A121548 - OEIS

A121548

Triangle read by rows: T(n,k) is the number of compositions of n into k Fibonacci numbers (1 <= k <= n; only one 1 is considered as a Fibonacci number).

1, 1, 1, 1, 2, 1, 0, 3, 3, 1, 1, 2, 6, 4, 1, 0, 3, 7, 10, 5, 1, 0, 2, 9, 16, 15, 6, 1, 1, 2, 9, 23, 30, 21, 7, 1, 0, 2, 10, 28, 50, 50, 28, 8, 1, 0, 3, 9, 34, 71, 96, 77, 36, 9, 1, 0, 2, 12, 36, 95, 156, 168, 112, 45, 10, 1, 0, 0, 12, 43, 115, 231, 308, 274, 156, 55, 11, 1, 1, 2, 9, 48, 140, 312, 504, 560, 423, 210, 66, 12, 1

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OFFSET

1,5

LINKS

Alois P. Heinz, Rows n = 0..150, flattened

FORMULA

G.f.: G(t,z) = 1 / (1 - t*Sum_{i>=2} z^Fibonacci(i)) - 1.

Sum of terms in row n = A076739(n).

T(n,1) = A010056(n) (the characteristic function of the Fibonacci numbers);

T(n,2) = A121549(n);

T(n,3) = A121550(n);

Sum_{k=1..n} k*T(n,k) = A121551(n).

EXAMPLE

T(5,3)=6 because we have [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1] and [3,1,1].

Triangle starts:

1, 1;

1, 2, 1;

0, 3, 3, 1;

1, 2, 6, 4, 1;

0, 3, 7, 10, 5, 1;

0, 2, 9, 16, 15, 6, 1;

...

MAPLE

with(combinat): G:=1/(1-t*sum(z^fibonacci(i), i=2..40))-1: Gser:=simplify(series(G, z=0, 25)): for n from 1 to 23 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 15 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form

# second Maple program:

g:= proc(n) g(n):= (t-> issqr(t+4) or issqr(t-4))(5*n^2) end:

T:= proc(n, t) option remember;

`if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(

`if`(g(j), T(n-j, t-1), 0), j=1..n)))

end:

seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Oct 10 2022

MATHEMATICA

nmax = 14;

T = Rest@CoefficientList[#, t]& /@ Rest@(1/(1 - t*Sum[z^Fibonacci[i],

{i, 2, nmax}]) - 1 + O[z]^(nmax+1) // CoefficientList[#, z]&);

Table[T[[n, k]], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 02 2022 *)