OFFSET
0,5
COMMENTS
Row n contains 1+floor(n/3) terms. Row sums yield A001470. Given column vector V = A118932, then V is invariant under matrix product T*V = V, or, A118932(n) = Sum_{k=0..n} T(n,k)*A118932(k). Given C = Pascal's triangle and T = this triangle, then matrix product M = C^-1*T yields M(3n,n) = (3*n)!/(n!*3^n), 0 otherwise (cf. A100861 formula due to Paul Barry).
LINKS
G. C. Greubel, Rows n = 0..150 of the triangle, flattened
FORMULA
E.g.f.: A(x,y) = exp(x + y*x^3/3).
EXAMPLE
Triangle T begins:
1;
1;
1;
1, 2;
1, 8;
1, 20;
1, 40, 40;
1, 70, 280;
1, 112, 1120;
1, 168, 3360, 2240;
1, 240, 8400, 22400;
1, 330, 18480, 123200;
1, 440, 36960, 492800, 246400;
MAPLE
Trow := n -> seq(n!/(j!*(n - 3*j)!*(3^j)), j = 0..n/3):
seq(Trow(n), n = 0..14); # Peter Luschny, Jun 06 2021
MATHEMATICA
T[n_, k_]:= If[n<3*k, 0, n!/(3^k*k!*(n-3*k)!)];
Table[T[n, k], {n, 0, 20}, {k, 0, Floor[n/3]}]//Flatten (* G. C. Greubel, Mar 07 2021 *)
PROG
(PARI) T(n, k)=if(n<3*k, 0, n!/(k!*(n-3*k)!*3^k))
(Sage)
f=factorial;
flatten([[0 if n<3*k else f(n)/(3^k*f(k)*f(n-3*k)) for k in [0..n/3]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021
(Magma)
F:= Factorial;
[n lt 3*k select 0 else F(n)/(3^k*F(k)*F(n-3*k)): k in [0..Floor(n/3)], n in [0..20]]; // G. C. Greubel, Mar 07 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, May 06 2006
STATUS
approved