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A009963 Triangle of numbers n!(n-1)!...(n-k+1)!/(1!2!...k!). 14
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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..44.

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

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

row[n_] := Table[Product[i+j, {i, 1, n-k}, {j, 1, k}], {k, 0, n}];

Array[row, 9, 0] // Flatten (* Jean-Fran├žois Alcover, Jun 01 2019, after Peter Luschny *)

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

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

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Last modified June 1 18:43 EDT 2020. Contains 334762 sequences. (Running on oeis4.)