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A116383 Row sums of number triangle A116382. 3
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 (list; graph; refs; listen; history; text; internal format)
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 A222379 A202618

Adjacent sequences:  A116380 A116381 A116382 * A116384 A116385 A116386

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Feb 12 2006

STATUS

approved

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Last modified November 12 13:43 EST 2019. Contains 329058 sequences. (Running on oeis4.)