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A193465 Row sums of triangle A061312.

%I A193465 #31 Sep 01 2021 20:31:14
%S A193465 0,2,9,52,335,2466,20447,189064,1930959,21603430,262869959,3457226268,
%T A193465 48880169351,739429561066,11918051268255,203914545928336,
%U A193465 3691384616598047,70491995143458894,1416242276574905879,29862732908481855460,659413025994777460119
%N A193465 Row sums of triangle A061312.
%C A193465 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
%H A193465 G. C. Greubel, <a href="/A193465/b193465.txt">Table of n, a(n) for n = 0..445</a>
%F A193465 a(n) = Sum_{k=0..n} A061312(n,k).
%F A193465 a(n) = (n+1)*A180191(n+1).
%F A193465 a(n) = A002467(n+2) - (n+1)! (the game of mousetrap with n cards).
%F A193465 a(n) = (n+1)*(n+1)! - A000166(n+2) (rencontres numbers).
%F A193465 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.
%F A193465 E.g.f: (1 + x - (1 + x^2) / exp(x)) / (1 - x)^3. - _Michael Somos_, Jun 06 2012
%F A193465 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
%F A193465 a(n) = Sum_{k=1..n+1} A047920(n+1,k). - _Alois P. Heinz_, Sep 01 2021
%e A193465 2*x + 9*x^2 + 52*x^3 + 335*x^4 + 2466*x^5 + 20447*x^6 + 189064*x^7 + ...
%p A193465 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);
%t A193465 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 *)
%o A193465 (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 */
%Y A193465 Cf. A000166, A001563, A002467, A047920, A061312, A180191, A074909.
%K A193465 nonn,easy
%O A193465 0,2
%A A193465 _Johannes W. Meijer_, Jul 27 2011