A112293 Row sums of number triangle A112292.

1, 2, 7, 36, 253, 2278, 25059, 325768, 4886521, 83070858, 1578346303, 33145272364, 762341264373, 19058531609326, 514580353451803, 14922830250102288, 462607737753170929, 15266055345854640658, 534311937104912423031

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..404

Formula

a(n) = Sum_{k=0..n} (2n-1)!!/(2k-1)!!.

a(n) = Sum_{k=0..n} 2^(n-k)(n-1/2)!/(k-1/2)!.

a(n) = Sum_{k=0..n} n!C(2n, n)2^(k-n)/(k!C(2k, k)).

a(n) = Sum_{k=0..n} (n+1)!*C(n)2^(k-n)/((k+1)!*C(k)).

Conjecture: a(n) - 2*n*a(n-1) + (2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 28 2014

a(n) = floor((2*n-1)!! * C) for n>0, where C = 1 + sqrt(e*Pi/2)*erf(1/sqrt(2)). - Don Knuth, Mar 25 2018

a(n) = 2^n*(C*Gamma(n + 1/2) + Gamma(n + 1/2, 1/2))*sqrt(e/2) for n >= 0, where C = sqrt(2/(e*Pi)) - erfc(1/sqrt(2)). - Peter Luschny, Mar 25 2018

a(n) = a(n-1) * (2*n-1) + 1 for n > 0 and a(0) = 1; that proves the conjecture of R. J. Mathar from Nov 28 2014; G.f. A(x) satisfies the equation: A(x) = 1/(1-x)^2 + A'(x) * 2*x^2/(1-x), where A' is the first derivative of A. - Werner Schulte, Oct 18 2023

Mathematica

T[n_, k_] := If[k <= n, (2 n - 1)!! / (2 k - 1)!!, 0];
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jun 13 2019 *)

Crossrefs

Cf. A001147, A060196, A286286.

Sequence in context: A143805 A249637 A259793 * A090352 A123549 A009704

Adjacent sequences: A112290 A112291 A112292 * A112294 A112295 A112296

Keyword

easy,nonn

Author

Paul Barry, Sep 01 2005

Status

approved