A006480 -id:A006480 - cp's OEIS frontend

A172634 Number of n X 3 0..2 arrays with row sums 3 and column sums n.

1, 1, 7, 31, 175, 991, 5881, 35617, 219871, 1376095, 8710537, 55644337, 358198369, 2320792657, 15120204295, 98984058271, 650725327231, 4293779332927, 28425752310361, 188739799967425, 1256510215733185, 8385127334900305, 56078904057164215, 375796823748323215

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

Inverse binomial transform of the Franel numbers (A000172). - Paul D. Hanna, Feb 26 2012
a(n) is the constant term in the expansion of (1 + x + y + 1/x + 1/y + x/y + y/x)^n. - Seiichi Manyama, Oct 26 2019
a(n) is the constant term in the expansion of (-1 + (1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - Seiichi Manyama, Oct 27 2019
a(n) is the number of n step closed walks on the hexagonal lattice with loops at each node. A step along a loop leaves the position unchanged. The bijection is as follows: after subtracting 1 from each element in the array, values are -1, 0 or 1 and row and column sums are zero. There are only seven possibilities for each row. An all zero row corresponds with a step along the loop leaving the position unchanged and the others to a unit step in each of the six possible directions. This justifies that this sequence is the binomial transform of A002898. - Andrew Howroyd, May 09 2020

			G.f.: A(x) = 1 + x + 7*x^2 + 31*x^3 + 175*x^4 + 991*x^5 + 5881*x^6 +...
G.f.: A(x) = 1/(1-x) + 6*x^2*(1+x)/(1-x)^4 + 90*x^4*(1+x)^2/(1-x)^7 + 1680*x^6*(1+x)^3/(1-x)^10 + 34650*x^8*(1+x)^4/(1-x)^13 +...+ A006480(n)*x^(2*n)*(1+x)^n/(1-x)^(3*n+1) +...

Andrew Howroyd, Table of n, a(n) for n = 0..500 (terms 1..99 from R. H. Hardin)

Column k=3 of A328747 and A334549.

Cf. A000172, A002898, A006480, A328725.

Table[SeriesCoefficient[Sum[(3*k)!/k!^3*x^(2*k)*(1+x)^k/(1-x)^(3*k+1),{k,0,n}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 20 2012 *)

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)*(1+x)^m/(1-x + x*O(x^n))^(3*m+1)),n)} \\ Paul D. Hanna, Feb 26 2012

a(n)={sum(i=0, n, sum(j=0, i, (-1)^(n-i)*binomial(n, i)*binomial(i, j)^3))} \\ Andrew Howroyd, May 09 2020

From Paul D. Hanna, Feb 26 2012: (Start)
G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1+x)^n / (1-x)^(3*n+1).
Equals the binomial transform of A002898.
a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(n, k) * A000172(k), where A000172(k) = Sum_{j=0..k} binomial(k,j)^3 forms the Franel numbers.
(End)
Recurrence: n^2*a(n) = (2*n-1)^2*a(n-1) + 19*(n-1)^2*a(n-2) + 14*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 7^(n+1)*sqrt(3)/(12*Pi*n). - Vaclav Kotesovec, Oct 20 2012
G.f.: hypergeom([1/3, 1/3],[1],-27*x*(x+1)^2/((1-7*x)^2*(1+2*x)))/((1+2*x)^(1/3)*(1-7*x)^(2/3)). - Mark van Hoeij, May 07 2013

a(0)=1 prepended by Andrew Howroyd, May 09 2020

A364509 Square array read by ascending antidiagonals: T(n,k) = (2k)!/k!^2 ( (2nk)! * ((n + 2)k)! )/( (nk)! * ((n + 1)*k)!^2 ) for n, k > = 0.

Original entry on oeis.org

1, 1, 4, 1, 6, 36, 1, 16, 90, 400, 1, 50, 784, 1680, 4900, 1, 168, 8910, 48400, 34650, 63504, 1, 588, 113256, 2011100, 3312400, 756756, 853776, 1, 2112, 1528436, 96993024, 503909070, 240374016, 17153136, 11778624, 1, 7722, 21395520, 5056527000, 92279796840, 133954543800, 18116083216

Offset: 0

Peter Bala, Jul 28 2023

Given two sequences of integers c = (c_1, c_2, ..., c_K) and d = (d_1, d_2, ..., d_L) where c_1 + ... + c_K = d_1 + ... + d_L  we can define the factorial ratio sequence u_k(c, d) = (c_1*k)!*(c_2*k)!* ... *(c_K*k)!/ ( (d_1*k)!*(d_2*k)!* ... *(d_L*k)! ) and ask whether it is integral for all k >= 0. The integer L - K is called the height of the sequence. Bober completed the classification of integral factorial ratio sequences of height 1. Soundararajan gives many examples of two-parameter families of integral factorial ratio sequences of height 2.
Each row sequence of the present table is an integral factorial ratio sequence of height 2.
It is known that both row 0, the squares of the central binomial numbers, and row 1, the de Bruijn numbers, satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r. We conjecture that all the row sequences of the table satisfy the same supercongruences [added Oct 11 2024: follows from Meštrović, Section 6, equation 39, since T(n, k) = binomial(2*k, k) * binomial(2*n*k, n*k) * binomial((n+2)*k, k)/binomial((n+1)*k, k)].

			 Square array begins:
 n\k|  0    1        2           3               4                  5
  - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0 |  1    4       36         400            4900              63504 ... A002894
  1 |  1    6       90        1680           34650             756756 ... A006480
  2 |  1   16      784       48400         3312400          240374016 ... A364510
  3 |  1   50     8910     2011100       503909070       133954543800 ... A364511
  4 |  1  168   113256    96993024     92279796840     93172920645168 ...
  5 |  1  588  1528436  5056527000  18592935952500  72567511917065088 ...

Winston de Greef, Table of n, a(n) for n = 0..3240 (80 antidiagonals)
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc. (2) 79 2009, 422-444.
Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012), arXiv:1111.3057 [math.NT], (2011).
K. Soundararajan, Integral factorial ratios: irreducible examples with height larger than 1, Phil. Trans. Royal Soc., A378: 2018044, 2019.
Wikipedia, Dixon's identity

A002894 (row 0), A006480 (row 1), A364510 (row 3), A364511 (row 4).

Cf. A364303, A364506.

 # display as a square array
T(n,k) := (2*k)!/k!^2 * ( (2*n*k)! * ((n + 2)*k)! )/( (n*k)! * ((n + 1)*k)!^2 ):
seq( print(seq(T(n,k), k = 0..10)), n = 0..10):
# display as a sequence
seq( seq(T(n-k,k), k = 0..n), n = 0..10);

T(n,k) = (2*k)!/k!^2 * ( (2*n*k)! * ((n + 2)*k)! )/( (n*k)! * ((n + 1)*k)!^2 ) \\ Winston de Greef, Oct 05 2023

T(n,k) = Sum_{i = -k..k} (-1)^i * binomial(2*k, k+i)^2 * binomial(2*n*k, n*k+i) (shows that the table entries are integers).
For n >= 1, T(n,k) = (-1)^k * binomial(2*n*k, (n+1)*k)^2 * hypergeom([-2*k, -2*k, -(n+1)*k], [1, 1 + (n-1)*k], 1) = (2*k)!/k!^2 * ( (2*n*k)! * ((n + 2)*k)! )/( (n*k)! * ((n + 1)*k)!^2 ) by Dixon's 3F2 summation theorem.
T(n,k) = (-1)^(n*k) * [x^((n+1)*k)] ( (1 - x)^(2*(n+1)*k) * Legendre_P(2*k, (1 + x)/(1 - x)) ). - Peter Bala, Aug 14 2023

A022916 Multinomial coefficient n!/([n/3]![(n+1)/3]![(n+2)/3]!).

Original entry on oeis.org

1, 1, 2, 6, 12, 30, 90, 210, 560, 1680, 4200, 11550, 34650, 90090, 252252, 756756, 2018016, 5717712, 17153136, 46558512, 133024320, 399072960, 1097450640, 3155170590, 9465511770, 26293088250, 75957810500, 227873431500, 638045608200, 1850332263780, 5550996791340

Offset: 0

Clark Kimberling, Jun 14 1998

Number of permutation patterns modulo 3. This matches the multinomial formula. - Olivier Gérard, Feb 25 2011
Also the number of permutations of n elements where p(k-3) < p(k) for all k. - Joerg Arndt, Jul 23 2011
Also the number of n-step walks on cubic lattice starting at (0,0,0), ending at (floor(n/3), floor((n+1)/3), floor((n+2)/3)), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), and (1,0,0). - Alois P. Heinz, Oct 11 2019

			Starting from n=4, several permutations have the same pattern. Both (3,1,4,2) and (3,4,1,2) have pattern (0, 1, 1, 2) modulo 3.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 101 terms from Vincenzo Librandi)

A006480(n) = a(3*n).
Cf. A001405 (permutation patterns mod 2).
Cf. A022917 (permutation patterns mod 4).

a:= n-> combinat[multinomial](n, floor((n+i)/3)$i=0..2):
seq(a(n), n=0..24);  # Alois P. Heinz, Oct 11 2019

Table[ n!/(Quotient[n, 3]!*Quotient[n + 1, 3]!*Quotient[n + 2, 3]!), {n, 0, 30}]
Table[n!/Times@@(Floor/@((n+{0,1,2})/3)!),{n,0,30}] (* Harvey P. Dale, Jul 13 2012 *)
Table[Multinomial[Floor[n/3], Floor[(n+1)/3], Floor[(n+2)/3]], {n, 0, 30}] (* Jean-François Alcover, Jun 24 2015 *)

PARI
```
a(n)=n!/((n\3)!*((n+1)\3)!*((n+2)\3)!)
```

{a(n)= if(n<0, 0, n!/(n\3)!/((n+1)\3)!/((n+2)\3)!)} /* Michael Somos, Jun 20 2007 */

Recurrence: (n+1)*(n+2)*(3*n+1)*a(n) = 3*(3*n^2 + 3*n + 2)*a(n-1) + 27*(n-1)*(n+2)*a(n-2) + 27*(n-2)*(n-1)*(3*n+4)*a(n-3). - Vaclav Kotesovec, Feb 26 2014

a(n) ~ 3^(n+3/2) / (2*Pi*n). - Vaclav Kotesovec, Feb 26 2014

Corrected by Michael Somos, Jun 20 2007

A050984 de Bruijn's S(5,n) = Sum_{k = 0..2n} (-1)^(n+k)binomial(2*n, k)^5.

Original entry on oeis.org

1, 30, 5730, 1696800, 613591650, 248832363780, 108702332138400, 50030418256790400, 23933662070438513250, 11795304320307625903500, 5952113838155498195161980, 3061813957188788125283450400, 1600318610176809076206888362400, 847745162264320796366122559544000

Offset: 0

Eric W. Weisstein

Generally (de Bruijn, 1958), S(s,n) is asymptotic to (2*cos(Pi/(2*s)))^(2*n*s+s-1)*2^(2-s)*(Pi*n)^((1-s)/2)*s^(-1/2). - Vaclav Kotesovec, Jul 09 2013

Andrews (1988) on page 162 states "If, however, we resort to the theory of hypergeometric series, we find that, for example, S(5,n) = - 5F_4[-2n,-2n,-2n,-2n,-2n 1,1,1,1 ; 1]". - _Michael Somos, Jul 24 2013

			1 + 30*x + 5730*x^2 + 1696800*x^3 + 613591650*x^4 + ...

G. E. Andrews "Application of SCRATCHPAD to problems in special functions and combinatorics" Trends in Computer Algebra, R. Janssen, ed., Springer Lecture Notes in Comp.Sci., No. 296, pp. 159-166 (1988)
N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapters 4 and 6.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Binomial Sums

Cf. A000984, A006480, A050983, A227357.

Sum[ (-1)^(k+n)Binomial[ 2n, k ]^5, {k, 0, 2n} ]
a[ n_] := If[ n < 0, 0, (-1)^n HypergeometricPFQ[-2 n {1, 1, 1, 1, 1}, {1, 1, 1, 1}, 1]] (* Michael Somos, Jul 24 2013 *)

a(n)=sum(k=0,2*n,(-1)^(k+n)*binomial(2*n,k)^5) \\ Charles R Greathouse IV, Dec 21 2011

E.g.f.: Sum(n>=0,I^n*x^n/n!^5) * Sum(n>=0,(-I)^n*x^n/n!^5) = Sum(n>=0,a(n)*x^(2*n)/n!^5) where I^2=-1. - Paul D. Hanna, Dec 21 2011
a(n) ~ (5+sqrt(5))^(5*n+2)/(sqrt(5)*Pi^2*n^2*2^(5*(n+1))). - Vaclav Kotesovec, Jul 09 2013
Recurrence: n^4*(2*n - 1)^2*(220*n^3 - 858*n^2 + 1119*n - 488)*a(n) = 5*(110000*n^9 - 759000*n^8 + 2252400*n^7 - 3766690*n^6 + 3908325*n^5 - 2609510*n^4 + 1122418*n^3 - 300699*n^2 + 45738*n - 3024)*a(n-1) - 5*(2*n - 3)^2*(5*n - 8)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(220*n^3 - 198*n^2 + 63*n - 7)*a(n-2). - Vaclav Kotesovec, Sep 27 2016
For n >= 1, a(n) = 2 * Sum_{k = 0..2*n-1} (-1)^(n+k) * binomial(2*n, k)^4 * binomial(2*n-1, k) = (1/n) * Sum_{k = 0..2*n} (-1)^(n+k) * k * binomial(2*n, k)^5. - Peter Bala, Oct 31 2024

A060538 Square array read by antidiagonals of number of ways of dividing n*k labeled items into n labeled boxes with k items in each box.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 20, 90, 24, 1, 70, 1680, 2520, 120, 1, 252, 34650, 369600, 113400, 720, 1, 924, 756756, 63063000, 168168000, 7484400, 5040, 1, 3432, 17153136, 11732745024, 305540235000, 137225088000, 681080400, 40320, 1, 12870

Offset: 1

Henry Bottomley, Apr 02 2001

			       1        1        1        1
       2        6       20       70
       6       90     1680    34650
      24     2520   369600 63063000

Harry J. Smith, Table of n, a(n) for n = 1..210

Subtable of A187783.
Rows include A000012, A000984, A006480, A008977, A008978 etc.
Columns include A000142, A000680, A014606, A014608, A014609 etc.
Main diagonal is A034841.

T(n,k)=(n*k)!/k!^n;
for(n=1, 6, for(k=1, 6, print1(T(n,k), ", ")); print) \\ Harry J. Smith, Jul 06 2009

T(n, k) = (nk)!/k!^n = T(n-1, k)*binomial(nk, k) = T(n-1, k)*A060539(n, k) = A060540(n, k)*A000142(k).

A071550 a(n) = (8n)!/n!^8.

Original entry on oeis.org

1, 40320, 81729648000, 369398958888960000, 2390461829733887910000000, 18975581770994682860770223800320, 171889289584866507880743491472699801600

Offset: 0

Benoit Cloitre, May 30 2002

nonn

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

Sequences (k*n)!/n!^k: A000984 (k = 2), A006480 (k = 3), A008977 (k = 4), A008978 (k = 5), A008979 (k = 6), A071549 (k = 7), A071551 (k = 9), A071552 (k = 10).

[Factorial(8*n)/Factorial(n)^8: n in [0..20]]; // Vincenzo Librandi, Aug 13 2014

Table[(8 n)!/(n)!^8, {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2014 *)

A071551 a(n) = (9n)!/n!^9.

Original entry on oeis.org

1, 362880, 12504636144000, 1080491954750208000000, 140810154080474667338550000000, 23183587808948692737291767860055162880, 4439413043841128802009762476941510771390464000

Offset: 0

Benoit Cloitre, May 30 2002

nonn

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

Sequences (k*n)!/n!^k: A000984 (k = 2), A006480 (k = 3), A008977 (k = 4), A008978 (k = 5), A008979 (k = 6), A071549 (k = 7), A071550 (k = 8), A071552 (k = 10).

[Factorial(9*n)/Factorial(n)^9: n in [0..20]]; // Vincenzo Librandi, Aug 13 2014

Table[(9n)!/(n)!^9, {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2014 *)

A071552 a(n) = (10n)!/n!^10.

Original entry on oeis.org

1, 3628800, 2375880867360000, 4386797336285844480000000, 12868639981414579848070084500000000, 49120458506088132224064306071170476903628800

Offset: 0

Benoit Cloitre, May 30 2002

nonn

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

Sequences (k*n)!/n!^k: A000984 (k = 2), A006480 (k = 3), A008977 (k = 4), A008978 (k = 5), A008979 (k = 6), A071549 (k = 7), A071550 (k = 8), A071551 (k = 9).

[Factorial(10*n)/Factorial(n)^10: n in [0..20]]; // Vincenzo Librandi, Aug 13 2014

Table[(10n)!/(n)!^10, {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2014 *)

A364506 Square array read by ascending antidiagonals: T(n,k) = (2k)!/k! ( (2nk)! * ((2n+1)k)! )/( (nk)!^2 ((n+1)*k)!^2 ).

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 40, 90, 20, 1, 350, 5880, 1680, 70, 1, 3528, 594594, 1101100, 34650, 252, 1, 38808, 75088728, 1299170600, 229265400, 756756, 924, 1, 453024, 10861066216, 2066315135040, 3164045050530, 50678855040, 17153136, 3432, 1, 5521230, 1721929279200, 3943172216808000

Offset: 0

Peter Bala, Jul 27 2023

Given two sequences of integers c = (c_1, c_2, ..., c_K) and d = (d_1, d_2, ..., d_L) where c_1 + ... + c_K = d_1 + ... + d_L  we can define the factorial ratio sequence u_k(c, d) = (c_1*k)!*(c_2*k)!* ... *(c_K*k)!/ ( (d_1*k)!*(d_2*k)!* ... *(d_L*k)! ) and ask whether it is integral for all k >= 0. The integer L - K is called the height of the sequence. Bober completed the classification of integral factorial ratio sequences of height 1. Soundararajan gives many examples of two-parameter families of integral factorial ratio sequences of height 2.
Each row sequence of the present table is an integral factorial ratio sequence of height 2.
It is known that both row 0, the central binomial numbers, and row 1, the de Bruijn numbers, satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r. We conjecture that all the row sequences of the table satisfy the same supercongruences.

			 Square array begins:
 n\k|  0     1         2              3                  4                 5
  - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0 |  1     2         6             20                 70               252 ...
  1 |  1     6        90           1680              34650            756756 ...
  2 |  1    40      5880        1101100          229265400       50678855040 ...
  3 |  1   350    594594     1299170600      3164045050530  8188909171581600 ...
  4 |  1  3528  75088728  2066315135040  63464046079757400  ...
  5 |  1 38808  ...

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.
K. Soundararajan, Integral factorial ratios: irreducible examples with height larger than 1, Phil. Trans. Royal Soc., A378: 2018044, 2019.
Wikipedia, Dixon's identity

A000984 (row 0), A006480 (row 1), A364507 (row 2), A364508 (row 3). Cf. A364303, A364509, A365025.

# display as a square array
T(n,k) := (2*k)!/k! * ( (2*n*k)! * ((2*n+1)*k)! )/((n*k)!^2 * ((n+1)*k)!^2):
seq( print(seq(T(n,k), k = 0..10)), n = 0..10);
# display as a sequence
seq( seq(T(n-k,k), k = 0..n), n = 0..10);

T(n,k) = Sum_{i = -k..k} (-1)^i * binomial(2*k, k+i) * binomial(2*n*k, n*k+i)^2 (shows that the table entries are integers).
For n >= 1, T(n,k) = (-1)^k * binomial(2*n*k, (n+1)*k)^2 * hypergeom([-2*k, -(n+1)*k, -(n+1)*k], [1 + (n-1)*k, 1 + (n-1)*k], 1) = (2*k)!/k! * ( (2*n*k)! * ((2*n+1)*k)! )/( (n*k)!^2 * ((n+1)*k)!^2 ) by Dixon's 3F2 summation theorem.
T(n,k) = (-1)^k * [x^((n + 1)*k)] ( (1 - x)^(2*(n+1)*k) * Legendre_P(2*n*k, (1 + x)/(1 - x)) ). - Peter Bala, Aug 15 2023

A089659 a(n) = S1(n,2), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

Original entry on oeis.org

0, 2, 19, 104, 440, 1600, 5264, 16128, 46848, 130560, 352000, 923648, 2369536, 5963776, 14766080, 36044800, 86900736, 207224832, 489357312, 1145569280, 2660761600, 6136266752, 14060355584, 32027705344, 72561459200, 163577856000, 367068708864, 820204535808

Offset: 0

N. J. A. Sloane, Jan 04 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Columns : A000012, A000012, A000984, A006480, A008977, A008978.

Sequences of S1(n, t): A001792 (t=0), A089658 (t=1), this sequence (t=2), A089660 (t=3), A089661 (t=4), A089662 (t=5), A089663 (t=6).

I:=[0,2,19,104]; [n le 4 select I[n] else 8*Self(n-1)-24*Self(n-2)+32*Self(n-3)-16*Self(n-4): n in [1..41]]; // Vincenzo Librandi, Jun 22 2016

LinearRecurrence[{8,-24,32,-16}, {0,2,19,104}, 40] (* Vincenzo Librandi, Jun 22 2016 *)

[2^(n-3)*n*(7*n^2 + 12*n + 5)/3 for n in (0..40)] # G. C. Greubel, May 24 2022

a(n) = 2^(n-3)*n*(7*n^2 + 12*n + 5)/3. (see Wang and Zhang p. 333)
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n > 3.
G.f.: x*(2 + 3*x)/(1 - 2*x)^4. (End)
E.g.f.: x*(12 + 33*x + 14*x^2)*exp(2*x)/6. - Ilya Gutkovskiy, Jun 21 2016

cp's OEIS Frontend

A172634 Number of n X 3 0..2 arrays with row sums 3 and column sums n.

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A364509 Square array read by ascending antidiagonals: T(n,k) = (2*k)!/k!^2 * ( (2*n*k)! * ((n + 2)*k)! )/( (n*k)! * ((n + 1)*k)!^2 ) for n, k > = 0.

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A022916 Multinomial coefficient n!/([n/3]![(n+1)/3]![(n+2)/3]!).

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Maple

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A050984 de Bruijn's S(5,n) = Sum_{k = 0..2*n} (-1)^(n+k)*binomial(2*n, k)^5.

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A060538 Square array read by antidiagonals of number of ways of dividing n*k labeled items into n labeled boxes with k items in each box.

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A071550 a(n) = (8n)!/n!^8.

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Magma

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A071551 a(n) = (9n)!/n!^9.

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A364509 Square array read by ascending antidiagonals: T(n,k) = (2k)!/k!^2 ( (2nk)! * ((n + 2)k)! )/( (nk)! * ((n + 1)*k)!^2 ) for n, k > = 0.

A050984 de Bruijn's S(5,n) = Sum_{k = 0..2n} (-1)^(n+k)binomial(2*n, k)^5.

A364506 Square array read by ascending antidiagonals: T(n,k) = (2k)!/k! ( (2nk)! * ((2n+1)k)! )/( (nk)!^2 ((n+1)*k)!^2 ).