|
|
A005324
|
|
Column of Motzkin triangle A026300.
(Formerly M3902)
|
|
4
|
|
|
1, 5, 20, 70, 230, 726, 2235, 6765, 20240, 60060, 177177, 520455, 1524120, 4453320, 12991230, 37854954, 110218905, 320751445, 933149470, 2714401580, 7895719634, 22969224850, 66829893650, 194486929650, 566141346225, 1648500576021
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
4,2
|
|
COMMENTS
|
a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 0, s(n) = 4. - Clark Kimberling
a(n) = T(n,n-4), where T is the array in A026300.
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
|
|
FORMULA
|
G.f.: z^4*M^5, where M is g.f. of Motzkin numbers (A001006).
a(n) = (-5*I*(-1)^n*(n^4-6*n^3-43*n^2-24*n+36)*3^(1/2)*hypergeom([1/2, n+2],[1],4/3)+15*I*(-1)^n*(n^4+6*n^3+17*n^2+24*n-12)*3^(1/2)*hypergeom([1/2, n+1],[1],4/3))/(6*(n+3)*(n+2)*(n+4)*(n+5)*(n+6)). - Mark van Hoeij, Oct 29 2011
a(n) (n + 6) (n - 4) = n (2 n + 1) a(n - 1) + 3 n (n - 1) a(n - 2). - Simon Plouffe, Feb 09 2012, corrected for offset Aug 17 2022
a(n) = 5*sum(j=ceiling((n-4)/2)..(n+1), binomial(j,2*j-n+4)*binomial(n+1,j))/(n+1). - Vladimir Kruchinin, Mar 17 2014
|
|
MAPLE
|
if n <= 6 then
op(n-3, [1, 5, 20]) ;
else
n*(2*n+1)*procname(n-1)+3*n*(n-1)*procname(n-2) ;
%/(n+6)/(n-4) ;
end if;
end proc:
|
|
MATHEMATICA
|
T[n_, k_] := Sum[m = 2j+n-k; Binomial[n, m] (Binomial[m, j] - Binomial[m, j-1]), {j, 0, k/2}];
a[n_] := T[n, n-4];
|
|
PROG
|
(Maxima) a(n) := 5*sum(binomial(j, 2*j-n+4)*binomial(n+1, j), j, ceiling((n-4)/2), (n+1))/(n+1); /* Vladimir Kruchinin, Mar 18 2014 */
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|