A006321 Royal paths in a lattice. (Formerly M4535)

1, 8, 48, 264, 1408, 7432, 39152, 206600, 1093760, 5813000, 31019568, 166188552, 893763840, 4823997960, 26124870640, 141926904328, 773293020928, 4224773978632, 23139861329456, 127039971696392, 698993630524032, 3853860616119048, 21288789223825648

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..1000

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)

Formula

a(n) = (4/n)*sum(binomial(n, j)*binomial(n+3+j, n-1), j=0..n) (n>0). - Emeric Deutsch, Aug 19 2004

Recurrence: n*(n+4)*a(n) = (5*n^2+14*n+21)*a(n-1) + (5*n^2-4*n+12)*a(n-2) - (n-3)*(n+1)*a(n-3). - Vaclav Kotesovec, Oct 05 2012

a(n) ~ 2*sqrt(816+577*sqrt(2))*(3+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 05 2012

G.f.: (x^4-8*x^3+16*x^2-8*x+1+sqrt(x^2-6*x+1)*(x-1)*(x^2-4*x+1))/(2*x^4). - Mark van Hoeij, Apr 16 2013

Maple

1, seq(4*sum(binomial(n, j)*binomial(n+3+j, n-1), j=0..n)/n, n=1..17);

Mathematica

Flatten[{1, RecurrenceTable[{n*(n+4)*a[n] == (5*n^2+14*n+21)*a[n-1] + (5*n^2-4*n+12)*a[n-2] - (n-3)*(n+1)*a[n-3], a[1] == 8, a[2] == 48, a[3] == 264}, a, {n, 25}]}] (* Vaclav Kotesovec, Oct 05 2012 *)

Crossrefs

Fourth diagonal of A033877.

Sequence in context: A026761 A026706 A128734 * A371620 A295047 A295375

Adjacent sequences: A006318 A006319 A006320 * A006322 A006323 A006324

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vincenzo Librandi, May 03 2013

Status

approved