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.

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

Author

Werner Schulte, Apr 11 2024

Keywords

• nonn
• easy
• tabl

Examples

			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.
		

Crossrefs

Cf. A000012 (column 0), A007526 (column 1), A001044 (main diagonal).

Cf. A000166, A048993, A131689, A320031, A371766, A371767.

Programs

Mathematica

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 *)

PARI

T(n, k) = if(k==0, 1, if(k > n, 0, n*k*(T(n-1, k-1) + T(n-1, k))))

Formula

T(n, k) = Sum_{i=k..n} A131689(i, k) * n! / (n-i)!.

T(n, k) = n! * k! * (Sum_{i=0..n-k} A048993(n-i, k) / i!).

T(n, k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * A320031(n, i).

Conjecture: E.g.f. of column k is exp(t) * t^k * k! / (Prod_{i=0..k} (1 - i*t)).

Conjecture: Sum_{k=0..n} (-1)^(n-k) * T(n, k) = A000166(n).

T(n, k) = A371766(n, k) * A371767(n, k). - Peter Luschny, Apr 14 2024