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

1, 4, 26, 260, 3610, 64472, 1409006, 36432076, 1087911890, 36844580000, 1395429571222, 58439837713556, 2681526361893626, 133783187672365480, 7210345924097089790, 417482356526745344732, 25844171201928905477026, 1703359919973405018460976

Offset: 1

Author

Clark Kimberling, Oct 11 2014

Keywords

• nonn
• easy

Comments

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.

Examples

			The first six polynomials:
p(1,x) = 1
p(2,x) = 2 (x + 1)
p(3,x) = 9x^2 + 12 x +  5
p(4,x) = 4 (16 x^3 + 28 x^2 + 17 x + 4)
p(5,x) = 5 (125 x^4 + 275 x^3 + 225 x^2 + 84 x + 13)
p(6,x) = 2 (3888 x^5 + 10368 x^4 + 10800 x^3 + 5562 x^2 + 1455 x + 163), so that
a(1) = p(1,1) = 1, a(2) = p(2,1) = 4, a(3) = p(3,1) = 26.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..366

Crossrefs

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

Programs

Maple

with (combinat):
seq(add( k!*binomial(2*n-1,k),k = 0..n-1 ), n = 0..20);
# Peter Bala, Nov 14 2017

Mathematica

t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];
p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]
c[n_] := c[n] = CoefficientList[p[x, n], x];
TableForm[Table[c[n], {n, 1, 10}]] (* A248664 array *)
Table[Apply[Plus, c[n]], {n, 1, 60}] (* A248668 *)

PARI

a(n) = sum(k = 0, n-1, k!*binomial(2*n-1,k)); \\ Michel Marcus, Nov 15 2017

Formula

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

a(n) = Sum_{k = 0..n-1} k!*binomial(2*n-1,k). - Peter Bala, Nov 14 2017

a(n) = A294039(n) - Pochhammer(n, n)*A000522(n). - Peter Luschny, Nov 14 2017