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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.
20
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, 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, 1, 3, 4, 5, 4, 4, 2, 2, 1, 1, 0, 0, 0, 3, 5, 5, 6, 4, 4, 2, 2, 1, 1
OFFSET
0,13
COMMENTS
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.
LINKS
FORMULA
T(n,k) = [x^n y^k] 1/Product_{j>=2} (1-y*x^A000045(j)).
Sum_{k=1..n} k * T(n,k) = A281689(n).
T(A000045(n),n) = A319503(n).
EXAMPLE
T(14,3) = 2: 851, 833.
T(14,4) = 5: 8321, 8222, 5531, 5522, 5333.
T(14,5) = 6: 83111, 82211, 55211, 53321, 53222, 33332.
T(14,6) = 8: 821111, 551111, 533111, 532211, 522221, 333311, 333221, 332222.
T(14,7) = 7: 8111111, 5321111, 5222111, 3332111, 3322211, 3222221, 2222222.
T(14,8) = 6: 53111111, 52211111, 33311111, 33221111, 32222111, 22222211.
Triangle T(n,k) begins:
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, 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, 1, 3, 4, 5, 4, 4, 2, 2, 1, 1;
0, 0, 0, 3, 5, 5, 6, 4, 4, 2, 2, 1, 1;
0, 1, 1, 2, 4, 7, 6, 6, 4, 4, 2, 2, 1, 1;
0, 0, 1, 2, 5, 6, 8, 7, 6, 4, 4, 2, 2, 1, 1;
...
MAPLE
h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..20);
MATHEMATICA
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}];
Table[T[n, k], {n, 0, 40}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 28 2020 *)
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]]];
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[n_, k_] := b[n, h[n], k] - b[n, h[n], k - 1];
Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)
CROSSREFS
Columns k=0-10 give: A000007, A010056 (for n>0), A319395, A319396, A319397, A319398, A319399, A319400, A319401, A319402, A319403.
Row sums give A003107.
T(2n,n) gives A136343.
Sequence in context: A340379 A075685 A037906 * A278347 A120936 A335294
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 18 2018
STATUS
approved