A193520 -id:A193520 - 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

A193521 G.f.: A(x) = ( Sum_{n>=0} x^n/sf(n) )^3 where A(x) = Sum_{n>=0} a(n)*x^n/sf(n), and sf(n) = Product_{k=0..n} k! is the superfactorial of n (A000178).

Original entry on oeis.org

1, 3, 9, 51, 795, 43923, 10372323, 11996843043, 75315947454723, 2788806652875290883, 654625444656522114316803, 1045012738906587147509753740803, 12046169853230117709495421609499289603, 1053916215003128938522329980606467994425804803

Offset: 0

Paul D. Hanna, Jul 29 2011

nonn

			Let F(x) = 1 + x + x^2/(1!*2!) + x^3/(1!*2!*3!) + x^4/(1!*2!*3!*4!) + ... + x^n/sf(n) + ...
then F(x)^3 = 1 + 3*x + 9*x^2/(1!*2!) + 51*x^3/(1!*2!*3!) + 795*x^4/(1!*2!*3!*4!) + 43923*x^5/(1!*2!*3!*4!*5!) + ... + a(n)*x^n/sf(n) + ...

G. C. Greubel, Table of n, a(n) for n = 0..48

Cf. A000178, A009963, A193520.

A193521:= func< n | (&+[ A009963(n,k)*A193520(k): k in [0..n]]) >;
[A193521(n): n in [0..20]]; // G. C. Greubel, Jan 05 2022

a[n_]:= a[n]= Sum[BarnesG[n+2]/(BarnesG[j+2]*BarnesG[k-j+2]*BarnesG[n-k+2]), {k,0,n}, {j,0,k}];
Table[a[n], {n, 0, 20}] (* G. C. Greubel, Jan 05 2022 *)

{a(n) = prod(k=1,n,k!)*polcoeff((sum(m=0, n+1, x^m/prod(k=0, m, k!) + x*O(x^n))^3), n)}

@CachedFunction
def A009963(n,k): return product(factorial(n-j+1)/factorial(j) for j in (1..k))
def A193521(n): return sum(sum(A009963(n,k)*A009963(k,j) for j in (0..k)) for k in (0..n))
[A193521(n) for n in (0..20)] # G. C. Greubel, Jan 05 2022

From G. C. Greubel, Jan 05 2022: (Start)
a(n) = Sum_{k=0..n} Sum_{j=0..k} BarnesG(n+2)/(BarnesG(j+2)*BarnesG(k-j+2 )*BarnesG(n-k+2)).
a(n) = Sum_{k=0..n} A009963(n, k) * Sum_{j=0..k} A009963(k, j).
a(n) = Sum_{j=0..n} A009963(n, j)*A193520(j). (End)
a(n) ~ c(n) * A^2 * 3^(5/4 + n + n^2/6) * n^(-5/6 + n^2/3) / (2*Pi * exp(1/6 + n^2/2)), where c(n) = 1 if mod(n,3) = 0 and c(n) = 3^(4/3) / n^(1/3) if mod(n,3) = 1 or if mod(n,3) = 2, A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023

A335997 Triangle read by rows: T(n,k) = Product_{i=n-k+1..n} i! for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 6, 12, 12, 1, 24, 144, 288, 288, 1, 120, 2880, 17280, 34560, 34560, 1, 720, 86400, 2073600, 12441600, 24883200, 24883200, 1, 5040, 3628800, 435456000, 10450944000, 62705664000, 125411328000, 125411328000

Offset: 0

Werner Schulte, Jul 08 2020

Based on some integer sequence a(n), n>0, define triangular arrays A(a;n,k) by recurrence: A(a;0,0) = 1, and A(a;i,j) = 0 if j<0 or j>i, and A(a;n,k) = n! / (n-k)! * A(a;n-1,k) + a(n) * A(a;n-1,k-1) for 0<=k<=n. Then, Product_{i=1..n} (1 + (a(i) / i!) * x) = Sum_{k=0..n} A(a;n,k) / T(n,k) * x^k for n>=0 with empty product 1 (case n=0).

For the row reversed triangle R(n,k) = Product_{i=k+1..n} i! with empty product 1 (case k=n) the terms of the matrix inverse M are given by M(n,n) = 1 for n >= 0 and M(n,n-1) = -n! for n > 0 otherwise 0. - Werner Schulte, Oct 25 2022

			The triangle starts:
n\k :  0     1      2        3         4         5         6
============================================================
  0 :  1
  1 :  1     1
  2 :  1     2      2
  3 :  1     6     12       12
  4 :  1    24    144      288       288
  5 :  1   120   2880    17280     34560     34560
  6 :  1   720  86400  2073600  12441600  24883200  24883200
  etc.

Cf. A000012 (col_0), A000142 (col_1), A010790 (col_2), A176037 (col_3), A000178 (main diagonal and first subdiagonal).
Row sums equal A051399(n+1).
Cf. A009963, A093002, A193520.

T[n_, k_] := Product[i!, {i, n - k + 1, n}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 08 2020 *)

T(n,k) = T(n,1) * T(n-1,k-1) for 0 < k <= n.
T(2*n,n) = A093002(n+1) for n >= 0.
T(n,k)/T(k,k) = A009963(n,k) for 0 <= k <= n.
(Sum_{k=0..n} T(n,k) * T(n,n-k))/T(n,n) = A193520(n) for n >= 0.

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

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A193521 G.f.: A(x) = ( Sum_{n>=0} x^n/sf(n) )^3 where A(x) = Sum_{n>=0} a(n)*x^n/sf(n), and sf(n) = Product_{k=0..n} k! is the superfactorial of n (A000178).

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A335997 Triangle read by rows: T(n,k) = Product_{i=n-k+1..n} i! for 0 <= k <= n.

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