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A006603 Generalized Fibonacci numbers.
(Formerly M1771)
7
1, 2, 7, 26, 107, 468, 2141, 10124, 49101, 242934, 1221427, 6222838, 32056215, 166690696, 873798681, 4612654808, 24499322137, 130830894666, 702037771647, 3783431872018, 20469182526595, 111133368084892, 605312629105205, 3306633429423460, 18111655081108453 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The Kn21 sums, see A180662, of the Schroeder triangle A033877 equal A006603(n) while the Kn3 sums equal A006603(2*n). The Kn22 sums, see A227504, and the Kn23 sums, see A227505, are also related to the sequence given above. - _ Johannes W. Meijer_, Jul 15 2013

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.

FORMULA

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

a(n) = sum(k=1..n/2+1, (k*sum(i=0..n-2*k+2, C(n-k+2,i)*C(2*n-3*k-i+3,n-k+1))) / (n-k+2)). - Vladimir Kruchinin, Oct 23 2011

MAPLE

A006603 := n-> add((k*add(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i= 0.. n-2*k+2))/(n-k+2), k= 1.. n/2+1): seq(A006603(n), n=0..24); # Johannes W. Meijer, Jul 15 2013

MATHEMATICA

CoefficientList[Series[(1-x-2x^2-Sqrt[1-6x+x^2])/(2x(1-x+x^2+x^3)), {x, 0, 30}], x] (* Harvey P. Dale, Jun 12 2016 *)

PROG

(Maxima) a(n):=sum((k*sum(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i, 0, n-2*k+2))/(n-k+2), k, 1, n/2+1); [Vladimir Kruchinin, Oct 23 2011]

CROSSREFS

a(n) = abs(A080244(n-1)).

Sequence in context: A150568 A102319 * A080244 A124542 A003447 A150569

Adjacent sequences:  A006600 A006601 A006602 * A006604 A006605 A006606

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

More terms from Emeric Deutsch, Feb 28 2004

STATUS

approved

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Last modified October 20 17:39 EDT 2017. Contains 293648 sequences.