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A118931
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Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3*k>=0.
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5
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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, 1, 572, 68640, 1601600, 3203200, 1, 728, 120120, 4484480, 22422400, 1, 910, 200200, 11211200, 112112000, 44844800
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OFFSET
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0,5
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COMMENTS
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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).
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LINKS
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FORMULA
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E.g.f.: A(x,y) = exp(x + y*x^3/3).
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EXAMPLE
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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;
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MAPLE
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Trow := n -> seq(n!/(j!*(n - 3*j)!*(3^j)), j = 0..n/3):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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