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A178126 Triangle T(n, k) = coefficients of (n+1)!*(binomial(x+n+1, n+1) - binomial(x, n+1)), read by rows.

%I A178126 #11 Aug 03 2021 01:52:47
%S A178126 1,2,4,6,9,9,24,56,24,16,120,250,275,50,25,720,1884,1350,960,90,36,
%T A178126 5040,12348,14896,5145,2695,147,49,40320,114624,105056,80416,15680,
%U A178126 6496,224,64,362880,986256,1282284,605556,336609,40824,13986,324,81
%N A178126 Triangle T(n, k) = coefficients of (n+1)!*(binomial(x+n+1, n+1) - binomial(x, n+1)), read by rows.
%D A178126 Brendan Hassett, Introduction to algebraic Geometry, Cambridge University Press, New York, 2007, page 214
%H A178126 G. C. Greubel, <a href="/A178126/b178126.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A178126 T(n, k) = coefficients of n!*(binomial(x+n+1, n+1) - binomial(x, n+1)).
%F A178126 From _G. C. Greubel_, Apr 14 2021: (Start)
%F A178126 T(n, k) = coefficients of Sum_{j=0..n+1} Stirling1(n+1, j)*( (x+n+1)^j - x^j ).
%F A178126 T(n, 0) = (n+1)!.
%F A178126 T(n, n) = (n+1)^2.
%F A178126 Sum_{k=0..n} T(n,k) = (n+2)! - [n=0]. (End)
%e A178126 Triangle begins as:
%e A178126         1;
%e A178126         2,        4;
%e A178126         6,        9,        9;
%e A178126        24,       56,       24,      16;
%e A178126       120,      250,      275,      50,      25;
%e A178126       720,     1884,     1350,     960,      90,      36;
%e A178126      5040,    12348,    14896,    5145,    2695,     147,    49;
%e A178126     40320,   114624,   105056,   80416,   15680,    6496,   224,    64;
%e A178126    362880,   986256,  1282284,  605556,  336609,   40824, 13986,   324,  81;
%e A178126   3628800, 10991520, 11727000, 9582200, 2693250, 1171380, 94500, 27600, 450, 100;
%t A178126 T[n_, k_]:= SeriesCoefficient[Series[Sum[StirlingS1[n+1, j]*((x+n+1)^j -x^j), {j, 0, n+1}], {x, 0, n+1}], k];
%t A178126 Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Apr 14 2021 *)
%o A178126 (Sage)
%o A178126 def T(n,k): return ( sum((-1)^(n+j+1)*stirling_number1(n+1, j)*((x+n+1)^j - x^j) for j in (0..n+1)) ).series(x,n+1).list()[k]
%o A178126 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 14 2021
%Y A178126 Cf. A048994, A139167.
%K A178126 nonn,tabl
%O A178126 0,2
%A A178126 _Roger L. Bagula_, May 20 2010
%E A178126 Edited by _G. C. Greubel_, Apr 14 2021