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A193465 Row sums of triangle A061312. 2
0, 2, 9, 52, 335, 2466, 20447, 189064, 1930959, 21603430, 262869959, 3457226268, 48880169351, 739429561066, 11918051268255, 203914545928336, 3691384616598047, 70491995143458894, 1416242276574905879, 29862732908481855460, 659413025994777460119 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

LINKS

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

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

EXAMPLE

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

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 *)

PROG

(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 */

CROSSREFS

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

Sequence in context: A246464 A009310 A091319 * A003584 A301928 A069271

Adjacent sequences:  A193462 A193463 A193464 * A193466 A193467 A193468

KEYWORD

nonn,easy

AUTHOR

Johannes W. Meijer, Jul 27 2011

STATUS

approved

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)