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A006604 Generalized Fibonacci numbers.
(Formerly M3469)
1
1, 1, 4, 13, 53, 228, 1037, 4885, 23640, 116793, 586633, 2986616, 15377097, 79927913, 418852716, 2210503285, 11738292397, 62673984492, 336260313765, 1811960161517, 9802082905840, 53213718977777, 289817858570513, 1583076422786096, 8670574105626961 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

D. G. Rogers, A Schroeder triangle: three combinatorial problems, in "Combinatorial Mathematics V (Melbourne 1976)", Lect. Notes Math. 622 (1976), pp. 175-196.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=0..24.

Plouffe, Simon, Approximations of generating functions and a few conjectures, Master's Thesis, arXiv:0911.4975 [math.NT], 2009.

FORMULA

G.f.: (1+x-2*x^2-sqrt(1-6*x+x^2))/(2*(2*x-x^2-x^3+x^4)).

n*a(n) = (-1/2*n+3/2)*a(n-5)+(7/2*n-6)*a(n-4) +(13/2*n-9)*a(n-1) +(-7/2*n+15/2) *a(n-2) +(-3*n+3)*a(n-3). - Simon Plouffe, Master's Thesis, UQAM, 1992

a(n) = sum(k=1..n/2+1, (k*sum(j=0..n-2*k+2, (-1)^j*2^(n-2*k-j+2)*C(n-k+2,j) * C(2*n-3*k-j+3,n-k+1)))/((n-k+2))). - Vladimir Kruchinin, Oct 22 2011

MATHEMATICA

CoefficientList[Series[(1+x-2 x^2-Sqrt[1-6 x+x^2])/(2 (2 x-x^2-x^3+x^4)), {x, 0, 60}], x]  (* Harvey P. Dale, Mar 23 2011 *)

PROG

(Maxima) a(n):=sum((k*sum((-1)^j*2^(n-2*k-j+2)*binomial(n-k+2, j)*binomial(2*n-3*k-j+3, n-k+1), j, 0, n-2*k+2))/((n-k+2)), k, 1, n/2+1); // Vladimir Kruchinin, Oct 22 2011

CROSSREFS

Sequence in context: A140454 A149465 A149466 * A082570 A145208 A149467

Adjacent sequences:  A006601 A006602 A006603 * A006605 A006606 A006607

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Harvey P. Dale, Mar 23 2011

STATUS

approved

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Last modified June 23 17:39 EDT 2021. Contains 345402 sequences. (Running on oeis4.)