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

1,2

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.

Links

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

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A098620 A215266 A002465 * A079473 A145164 A113078

Adjacent sequences: A248665 A248666 A248667 * A248669 A248670 A248671

Keyword

nonn,easy

Author

Clark Kimberling, Oct 11 2014

Status

approved