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%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
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