cp's OEIS Frontend

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.

%I A372921 #8 May 18 2024 15:24:06
%S A372921 1,0,1,3,6,2,18,63,36,6,189,828,684,216,24,2484,13365,14400,6660,1440,
%T A372921 120,40095,255474,339390,206280,65880,10800,720,766422,5645619,
%U A372921 8915508,6707610,2827440,687960,90720,5040,16936857,141626232,259137144,232306704,121519440,39130560,7680960,846720,40320
%N 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.
%F A372921 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 A372921 E.g.f. of column k: exp(-t) / (1-3*t)^(1/3) * (t / (1-3*t))^k.
%F A372921 E.g.f.: exp(x*t / (1-3*t) - t) / (1-3*t)^(1/3).
%e A372921 Triangle T(n, k) starts:
%e A372921 n\k :       0        1        2        3        4       5      6     7
%e A372921 ======================================================================
%e A372921   0 :       1
%e A372921   1 :       0        1
%e A372921   2 :       3        6        2
%e A372921   3 :      18       63       36        6
%e A372921   4 :     189      828      684      216       24
%e A372921   5 :    2484    13365    14400     6660     1440     120
%e A372921   6 :   40095   255474   339390   206280    65880   10800    720
%e A372921   7 :  766422  5645619  8915508  6707610  2827440  687960  90720  5040
%e A372921   etc.
%t A372921 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 A372921 (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 A372921 Cf. A007559, A033030 (column 0), A000142 (main diagonal).
%K A372921 nonn,easy,tabl
%O A372921 0,4
%A A372921 _Werner Schulte_, May 16 2024