A116383 Row sums of number triangle A116382.

1, 1, 4, 6, 19, 33, 93, 175, 460, 910, 2286, 4676, 11388, 23842, 56808, 120926, 283611, 611065, 1416625, 3079635, 7078263, 15490553, 35374519, 77805481, 176813809, 390379483, 883861033, 1957097715, 4418562265, 9805546875, 22090136885

Offset

0,3

Comments

Binomial transform is A116387.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

G.f.: (1+x-4*x^2)/(2*(1-5*x^2)*sqrt(1-4*x^2)) + (1+x)/(2*(1-5*x^2)).

a(n) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*C(n,j)*Sum_{i=0..j} C(j,i-k) * C(i,j-i).

Conjecture: n*a(n) + (n-2)*a(n-1) + (-13*n+20)*a(n-2) + (-9*n+22)*a(n-3) + 4*(14*n-45)*a(n-4) + 20*(n-3)*a(n-5) + 80*(-n+5)*a(n-6) = 0. - R. J. Mathar, Nov 24 2012

Recurrence: n*(n^2 - 4*n - 1)*a(n) = 4*(2*n-3)*a(n-1) + (9*n^3 - 40*n^2 + 3*n + 48)*a(n-2) - 20*(2*n-3)*a(n-3) - 20*(n-3)*(n^2 - 2*n - 4)*a(n-4). - Vaclav Kotesovec, Feb 12 2014

a(n) ~ (5+sqrt(5))/10 * 5^(n/2). - Vaclav Kotesovec, Feb 12 2014

Mathematica

CoefficientList[Series[(1+x-4*x^2)/(2*(1-5*x^2)*Sqrt[1-4*x^2])+(1+x)/(2*(1-5*x^2)), {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

Prog

(PARI) my(x='x+O('x^40)); Vec((1+x-4*x^2)/(2*(1-5*x^2)*sqrt(1-4*x^2)) + (1+x)/(2*(1-5*x^2))) \\ G. C. Greubel, May 22 2019
(Magma)
T:= func< n, k | (&+[(-1)^(n-j)*Binomial(n, j)*(&+[Binomial(j, m-k)* Binomial(m, j-m): m in [0..j]]): j in [0..n]]) >;
[(&+[T(n, k): k in [0..n]]): n in [0..40]]; // G. C. Greubel, May 22 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1+x-4*x^2)/(2*(1-5*x^2)*Sqrt(1-4*x^2)) + (1+x)/(2*(1-5*x^2)) )); // G. C. Greubel, May 22 2019
(Sage) ((1+x-4*x^2)/(2*(1-5*x^2)*sqrt(1-4*x^2)) + (1+x)/(2*(1-5*x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 22 2019
(GAP) List([0..40], n-> Sum([0..n], k-> Sum([0..n], j-> (-1)^(n-j)* Binomial(n, j)*Sum([0..j], m-> Binomial(j, m-k)*Binomial(m, j-m) )))) # G. C. Greubel, May 22 2019

Crossrefs

Sequence in context: A024697 A024874 A095383 * A026521 A343677 A222379

Adjacent sequences: A116380 A116381 A116382 * A116384 A116385 A116386

Keyword

easy,nonn

Author

Paul Barry, Feb 12 2006

Status

approved