A010796 -id:A010796 - cp's OEIS frontend

A009963 Triangle of numbers n!(n-1)!...(n-k+1)!/(1!2!...k!).

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 24, 72, 24, 1, 1, 120, 1440, 1440, 120, 1, 1, 720, 43200, 172800, 43200, 720, 1, 1, 5040, 1814400, 36288000, 36288000, 1814400, 5040, 1, 1, 40320, 101606400, 12192768000, 60963840000, 12192768000, 101606400, 40320, 1

Offset: 0

N. J. A. Sloane

Product of all matrix elements of n X k matrix M(i,j) = i+j (i=1..n-k, j=1..k). - Peter Luschny, Nov 26 2012

These are the generalized binomial coefficients associated to the sequence A000178. - Tom Edgar, Feb 13 2014

			Rows start:
  1;
  1,   1;
  1,   2,    1;
  1,   6,    6,    1;
  1,  24,   72,   24,   1;
  1, 120, 1440, 1440, 120, 1;  etc.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000178, A007318, A060854, A090441.
Central column is A079478.
Columns include A010796, A010797, A010798, A010799, A010800.
Row sums give A193520.

A009963:= func< n,k | (1/Factorial(n+1))*(&*[ Factorial(n-j+1)/Factorial(j): j in [0..k]]) >;
[A009963(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 04 2022

(* First program *)
row[n_]:= Table[Product[i+j, {i,1,n-k}, {j,1,k}], {k,0,n}];
Array[row, 9, 0] // Flatten (* Jean-François Alcover, Jun 01 2019, after Peter Luschny *)
(* Second program *)
T[n_, k_]:= BarnesG[n+2]/(BarnesG[k+2]*BarnesG[n-k+2]);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 04 2022 *)

def A009963_row(n):
    return [mul(mul(i+j for j in (1..k)) for i in (1..n-k)) for k in (0..n)]
for n in (0..7): A009963_row(n)  # Peter Luschny, Nov 26 2012

def triangle_to_n_rows(n): #changing n will give you the triangle to row n.
    N=[[1]+n*[0]]
    for i in [1..n]:
        N.append([])
        for j in [0..n]:
            if i>=j:
                N[i].append(factorial(i-j)*binomial(i-1,j-1)*N[i-1][j-1]+factorial(j)*binomial(i-1,j)*N[i-1][j])
            else:
                N[i].append(0)
    return [[N[i][j] for j in [0..i]] for i in [0..n]]
    # Tom Edgar, Feb 13 2014

T(n,k) = T(n-1,k-1)*A008279(n,n-k) = A000178(n)/(A000178(k)*A000178(n-k)) i.e., a "supercombination" of "superfactorials". - Henry Bottomley, May 22 2002
Equals ConvOffsStoT transform of the factorials starting (1, 2, 6, 24, ...); e.g., ConvOffs transform of (1, 2, 6, 24) = (1, 24, 72, 24, 1). Note that A090441 = ConvOffsStoT transform of the factorials, A000142. - Gary W. Adamson, Apr 21 2008
Asymptotic: T(n,k) ~ exp((3/2)*k^2 - zeta'(-1) + 3/4 - (3/2)*n*k)*(1+n)^((1/2)*n^2 + n + 5/12)*(1+k)^(-(1/2)*k^2 - k - 5/12)*(1 + n - k)^(-(1/2)*n^2 + n*k - (1/2)*k^2 - n + k - 5/12)/(sqrt(2*Pi). - Peter Luschny, Nov 26 2012
T(n,k) = (n-k)!*C(n-1,k-1)*T(n-1,k-1) + k!*C(n-1,k)*T(n-1,k) where C(i,j) is given by A007318. - Tom Edgar, Feb 13 2014
T(n,k) = Product_{i=1..k} (n+1-i)!/i!. - Alois P. Heinz, Jun 07 2017
T(n,k) = BarnesG(n+2)/(BarnesG(k+2)*BarnesG(n-k+2)). - G. C. Greubel, Jan 04 2022

A002397 a(n) = n! * lcm({1, 2, ..., n+1}).

Original entry on oeis.org

1, 2, 12, 72, 1440, 7200, 302400, 4233600, 101606400, 914457600, 100590336000, 1106493696000, 172613016576000, 2243969215488000, 31415569016832000, 942467070504960000, 256351043177349120000, 4357967734014935040000, 1490424965033107783680000

Offset: 0

N. J. A. Sloane

nonn

This term appears in the numerator of several sequences of coefficients used in numerical solutions of ordinary differential equations.

			5! is 120, and the least common multiple of 2, 3, 4, 5 and 6 is 60, so a(5) = 7200.

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).

Jack W Grahl, Table of n, a(n) for n = 0..100
Jack W Grahl, Explanation of the use of this sequence.
Jack W Grahl, Python code to calculate this and related sequences.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233.
W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233. [Annotated scanned copy]

Cf. A010796. Row sums of A260780, also of A260781.

The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.

a(n) = n!*lcm([1..n+1]); \\ Michel Marcus, Oct 15 2023

a(n) = n! * lcm{1,2,...,n+1} = n!*A003418(n+1). - Sean A. Irvine, Nov 07 2013

More terms from Sean A. Irvine, Nov 07 2013

More terms from Jack W Grahl, Feb 27 2021

A091543 Triangle built from first column sequences of generalized Stirling2 arrays (m+2,2)-Stirling2, m >= 0.

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 8, 72, 12, 1, 16, 1440, 360, 20, 1, 32, 43200, 20160, 1120, 30, 1, 64, 1814400, 1814400, 123200, 2700, 42, 1, 128, 101606400, 239500800, 22422400, 491400, 5544, 56, 1, 256, 7315660800, 43589145600, 6098892800, 150368400

Offset: 1

Wolfdieter Lang, Feb 13 2004

P. Blasiak, K. A. Penson, and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Blasiak, K. A. Penson, and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Wolfdieter Lang, First 10 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A091547 (row sums), A091548 (alternating row sums).

For m = 0, 1, ..., 6 the column sequences are (without leading zeros): A000079 (powers of 2), A010796, A002674, A091535, A091544-6.

a(n, m) = m^(2*(n-m))*Pochhammer(1/m, n-m)*Pochhammer(2/m, n-m)/2 if n-1 >= m >= 1; a(n, 0) = 2^(n-1); otherwise 0.
E.g.f. for m = 1, 2, ... column (without leading zeros and offset n=1): (hypergeom([1/m, 2/m], [], (m^2)*x)-1)/2.
G.f. for m=1 column: x/(1-2*x); e.g.f.: (exp(2*x)-1)/2.
a(n, m) = (1/2)*Product_{j=0..n-m-1} (m*j+2)*(m*j+1), n >= m+1 >= 1, otherwise 0. From eq. 12 of the Blasiak et al. reference with r=m+2, s=2, k=2.

A291909 Triangle read by rows: T(n,k) is the coefficient of x^(2*k) in the cycle polynomial of the complete bipartite graph K_{n,n}, 1 <= k <= n.

Original entry on oeis.org

0, 0, 1, 0, 9, 6, 0, 36, 96, 72, 0, 100, 600, 1800, 1440, 0, 225, 2400, 16200, 51840, 43200, 0, 441, 7350, 88200, 635040, 2116800, 1814400, 0, 784, 18816, 352800, 4515840, 33868800, 116121600, 101606400, 0, 1296, 42336, 1143072, 22861440, 304819200, 2351462400, 8230118400, 7315660800

Offset: 1

Eric W. Weisstein, Sep 05 2017

Also the coefficients of x^(2*k) in the chordless cycle polynomial of the n X n rook graph. - Eric W. Weisstein, Feb 21 2018

			Cycle polynomials are
        0
      x^4
    9 x^4 +   6 x^6
   36 x^4 +  96 x^6 +   72 x^8
  100 x^4 + 600 x^6 + 1800 x^8 + 1440 x^10
  ...
so the first few rows are
  0;
  0,  1;
  0,  9,  6;
  0, 36, 96, 72;
  ...

Pontus von Brömssen, Rows n = 1..100, flattened (rows n = 1..60 from Vincenzo Librandi).
Eric Weisstein's World of Mathematics, Chordless Cycle
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Cycle Polynomial

Cf. A070968 (row sums), A010796 (main diagonal).

CoefficientList[Table[Sum[Binomial[n, k]^2 k! (k - 1)! x^k, {k, 2, n}]/2, {n, 10}], x] // Flatten
Join[{0}, CoefficientList[Table[n^2 (HypergeometricPFQ[{1, 1, 1 - n, 1 - n}, {2}, x] - 1)/2, {n, 2, 10}], x]] // Flatten (* Eric W. Weisstein, Feb 21 2018 *)

T(n, k) = if(k>1, binomial(n, k)^2*k!*(k - 1)!/2, 0) \\ Andrew Howroyd, Apr 29 2018

T(n, k) = binomial(n, k)^2*k!*(k - 1)!/2 for k > 1.

Terms T(n,0) for n >= 3 deleted (in order to have a regular triangle) by Pontus von Brömssen, Sep 06 2022

A112578 Number of indecomposable 3-D arrays of 0's and 1's with plane sums 2.

Original entry on oeis.org

0, 8, 900, 359424, 370828800, 820150272000, 3435918974208000, 24957654229057536000, 294060698786444083200000, 5334667831784096818790400000, 42889554205720574193041408000000

Offset: 1

Peter J. Cameron, Sep 14 2005

			a(2)=8: six pairs of opposite edges and two inscribed tetrahedra.

P. J. Cameron and T. W. Mueller, Decomposable functors and the exponential principle, II, in preparation

P. J. Cameron and T. W. Mueller, Decomposable functors and the exponential principle, II, arXiv:0911.3760 [math.CO]

Cf. A010796 (2-D case), A112579, A112580.

a(1)=0, a(n) = (n!)^2*b(n)/2^n for n>1, where b(0)=1 and sum (n-1 choose k-1)*((2n-2k-1)!!)^2*b(k) = ((2n-1)!!)^2.

A357003 Number of Hamiltonian cycles in the cyclic Haar graph with index n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 6, 0, 1, 0, 6, 1, 6, 6, 72, 0, 1, 1, 8, 1, 8, 8, 156, 1, 8, 8, 156, 8, 156, 156, 1440, 0, 1, 0, 8, 0, 12, 12, 335, 0, 12, 0, 300, 12, 352, 300, 4800, 1, 8, 12, 335, 12, 300, 352, 4800, 8, 335, 300, 4800, 335, 4800, 4800, 43200, 0, 1, 1, 10, 1

Offset: 1

Pontus von Brömssen, Sep 08 2022

nonn

a(n) > 0 for all odd n >= 3 (Hladnik, Marušič, and Pisanski, 2002).

Pontus von Brömssen, Table of n, a(n) for n = 1..2047
Milan Hladnik, Dragan Marušič, and Tomaž Pisanski, Cyclic Haar graphs, Discrete Mathematics 244 (2002), 137-152.
Eric Weisstein's World of Mathematics, Haar Graph.

Cf. A010796, A291165, A357000, A357004.

a(2^k-1) = A010796(k-1) for k >= 2.
a(A291165(n)) = 0.
a(n) = a(A357004(n)).

A182062 T(n,k) = C(n+1-k,k)k!(n-k)!, the number of ways for k men and n-k women to form a queue in which no man is next to another man.

Original entry on oeis.org

1, 1, 1, 2, 2, 0, 6, 6, 2, 0, 24, 24, 12, 0, 0, 120, 120, 72, 12, 0, 0, 720, 720, 480, 144, 0, 0, 0, 5040, 5040, 3600, 1440, 144, 0, 0, 0, 40320, 40320, 30240, 14400, 2880, 0, 0, 0, 0, 362880, 362880, 282240, 151200, 43200, 2880, 0, 0, 0, 0, 3628800, 3628800

Offset: 0

Dennis P. Walsh, Apr 09 2012

Triangle T(n,k), 0<=k<=n, is readily derived since there are C(n+1-k,k) ways to form a sequence of k zeros and n-k ones in which no zeros are consecutive and there are k!(n-k)! ways to permute k labeled zeros and n-k labeled ones. This triangle contains several known sequences, notably A000142 (factorial numbers), A062119 (number of multiplications performed in a determinant), and A010796.

			T(4,2)=12 since there are 12 ways to line up two men {M,m} and two women {W,w} so that no man is next to another man, namely, MWmw, MWwm, MwmW, MwWm, mWMw, mWwM, mwMW, mwWM, WMwm, WmwM, wMWm, and wmWM.
Triangle T(n,k) begins
1,
1, 1,
2, 2, 0,
6, 6, 2, 0,
24, 24, 12, 0, 0,
120, 120, 72, 12, 0, 0,
720, 720, 480, 144, 0, 0, 0,
5040, 5040, 3600, 1440, 144, 0, 0, 0,
40320, 40320, 30240, 14400, 2880, 0, 0, 0, 0,
362880,362880,282240,151200,43200,2880,0,0,0,0,
3628800,3628800,2903040,1693440,604800,86400,0,0,0,0,0

T. D. Noe, Rows n = 0..100, flattened
Dennis Walsh, Notes on isolating men in a line-up [dead link]

T(n,1) = A000142(n);
T(n,2) = A062119(n-1);
T(2n-1,n-1) = A010796;
Cf. A169803

seq(seq(binomial(n+1-k,k)*k!*(n-k)!,k=0..n),n=0..10);

Flatten[Table[Binomial[n+1-k,k]k!(n-k)!,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 15 2012 *)

binomial(n+1-k,k)*k!*(n-k)!
G.f. (fixed k): (1-k)*hypergeom([1, 1-k, 2-k],[2-2*k],t)*GAMMA(1-k)^2/GAMMA(2-2*k)
T(n,k)=(n+2-2k)*T(n-1,k-1)

cp's OEIS Frontend

A009963 Triangle of numbers n!(n-1)!...(n-k+1)!/(1!2!...k!).

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A002397 a(n) = n! * lcm({1, 2, ..., n+1}).

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A091543 Triangle built from first column sequences of generalized Stirling2 arrays (m+2,2)-Stirling2, m >= 0.

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A291909 Triangle read by rows: T(n,k) is the coefficient of x^(2*k) in the cycle polynomial of the complete bipartite graph K_{n,n}, 1 <= k <= n.

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Mathematica

PARI

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Extensions

A112578 Number of indecomposable 3-D arrays of 0's and 1's with plane sums 2.

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A357003 Number of Hamiltonian cycles in the cyclic Haar graph with index n.

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A182062 T(n,k) = C(n+1-k,k)*k!*(n-k)!, the number of ways for k men and n-k women to form a queue in which no man is next to another man.

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Maple

Mathematica

Formula

A182062 T(n,k) = C(n+1-k,k)k!(n-k)!, the number of ways for k men and n-k women to form a queue in which no man is next to another man.