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A248668 Sum of the numbers in the n-th row of the array at A248664.

%I A248668 #18 May 01 2021 05:11:37
%S A248668 1,4,26,260,3610,64472,1409006,36432076,1087911890,36844580000,
%T A248668 1395429571222,58439837713556,2681526361893626,133783187672365480,
%U A248668 7210345924097089790,417482356526745344732,25844171201928905477026,1703359919973405018460976
%N A248668 Sum of the numbers in the n-th row of the array at A248664.
%C A248668 The polynomial p(n,x) is defined as the numerator when the sum 1 + 1/(n*x + 1) + 1/((n*x + 1)(n*x + 2)) + ... + 1/((n*x + 1)(n*x + 2)...(n*x + n - 1)) is written as a fraction with denominator (n*x + 1)(n*x + 2)...(n*x + n - 1). For more, see A248664.
%H A248668 Seiichi Manyama, <a href="/A248668/b248668.txt">Table of n, a(n) for n = 1..366</a>
%F A248668 a(n) = p(n,1), where p(n,x) is defined at A248664.
%F A248668 a(n) = Sum_{k = 0..n-1} k!*binomial(2*n-1,k). - _Peter Bala_, Nov 14 2017
%F A248668 a(n) = A294039(n) - Pochhammer(n, n)*A000522(n). - _Peter Luschny_, Nov 14 2017
%e A248668 The first six polynomials:
%e A248668 p(1,x) = 1
%e A248668 p(2,x) = 2 (x + 1)
%e A248668 p(3,x) = 9x^2 + 12 x +  5
%e A248668 p(4,x) = 4 (16 x^3 + 28 x^2 + 17 x + 4)
%e A248668 p(5,x) = 5 (125 x^4 + 275 x^3 + 225 x^2 + 84 x + 13)
%e A248668 p(6,x) = 2 (3888 x^5 + 10368 x^4 + 10800 x^3 + 5562 x^2 + 1455 x + 163), so that
%e A248668 a(1) = p(1,1) = 1, a(2) = p(2,1) = 4, a(3) = p(3,1) = 26.
%p A248668 with (combinat):
%p A248668 seq(add( k!*binomial(2*n-1,k),k = 0..n-1 ), n = 0..20);
%p A248668 # _Peter Bala_, Nov 14 2017
%t A248668 t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];
%t A248668 p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
%t A248668 TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]
%t A248668 c[n_] := c[n] = CoefficientList[p[x, n], x];
%t A248668 TableForm[Table[c[n], {n, 1, 10}]] (* A248664 array *)
%t A248668 Table[Apply[Plus, c[n]], {n, 1, 60}] (* A248668 *)
%o A248668 (PARI) a(n) = sum(k = 0, n-1, k!*binomial(2*n-1,k)); \\ _Michel Marcus_, Nov 15 2017
%Y A248668 Cf. A000522, A248664, A248665, A248666, A248669, A294039.
%K A248668 nonn,easy
%O A248668 1,2
%A A248668 _Clark Kimberling_, Oct 11 2014