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Sum of the numbers in the n-th row of the array at A248664.
8

%I #18 May 01 2021 05:11:37

%S 1,4,26,260,3610,64472,1409006,36432076,1087911890,36844580000,

%T 1395429571222,58439837713556,2681526361893626,133783187672365480,

%U 7210345924097089790,417482356526745344732,25844171201928905477026,1703359919973405018460976

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

%C 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 Seiichi Manyama, <a href="/A248668/b248668.txt">Table of n, a(n) for n = 1..366</a>

%F a(n) = p(n,1), where p(n,x) is defined at A248664.

%F a(n) = Sum_{k = 0..n-1} k!*binomial(2*n-1,k). - _Peter Bala_, Nov 14 2017

%F a(n) = A294039(n) - Pochhammer(n, n)*A000522(n). - _Peter Luschny_, Nov 14 2017

%e The first six polynomials:

%e p(1,x) = 1

%e p(2,x) = 2 (x + 1)

%e p(3,x) = 9x^2 + 12 x + 5

%e p(4,x) = 4 (16 x^3 + 28 x^2 + 17 x + 4)

%e p(5,x) = 5 (125 x^4 + 275 x^3 + 225 x^2 + 84 x + 13)

%e p(6,x) = 2 (3888 x^5 + 10368 x^4 + 10800 x^3 + 5562 x^2 + 1455 x + 163), so that

%e a(1) = p(1,1) = 1, a(2) = p(2,1) = 4, a(3) = p(3,1) = 26.

%p with (combinat):

%p seq(add( k!*binomial(2*n-1,k),k = 0..n-1 ), n = 0..20);

%p # _Peter Bala_, Nov 14 2017

%t t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];

%t p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];

%t TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]

%t c[n_] := c[n] = CoefficientList[p[x, n], x];

%t TableForm[Table[c[n], {n, 1, 10}]] (* A248664 array *)

%t Table[Apply[Plus, c[n]], {n, 1, 60}] (* A248668 *)

%o (PARI) a(n) = sum(k = 0, n-1, k!*binomial(2*n-1,k)); \\ _Michel Marcus_, Nov 15 2017

%Y Cf. A000522, A248664, A248665, A248666, A248669, A294039.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Oct 11 2014