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%I #10 Jun 05 2021 03:07:50
%S 1,1,3,9,35,168,967,6538,50831,446919,4383861,47451921,561715093,
%T 7217604520,100031995789,1487319385140,23613262336093,398673670050021,
%U 7132188802005991,134766129577134553,2681929390235577831
%N Diagonal sums of triangle A155856.
%H G. C. Greubel, <a href="/A155858/b155858.txt">Table of n, a(n) for n = 0..440</a>
%F G.f.: 1/(1 -x^2 -x/(1 -x^2 -x/(1 -x^2 -2*x/(1 -x^2 -2*x/(1 -x^2 -3*x/(1 -x^2 -3*x/(1 - ... (continued fraction);
%F a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k, k)*(n-2*k)!.
%F Conjecture: a(n) -(n-1)*a(n-1) -(n-2)*a(n-2) +(n-3)*a(n-3) +(n-10)*a(n-4) -5*a(n-5) +3*a(n-6) +3*a(n-7) = 0. - _R. J. Mathar_, Feb 05 2015
%F a(n) ~ n! * (1 + 2/n + 1/n^2 - 2/(3*n^3) - 22/(3*n^4) - 491/(15*n^5) - 11467/(90*n^6) - ...). - _Vaclav Kotesovec_, Jun 05 2021
%t Table[Sum[Binomial[2*n-3*k, k]*(n-2*k)!, {k,0,Floor[n/2]}], {n,0,30}] (* _G. C. Greubel_, Jun 05 2021 *)
%o (Sage) [sum( binomial(2*n-3*k, k)*factorial(n-2*k) for k in (0..n//2) ) for n in (0..30)] # _G. C. Greubel_, Jun 05 2021
%Y Cf. A155856.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Jan 29 2009
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