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

0,4

Links

Table of n, a(n) for n=0..44.

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).

Example

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.

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 *)

Prog

(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)) }

Crossrefs

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

Sequence in context: A090774 A147995 A351232 * A367028 A349863 A306189

Adjacent sequences: A372918 A372919 A372920 * A372922 A372923 A372924

Keyword

nonn,easy,tabl

Author

Werner Schulte, May 16 2024

Status

approved