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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.

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)

...M=[[N[i][j] for j in [0..i]] for i in [0..n]]

...return M # 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: A172373 A174411 A155795 * A008300 A173887 A288025

Adjacent sequences:  A009960 A009961 A009962 * A009964 A009965 A009966

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified February 21 00:50 EST 2018. Contains 299388 sequences. (Running on oeis4.)