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A006321
Royal paths in a lattice.
(Formerly M4535)
4
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
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
KEYWORD
nonn
EXTENSIONS
More terms from Vincenzo Librandi, May 03 2013
STATUS
approved