A153189 Triangle T(n,k) = Product_{j=0..k} n*j+1.
1, 1, 2, 1, 3, 15, 1, 4, 28, 280, 1, 5, 45, 585, 9945, 1, 6, 66, 1056, 22176, 576576, 1, 7, 91, 1729, 43225, 1339975, 49579075, 1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000, 1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625
Offset: 0
Examples
Triangle begins as: 1; 1, 2; 1, 3, 15; 1, 4, 28, 280; 1, 5, 45, 585, 9945; 1, 6, 66, 1056, 22176, 576576; 1, 7, 91, 1729, 43225, 1339975, 49579075; 1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000; 1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625;
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Crossrefs
Programs
-
Magma
[(&*[n*j+1: j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 15 2020
-
Maple
seq(seq(mul(n*j+1, j=0..k), k=0..n), n=0..10); # G. C. Greubel, Feb 15 2020
-
Mathematica
T[n_, k_]= If[n==0 && k==0, 1, Product[n*j+1, {j,0,k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 15 2020 *) T[n_, k_]:= T[n, k]= If[k<2, 1+k*n, ((1+n*k)*T[n, k-1] + (1+n*k)*(1+n*(k-1))* T[n, k-2])/2]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* Georg Fischer, Feb 17 2020 *) -
PARI
T(n,k)=prod(j=0,k,n*j+1) \\ M. F. Hasler, Oct 28 2014
-
Sage
[[ product(n*j+1 for j in (0..k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 15 2020
Formula
T(n, k) = n^(k+1)*Pochhammer(1/n, k+1).
From Vaclav Kotesovec, Oct 10 2016: (Start)
For fixed n > 0:
T(n, k) ~ sqrt(2*Pi) * n^k * k^(k + 1/2 + 1/n) / (Gamma(1 + 1/n) * exp(k)).
T(n, k) ~ k! * n^k * k^(1/n) / Gamma(1 + 1/n).
(End)
T(n, k) = Sum_{j=0..k+1} (-1)^(k-j+1)*Stirling1(k+1,j)*n^(k-j+1). - G. C. Greubel, Feb 17 2020
T(n, k) = ((1+n*k)*T(n, k-1) + (1+n*k)*(1+n*(k-1))*T(n, k-2))/2. - Georg Fischer, Feb 17 2020
Extensions
Edited and row 0 added by M. F. Hasler, Oct 28 2014
Comments