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A168621 Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,k) = ((n - 1)! + 1)*binomial(n, k) for 1 <= k <= n - 1, n >= 2.

%I A168621 #19 Dec 22 2018 15:57:01
%S A168621 1,1,1,1,4,1,1,9,9,1,1,28,42,28,1,1,125,250,250,125,1,1,726,1815,2420,
%T A168621 1815,726,1,1,5047,15141,25235,25235,15141,5047,1,1,40328,141148,
%U A168621 282296,352870,282296,141148,40328,1,1,362889,1451556,3386964,5080446,5080446,3386964,1451556,362889,1
%N A168621 Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,k) = ((n - 1)! + 1)*binomial(n, k) for 1 <= k <= n - 1, n >= 2.
%C A168621 Row 0 is 1, and row n gives the coefficients in the expansion of (x + 1)^n + (n - 1)!*((x + 1)^n - x^n -1). - _Franck Maminirina Ramaharo_, Dec 22 2018
%F A168621 From _Franck Maminirina Ramaharo_, Dec 22 2018: (Start)
%F A168621 T(n,k) = A007318(n,k) + A219570(n,k) for 1 <= k <= n - 1, n >= 2.
%F A168621 E.g.f.: exp((1 + x)*y) + log((1 - y)*(1 - x*y)/(1 - (1 + x)*y)). (End)
%e A168621 Triangle begins:
%e A168621   1;
%e A168621   1,     1;
%e A168621   1,     4,      1;
%e A168621   1,     9,      9,      1;
%e A168621   1,    28,     42,     28,      1;
%e A168621   1,   125,    250,    250,    125,      1;
%e A168621   1,   726,   1815,   2420,   1815,    726,      1;
%e A168621   1,  5047,  15141,  25235,  25235,  15141,   5047,     1;
%e A168621   1, 40328, 141148, 282296, 352870, 282296, 141148, 40328, 1;
%e A168621   ...
%t A168621 p[x_, n_] := If[n == 0, 1, (x + 1)^n + (n - 1)!*((x + 1)^n - x^n - 1)];
%t A168621 Table[CoefficientList[p[x, n], x], {n, 0, 12}] // Flatten (* _Franck Maminirina Ramaharo_, Dec 22 2018 *)
%o A168621 (Maxima) T(n, k) := if k = 0 or k = n then 1 else ((n - 1)! + 1)*binomial(n, k)$
%o A168621 create_list(T(n, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Dec 22 2018 */
%Y A168621 Row sums: A154252.
%Y A168621 Cf. A007318, A132047, A155863, A155864, A155865 A168619, A219570.
%K A168621 nonn,tabl,easy
%O A168621 0,5
%A A168621 _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009
%E A168621 Edited by _Franck Maminirina Ramaharo_, Dec 22 2018 (previous definition and examples were the same as A168620, but with different entries, as pointed out by _R. J. Mathar_, Oct 21 2012)