A373050 - OEIS

%I #8 May 21 2024 13:11:35

%S 1,2,1,7,6,2,32,39,24,6,181,284,252,120,24,1214,2325,2680,1860,720,

%T 120,9403,21234,30030,27240,15480,5040,720,82508,214459,358848,400890,

%U 299040,143640,40320,5040,808393,2375736,4586456,6077904,5599440,3541440,1471680,362880,40320

%N Triangle read by rows: T(n, k) = (Sum_{i=0..n-k} (-1)^i * binomial(n-k, i) * (n+2-i)!) * binomial(n, k) / ((k+1) * (k+2)) for 0 <= k <= n.

%F T(n, k) = n * (T(n-1, k-1) + T(n-1, k)) for 0 < k < n with initial values T(n, 0) = A000153(n+1) and T(n, n) = A000142(n).

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

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

%e Triangle T(n, k) starts:

%e n\k :      0       1       2       3       4       5       6      7     8

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

%e   0 :      1

%e   1 :      2       1

%e   2 :      7       6       2

%e   3 :     32      39      24       6

%e   4 :    181     284     252     120      24

%e   5 :   1214    2325    2680    1860     720     120

%e   6 :   9403   21234   30030   27240   15480    5040     720

%e   7 :  82508  214459  358848  400890  299040  143640   40320   5040

%e   8 : 808393 2375736 4586456 6077904 5599440 3541440 1471680 362880 40320

%e   etc.

%o (PARI) T(n, k) = { sum(i=0, n-k, (-1)^i * binomial(n-k, i) * (n+2-i)!) * binomial(n, k) / ((k+1) * (k+2)) }

%Y Cf. A000153 (column 0), A000142 (main diagonal and 1st subdiagonal).

%Y Cf. A000255 (alt. row sums).

%K nonn,easy,tabl

%O 0,2

%A _Werner Schulte_, May 20 2024