A372921 - OEIS

%I #8 May 18 2024 15:24:06

%S 1,0,1,3,6,2,18,63,36,6,189,828,684,216,24,2484,13365,14400,6660,1440,

%T 120,40095,255474,339390,206280,65880,10800,720,766422,5645619,

%U 8915508,6707610,2827440,687960,90720,5040,16936857,141626232,259137144,232306704,121519440,39130560,7680960,846720,40320

%N 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.

%F 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).

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

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

%e Triangle T(n, k) starts:

%e n\k :       0        1        2        3        4       5      6     7

%e ======================================================================

%e   0 :       1

%e   1 :       0        1

%e   2 :       3        6        2

%e   3 :      18       63       36        6

%e   4 :     189      828      684      216       24

%e   5 :    2484    13365    14400     6660     1440     120

%e   6 :   40095   255474   339390   206280    65880   10800    720

%e   7 :  766422  5645619  8915508  6707610  2827440  687960  90720  5040

%e   etc.

%t 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 *)

%o (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)) }

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

%K nonn,easy,tabl

%O 0,4

%A _Werner Schulte_, May 16 2024