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

0,5

Comments

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

Links

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

Formula

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

Example

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

Mathematica

(* 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 *)

Prog

(Sage)
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
(Sage)
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
(Magma)
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

Crossrefs

Cf. A000178, A007318, A060854, A090441.

Central column is A079478.

Columns include A010796, A010797, A010798, A010799, A010800.

Row sums give A193520.

Sequence in context: A174411 A322620 A155795 * A008300 A321789 A173887

Adjacent sequences: A009960 A009961 A009962 * A009964 A009965 A009966

Keyword

nonn,tabl

Author

N. J. A. Sloane

Status

approved