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

%I A009963 #60 Apr 13 2022 01:14:45
%S A009963 1,1,1,1,2,1,1,6,6,1,1,24,72,24,1,1,120,1440,1440,120,1,1,720,43200,
%T A009963 172800,43200,720,1,1,5040,1814400,36288000,36288000,1814400,5040,1,1,
%U A009963 40320,101606400,12192768000,60963840000,12192768000,101606400,40320,1
%N A009963 Triangle of numbers n!(n-1)!...(n-k+1)!/(1!2!...k!).
%C A009963 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
%C A009963 These are the generalized binomial coefficients associated to the sequence A000178. - _Tom Edgar_, Feb 13 2014
%H A009963 G. C. Greubel, <a href="/A009963/b009963.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A009963 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
%F A009963 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
%F A009963 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
%F A009963 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
%F A009963 T(n,k) = Product_{i=1..k} (n+1-i)!/i!. - _Alois P. Heinz_, Jun 07 2017
%F A009963 T(n,k) = BarnesG(n+2)/(BarnesG(k+2)*BarnesG(n-k+2)). - _G. C. Greubel_, Jan 04 2022
%e A009963 Rows start:
%e A009963   1;
%e A009963   1,   1;
%e A009963   1,   2,    1;
%e A009963   1,   6,    6,    1;
%e A009963   1,  24,   72,   24,   1;
%e A009963   1, 120, 1440, 1440, 120, 1;  etc.
%t A009963 (* First program *)
%t A009963 row[n_]:= Table[Product[i+j, {i,1,n-k}, {j,1,k}], {k,0,n}];
%t A009963 Array[row, 9, 0] // Flatten (* _Jean-François Alcover_, Jun 01 2019, after _Peter Luschny_ *)
%t A009963 (* Second program *)
%t A009963 T[n_, k_]:= BarnesG[n+2]/(BarnesG[k+2]*BarnesG[n-k+2]);
%t A009963 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 04 2022 *)
%o A009963 (Sage)
%o A009963 def A009963_row(n):
%o A009963     return [mul(mul(i+j for j in (1..k)) for i in (1..n-k)) for k in (0..n)]
%o A009963 for n in (0..7): A009963_row(n)  # _Peter Luschny_, Nov 26 2012
%o A009963 (Sage)
%o A009963 def triangle_to_n_rows(n): #changing n will give you the triangle to row n.
%o A009963     N=[[1]+n*[0]]
%o A009963     for i in [1..n]:
%o A009963         N.append([])
%o A009963         for j in [0..n]:
%o A009963             if i>=j:
%o A009963                 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])
%o A009963             else:
%o A009963                 N[i].append(0)
%o A009963     return [[N[i][j] for j in [0..i]] for i in [0..n]]
%o A009963     # _Tom Edgar_, Feb 13 2014
%o A009963 (Magma)
%o A009963 A009963:= func< n,k | (1/Factorial(n+1))*(&*[ Factorial(n-j+1)/Factorial(j): j in [0..k]]) >;
%o A009963 [A009963(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 04 2022
%Y A009963 Cf. A000178, A007318, A060854, A090441.
%Y A009963 Central column is A079478.
%Y A009963 Columns include A010796, A010797, A010798, A010799, A010800.
%Y A009963 Row sums give A193520.
%K A009963 nonn,tabl
%O A009963 0,5
%A A009963 _N. J. A. Sloane_