A320329 Row sums of A174790.

1, 2, 7, 76, 1301, 26406, 619207, 16652168, 508596489, 17457431050, 666726681611, 28076838451212, 1293333060096013, 64713740778086414, 3495868307630899215, 202800355058036736016, 12574907509808996352017, 829987773918052958208018, 58100729276791270637568019

Offset

0,2

Comments

Except for n = 2, 3, 4 and 9, the A055642(n) least significant digits of a(n) give the decimal expansion of n + 1. - Stefano Spezia, Jul 02 2021

Links

Stefano Spezia, Table of n, a(n) for n = 0..300

Formula

a(n) = Sum_{m=0..n} 1 + (binomial(n, m) - 1)*(n!)^2/(m!*(n - m)!).

a(0) = 1, a(n) = 1 + n - 2^n*n! + 2*(2*n - 1)!/(n - 1)! for n > 0.

From Stefano Spezia, Jul 02 2021: (Start)

E.g.f.: 1/sqrt(1 - 4*x) + exp(x)*(1 + x) + 1/(2*x - 1).

a(n) ~ sqrt(2)*4^n*exp(-n)*n^n. (End)

Maple

a := n -> add(1+(binomial(n, m)-1)*(n!)^2/(m!*(n-m)!), m = 0 .. n): seq(a(n), n = 0 .. 20);

Mathematica

T[n_, m_] = 1+((Binomial[n, m]-1)(n!)^2)/(m!(n-m)!); Table[Sum[T[n, m], {m, 0, n}], {n, 0, 20}] (* or *)
a[n_]:=1+n-2^n n!+2(2n-1)!/(n-1)!; Join[{1}, Array[a, 20]]

Prog

(GAP) List([0..20], n->Sum([0..n], m->1+((Binomial(n, m)-1)*(Factorial(n)^2)/(Factorial(m)*Factorial(n-m)))));
(PARI) a(n) = sum(m=0, n, 1+(binomial(n, m)-1)*(n!)^2/(m!*(n-m)!));

Crossrefs

Cf. A000079, A000142, A000302, A174790.

Sequence in context: A144905 A349500 A266880 * A083455 A083829 A067963

Adjacent sequences: A320326 A320327 A320328 * A320330 A320331 A320332

Keyword

nonn

Author

Stefano Spezia, Dec 18 2018

Status

approved