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A319394 Number T(n,k) of partitions of n into exactly k positive Fibonacci numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A319394 #31 Dec 17 2020 15:00:57
%S A319394 1,0,1,0,1,1,0,1,1,1,0,0,2,1,1,0,1,1,2,1,1,0,0,2,2,2,1,1,0,0,1,3,2,2,
%T A319394 1,1,0,1,1,2,4,2,2,1,1,0,0,1,3,3,4,2,2,1,1,0,0,2,2,4,4,4,2,2,1,1,0,0,
%U A319394 1,3,4,5,4,4,2,2,1,1,0,0,0,3,5,5,6,4,4,2,2,1,1
%N A319394 Number T(n,k) of partitions of n into exactly k positive Fibonacci numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A319394 T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k <= n.  T(n,k) = 0 for k > n.
%H A319394 Alois P. Heinz, <a href="/A319394/b319394.txt">Rows n = 0..200, flattened</a>
%F A319394 T(n,k) = [x^n y^k] 1/Product_{j>=2} (1-y*x^A000045(j)).
%F A319394 Sum_{k=1..n} k * T(n,k) = A281689(n).
%F A319394 T(A000045(n),n) = A319503(n).
%e A319394 T(14,3) = 2: 851, 833.
%e A319394 T(14,4) = 5: 8321, 8222, 5531, 5522, 5333.
%e A319394 T(14,5) = 6: 83111, 82211, 55211, 53321, 53222, 33332.
%e A319394 T(14,6) = 8: 821111, 551111, 533111, 532211, 522221, 333311, 333221, 332222.
%e A319394 T(14,7) = 7: 8111111, 5321111, 5222111, 3332111, 3322211, 3222221, 2222222.
%e A319394 T(14,8) = 6: 53111111, 52211111, 33311111, 33221111, 32222111, 22222211.
%e A319394 Triangle T(n,k) begins:
%e A319394   1;
%e A319394   0, 1;
%e A319394   0, 1, 1;
%e A319394   0, 1, 1, 1;
%e A319394   0, 0, 2, 1, 1;
%e A319394   0, 1, 1, 2, 1, 1;
%e A319394   0, 0, 2, 2, 2, 1, 1;
%e A319394   0, 0, 1, 3, 2, 2, 1, 1;
%e A319394   0, 1, 1, 2, 4, 2, 2, 1, 1;
%e A319394   0, 0, 1, 3, 3, 4, 2, 2, 1, 1;
%e A319394   0, 0, 2, 2, 4, 4, 4, 2, 2, 1, 1;
%e A319394   0, 0, 1, 3, 4, 5, 4, 4, 2, 2, 1, 1;
%e A319394   0, 0, 0, 3, 5, 5, 6, 4, 4, 2, 2, 1, 1;
%e A319394   0, 1, 1, 2, 4, 7, 6, 6, 4, 4, 2, 2, 1, 1;
%e A319394   0, 0, 1, 2, 5, 6, 8, 7, 6, 4, 4, 2, 2, 1, 1;
%e A319394   ...
%p A319394 h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
%p A319394       issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
%p A319394     end:
%p A319394 b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
%p A319394       b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
%p A319394     end:
%p A319394 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
%p A319394 seq(T(n), n=0..20);
%t A319394 T[n_, k_] := SeriesCoefficient[1/Product[(1 - y x^Fibonacci[j]) + O[x]^(n+1) // Normal, {j, 2, n+1}], {x, 0, n}, {y, 0, k}];
%t A319394 Table[T[n, k], {n, 0, 40}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 28 2020 *)
%t A319394 h[n_] := h[n] = If[n < 1, 0, If[Function[t, IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4]][5 n^2], n, h[n - 1]]];
%t A319394 b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1 || t < 1, 0, b[n, h[i - 1], t] + b[n - i, h[Min[n - i, i]], t - 1]]];
%t A319394 T[n_, k_] :=  b[n, h[n], k] - b[n, h[n], k - 1];
%t A319394 Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A319394 Columns k=0-10 give: A000007, A010056 (for n>0), A319395, A319396, A319397, A319398, A319399, A319400, A319401, A319402, A319403.
%Y A319394 Row sums give A003107.
%Y A319394 T(2n,n) gives A136343.
%Y A319394 Cf. A000045, A281689, A319503.
%K A319394 nonn,tabl
%O A319394 0,13
%A A319394 _Alois P. Heinz_, Sep 18 2018