A153187 Triangle sequence: T(n, k) = -Product_{j=0..k+1} ((n+1)*j - 1).
0, 1, 3, 2, 10, 80, 3, 21, 231, 3465, 4, 36, 504, 9576, 229824, 5, 55, 935, 21505, 623645, 21827575, 6, 78, 1560, 42120, 1432080, 58715280, 2818333440, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 475493443425, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, 1270453824640, 101636305971200
Offset: 0
Examples
Triangle begins as: 0; 1, 3; 2, 10, 80; 3, 21, 231, 3465; 4, 36, 504, 9576, 229824; 5, 55, 935, 21505, 623645, 21827575; 6, 78, 1560, 42120, 1432080, 58715280, 2818333440; 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 475493443425;
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
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GAP
Flat(List([0..10], n-> List([0..n], k-> (-1)*Product([0..k+1], j-> j*(n+1) -1) ))); # G. C. Greubel, Mar 05 2020
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Magma
[-(&*[j*(n+1)-1: j in [0..k+1]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 05 2020
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Maple
seq(seq(-mul(j*(n+1)-1, j = 0..k+1), k = 0..n), n = 0..10); # G. C. Greubel, Mar 05 2020
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Mathematica
T[n_, m_] = -Product[(n+1)*j -1, {j,0,m+1}]; Table[T[n, m], {n,0,10}, {m,0,n}]//Flatten Table[-(n+1)^(k+2)*Pochhammer[-1/(n+1), k+2], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 05 2020 *) -
PARI
T(n,k) = (-1)*prod(j=0, k+1, j*(n+1)-1); for(j=0, 10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Mar 05 2020
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Sage
[[-(n+1)^(k+2)*rising_factorial(-1/(n+1), k+2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 05 2020
Formula
T(n, k) = -Product_{j=0..k+1} (j*(n+1) - 1).
T(n, k) = -(n+1)^(k+2) * Pochhammer(-1/(n+1), k+2).
Extensions
Edited by G. C. Greubel, Mar 05 2020
Comments