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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. 1
 1, 2, 4, 6, 9, 9, 24, 56, 24, 16, 120, 250, 275, 50, 25, 720, 1884, 1350, 960, 90, 36, 5040, 12348, 14896, 5145, 2695, 147, 49, 40320, 114624, 105056, 80416, 15680, 6496, 224, 64, 362880, 986256, 1282284, 605556, 336609, 40824, 13986, 324, 81 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A special case of Hilbert polynomials when the degree is equal to n+1. REFERENCES Brendan Hassett, Introduction to algebraic Geometry,Cambridge University Press. New York,2007, page 214 LINKS G. C. Greubel, Rows n = 0..50 of the triangle, flattened FORMULA T(n, k) = coefficients of n!*(binomial(x+n+1, n+1) - binomial(x, n+1)). From G. C. Greubel, Apr 14 2021: (Start) T(n, k) = coefficients of Sum_{j=0..n+1} StirlingS1(n+1, j)*( (x+n+1)^j - x^j ). T(n, 0) = (n+1)!. T(n, n) = (n+1)^2. Sum_{k=0..n} T(n,k) = (n+2)! - [n=0]. (End) EXAMPLE Triangle begins as:         1;         2,        4;         6,        9,        9;        24,       56,       24,      16;       120,      250,      275,      50,      25;       720,     1884,     1350,     960,      90,      36;      5040,    12348,    14896,    5145,    2695,     147,    49;     40320,   114624,   105056,   80416,   15680,    6496,   224,    64;    362880,   986256,  1282284,  605556,  336609,   40824, 13986,   324,  81;   3628800, 10991520, 11727000, 9582200, 2693250, 1171380, 94500, 27600, 450, 100; MATHEMATICA 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]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 14 2021 *) PROG (Sage) 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] flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 14 2021 CROSSREFS Cf. A048994, A139167. Sequence in context: A084407 A114526 A333387 * A162202 A210380 A228359 Adjacent sequences:  A178123 A178124 A178125 * A178127 A178128 A178129 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, May 20 2010 EXTENSIONS Edited by G. C. Greubel, Apr 14 2021 STATUS approved

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Last modified June 24 18:46 EDT 2021. Contains 345419 sequences. (Running on oeis4.)