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A257635 Triangle with n-th row polynomial equal to Product_{k = 1..n} (x + n + k).

%I A257635 #23 Mar 21 2022 11:46:34
%S A257635 1,2,1,12,7,1,120,74,15,1,1680,1066,251,26,1,30240,19524,5000,635,40,
%T A257635 1,665280,434568,117454,16815,1345,57,1,17297280,11393808,3197348,
%U A257635 495544,45815,2527,77,1,518918400,343976400,99236556,16275700,1659889,107800,4354,100,1
%N A257635 Triangle with n-th row polynomial equal to Product_{k = 1..n} (x + n + k).
%C A257635 The row polynomials are a Sheffer sequence. For the associated polynomial sequence of binomial type see A038455.
%H A257635 R. Sprugnoli, <a href="https://web.archive.org/web/20170401103408/http://www.dsi.unifi.it/~resp/Handbook.pdf">An Introduction to Mathematical Methods in Combinatorics, Section 5.6</a> CreateSpace Independent Publishing Platform 2006, ISBN-13: 978-1502925244.
%H A257635 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sheffer_sequence">Sheffer sequence</a>
%F A257635 E.g.f.: A(x,t) = B(t)*C(t)^x = 1 + (2 + x)*t + (3 + x)*(4 + x)*t^2/2! + (4 + x)*(5 + x)*(6 + x)*t^3/3! + ..., where B(t) = 1/sqrt(1 - 4*t) is the o.g.f. for A000984 and C(t) = (1 - sqrt(1 - 4*t))/(2*t) is the o.g.f. for A000108.
%F A257635 n-th row polynomial: n!*binomial(2*n + x,n).
%F A257635 T(n, k) = (-1)^k*n!*[x^k] hypergeom([-n, -x + n], [-n], 1). - _Peter Luschny_, Nov 27 2021
%F A257635 T(n, k) = [x^k] Gamma(2*n + x + 1) / Gamma(n + x + 1). - _Peter Luschny_, Mar 21 2022
%e A257635 Triangle begins:
%e A257635 [0]       1;
%e A257635 [1]       2,      1;
%e A257635 [2]      12,      7,      1;
%e A257635 [3]     120,     74,     15,     1;
%e A257635 [4]    1680,   1066,    251,    26,    1;
%e A257635 [5]   30240,  19524,   5000,   635,   40,  1;
%e A257635 [6]  665280, 434568, 117454, 16815, 1345, 57, 1;
%e A257635   ...
%p A257635 seq(seq(coeff(product(n + x + k, k = 1 .. n), x, i), i = 0..n), n = 0..8);
%p A257635 # Alternative:
%p A257635 p := n -> n!*hypergeom([-n, -x + n], [-n], 1):
%p A257635 seq(seq((-1)^k*coeff(simplify(p(n)), x, k), k=0..n), n=0..6); # _Peter Luschny_, Nov 27 2021
%t A257635 p[n_, x_] := FunctionExpand[Gamma[2*n + x + 1] / Gamma[n + x + 1]];
%t A257635 Table[CoefficientList[p[n, x], x], {n,0,8}] // Flatten (* _Peter Luschny_, Mar 21 2022 *)
%Y A257635 Cf. A001813 (column 0), A005449 (first subdiagonal), A098118 (column 1).
%Y A257635 Cf. A006963 (row sums), A000407 (alternating row sum).
%Y A257635 Cf. A054649, A000108, A000984, A038455, A092932.
%K A257635 nonn,tabl,easy
%O A257635 0,2
%A A257635 _Peter Bala_, Nov 05 2015