A372921 Triangle read by rows: T(n, k) = (Sum_{i=0..n-k} (-1)^i * binomial(n-k, i) * A007559(n-i)) * n! / ((n-k)! * A007559(k)) for 0 <= k <= n.

1, 0, 1, 3, 6, 2, 18, 63, 36, 6, 189, 828, 684, 216, 24, 2484, 13365, 14400, 6660, 1440, 120, 40095, 255474, 339390, 206280, 65880, 10800, 720, 766422, 5645619, 8915508, 6707610, 2827440, 687960, 90720, 5040, 16936857, 141626232, 259137144, 232306704, 121519440, 39130560, 7680960, 846720, 40320

Offset: 0

Author

Werner Schulte, May 16 2024

Keywords

• nonn
• easy
• tabl

Examples

			Triangle T(n, k) starts:
n\k :       0        1        2        3        4       5      6     7
======================================================================
  0 :       1
  1 :       0        1
  2 :       3        6        2
  3 :      18       63       36        6
  4 :     189      828      684      216       24
  5 :    2484    13365    14400     6660     1440     120
  6 :   40095   255474   339390   206280    65880   10800    720
  7 :  766422  5645619  8915508  6707610  2827440  687960  90720  5040
  etc.
		

Crossrefs

Cf. A007559, A033030 (column 0), A000142 (main diagonal).

Programs

Mathematica

T[n_,k_]:=n!SeriesCoefficient[Exp[-t]/ (1-3*t)^(1/3) * (t / (1-3*t))^k,{t,0,n}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, May 18 2024 *)

PARI

T(n, k) = { sum(i=0, n-k, (-1)^i * binomial(n-k, i) * prod(j=1, n-i, 3*j-2)) * n! / ((n-k)! * prod(m=1, k, 3*m-2)) }

Formula

T(n, k) = (T(n-1, k-1) + 3 * T(n-1, k)) * n for 0 < k < n with initial values T(n, 0) = A033030(n) and T(n, n) = A000142(n).

E.g.f. of column k: exp(-t) / (1-3*t)^(1/3) * (t / (1-3*t))^k.

E.g.f.: exp(x*t / (1-3*t) - t) / (1-3*t)^(1/3).