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A006320 Royal paths in a lattice.
(Formerly M4200)
5
1, 6, 30, 146, 714, 3534, 17718, 89898, 461010, 2386390, 12455118, 65478978, 346448538, 1843520670, 9859734630, 52974158938, 285791932578, 1547585781414, 8408765223294 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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

LINKS

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

G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).

G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

S.-n. Zheng and S.-l. Yang, On the-Shifted Central Coefficients of Riordan Matrices, Journal of Applied Mathematics, Volume 2014, Article ID 848374, 8 pages.

FORMULA

3-fold convolution of the large Schroeder numbers (A006318). G.f.=R^3, where R=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of A006318. - Emeric Deutsch, Mar 15 2004

a(n) = (3/n)*sum(binomial(n, j)*binomial(n+2+j, n-1), j=0..n) (n>0). - Emeric Deutsch, Aug 19 2004

Recurrence: (n+3)*(5*n-1)*a(n) = 2*(15*n^2+20*n+13)*a(n-1) - (5*n^2+5*n-24)*a(n-2) + (n-3)*a(n-3). - Vaclav Kotesovec, Oct 05 2012

a(n) ~ 3 * (1 + sqrt(2))^(2*n+3) / (2^(3/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 05 2012, simplified Dec 24 2017

MAPLE

1, seq(3*sum(binomial(n, j)*binomial(n+2+j, n-1), j=0..n)/n, n=1..18);

MATHEMATICA

Table[SeriesCoefficient[(1-x-Sqrt[1-6*x+x^2])^3/(8*x^3), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 05 2012 *)

CROSSREFS

Third diagonal of A033877.

Cf. A006318.

Sequence in context: A307494 A316593 A089817 * A319377 A079738 A127741

Adjacent sequences:  A006317 A006318 A006319 * A006321 A006322 A006323

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 24 18:55 EST 2021. Contains 341584 sequences. (Running on oeis4.)