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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).

%I A121548 #23 Oct 10 2022 10:42:45
%S A121548 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,
%T A121548 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,
%U A121548 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
%N 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).
%H A121548 Alois P. Heinz, <a href="/A121548/b121548.txt">Rows n = 0..150, flattened</a>
%F A121548 G.f.: G(t,z) = 1 / (1 - t*Sum_{i>=2} z^Fibonacci(i)) - 1.
%F A121548 Sum of terms in row n = A076739(n).
%F A121548 T(n,1) = A010056(n) (the characteristic function of the Fibonacci numbers);
%F A121548 T(n,2) = A121549(n);
%F A121548 T(n,3) = A121550(n);
%F A121548 Sum_{k=1..n} k*T(n,k) = A121551(n).
%e A121548 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].
%e A121548 Triangle starts:
%e A121548   1;
%e A121548   1,  1;
%e A121548   1,  2,  1;
%e A121548   0,  3,  3,  1;
%e A121548   1,  2,  6,  4,  1;
%e A121548   0,  3,  7, 10,  5,  1;
%e A121548   0,  2,  9, 16, 15,  6,  1;
%e A121548   ...
%p A121548 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
%p A121548 # second Maple program:
%p A121548 g:= proc(n) g(n):= (t-> issqr(t+4) or issqr(t-4))(5*n^2) end:
%p A121548 T:= proc(n, t) option remember;
%p A121548       `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
%p A121548       `if`(g(j), T(n-j, t-1), 0), j=1..n)))
%p A121548     end:
%p A121548 seq(seq(T(n, k), k=1..n), n=1..14); # _Alois P. Heinz_, Oct 10 2022
%t A121548 nmax = 14;
%t A121548 T = Rest@CoefficientList[#, t]& /@ Rest@(1/(1 - t*Sum[z^Fibonacci[i],
%t A121548      {i, 2, nmax}]) - 1 + O[z]^(nmax+1) // CoefficientList[#, z]&);
%t A121548 Table[T[[n, k]], {n, 1, nmax}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, May 02 2022 *)
%Y A121548 Cf. A076739, A010056, A121549, A121550, A121551.
%Y A121548 T(2n,n) gives A341072.
%K A121548 nonn,tabl
%O A121548 1,5
%A A121548 _Emeric Deutsch_, Aug 07 2006