A193465 Row sums of triangle A061312.

0, 2, 9, 52, 335, 2466, 20447, 189064, 1930959, 21603430, 262869959, 3457226268, 48880169351, 739429561066, 11918051268255, 203914545928336, 3691384616598047, 70491995143458894, 1416242276574905879, 29862732908481855460, 659413025994777460119

Offset: 0

Author

Johannes W. Meijer, Jul 27 2011

Keywords

• nonn
• easy

Comments

a(n) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = A001563(k) for k = 0, 1, ..., n. - Michael Somos, Jun 06 2012

Examples

			2*x + 9*x^2 + 52*x^3 + 335*x^4 + 2466*x^5 + 20447*x^6 + 189064*x^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445

Crossrefs

Cf. A000166, A001563, A002467, A047920, A061312, A180191, A074909.

Programs

Maple

A193465 := proc(n): add(A061312(n,k), k=0..n) end: A061312:=proc(n,k): add(((-1)^j)*binomial(k+1,j)*(n+1-j)!, j=0..k+1) end: seq(A193465(n), n=0..20);

Mathematica

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (1 + x - (1 + x^2) / Exp[ x ]) / (1 - x)^3, {x, 0, n}]] (* Michael Somos, Jun 06 2012 *)

PARI

{a(n) = if( n<0, 0, n! * polcoeff( (1 + x - (1 + x^2) / exp(x + x * O(x^n))) / (1 - x)^3, n))} /* Michael Somos, Jun 06 2012 */

Formula

a(n) = Sum_{k=0..n} A061312(n,k).

a(n) = (n+1)*A180191(n+1).

a(n) = A002467(n+2) - (n+1)! (the game of mousetrap with n cards).

a(n) = (n+1)*(n+1)! - A000166(n+2) (rencontres numbers).

a(n) = ((n-n^3)*a(n-3) + (2*n+n^2-n^3)*a(n-2) - (1-n-2*n^2)*a(n-1))/n with a(0) = 0, a(1) = 2 and a(2) = 9.

E.g.f: (1 + x - (1 + x^2) / exp(x)) / (1 - x)^3. - Michael Somos, Jun 06 2012

a(n) = Sum_{k=0..n} C(n+1,k)*A000166(k+1) = Sum_{k=0..n} A074909(n,k)*A000166(k+1). - Anton Zakharov, Sep 26 2016

a(n) = Sum_{k=1..n+1} A047920(n+1,k). - Alois P. Heinz, Sep 01 2021