A371766
Triangle read by rows: T(n, k) = A371898(n, k) / A371767(n, k).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 16, 21, 7, 1, 1, 65, 142, 63, 11, 1, 1, 326, 1201, 709, 151, 16, 1, 1, 1957, 12336, 9709, 2521, 311, 22, 1, 1, 13700, 149989, 157971, 50045, 7186, 575, 29, 1, 1, 109601, 2113546, 2993467, 1158871, 193765, 17536, 981, 37, 1
Offset: 0
Triangle starts:
[0] 1;
[1] 1, 1;
[2] 1, 2, 1;
[3] 1, 5, 4, 1;
[4] 1, 16, 21, 7, 1;
[5] 1, 65, 142, 63, 11, 1;
[6] 1, 326, 1201, 709, 151, 16, 1;
[7] 1, 1957, 12336, 9709, 2521, 311, 22, 1;
[8] 1, 13700, 149989, 157971, 50045, 7186, 575, 29, 1;
Antidiagonally read subtriangle of
A181783.
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A371766 := (n, k) -> local j; add((-1)^(k-j)*binomial(k, j)*hypergeom([1, -n],
[], -j), j = 0..k)/((k! * n!)/(n - k)!):
seq(print(seq(simplify(A371766(n, k)), k = 0..n)), n = 0..8);
A371898
Triangle read by rows: T(n, k) = n * k * (T(n-1, k-1) + T(n-1, k)) for k > 0 with initial values T(n, 0) = 1 and T(i, j) = 0 for j > i.
Original entry on oeis.org
1, 1, 1, 1, 4, 4, 1, 15, 48, 36, 1, 64, 504, 1008, 576, 1, 325, 5680, 22680, 31680, 14400, 1, 1956, 72060, 510480, 1304640, 1382400, 518400, 1, 13699, 1036224, 12233340, 50823360, 94046400, 79833600, 25401600, 1, 109600, 16798768, 318469536, 2017814400, 5794790400, 8346240000, 5893171200, 1625702400
Offset: 0
Lower triangular array starts:
n\k : 0 1 2 3 4 5 6 7
==========================================================================
0 : 1
1 : 1 1
2 : 1 4 4
3 : 1 15 48 36
4 : 1 64 504 1008 576
5 : 1 325 5680 22680 31680 14400
6 : 1 1956 72060 510480 1304640 1382400 518400
7 : 1 13699 1036224 12233340 50823360 94046400 79833600 25401600
etc.
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T[n_, k_] := Sum[(-1)^(k - j)*Binomial[k, j]*HypergeometricPFQ[{1, -n}, {}, -j], {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Peter Luschny, Apr 12 2024 *)
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T(n, k) = if(k==0, 1, if(k > n, 0, n*k*(T(n-1, k-1) + T(n-1, k))))
Showing 1-2 of 2 results.