in an m X n matrix so that each row and each column is increasing.

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A079478 Coefficient of x^0 in P(n,x) = (Product_{i=0..n-1} i!^2)/matdet(M(n)) of degree n^2 where M(n) is the n X n matrix m(i,j) = 1/(i+j+x).

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A010796 a(n) = n!*(n+1)!/2.

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A090441 Symmetric triangle of certain normalized products of decreasing factorials.

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A193520 a(n) = Sum_{k=0..n} G(n)/(G(k)*G(n-k)) where G(n) = Product_{k=0..n} k!.

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A010797 a(n) = n! * (n+1)! * (n+2)! / (2! * 3!).

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A010798 a(n) = n! * (n+1)! * (n+2)! * (n+3)! / (2! * 3! * 4!).

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A010799 a(n) = n!(n+1)!(n+2)!(n+3)!(n+4)! / ( 2!3!4!*5! ).