A346415 Triangle T(n,k), n>=0, 0<=k<=n, read by rows, where column k is (1/k!) times the k-fold exponential convolution of Fibonacci numbers with themselves.
1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 3, 11, 6, 1, 0, 5, 35, 35, 10, 1, 0, 8, 115, 180, 85, 15, 1, 0, 13, 371, 910, 630, 175, 21, 1, 0, 21, 1203, 4494, 4445, 1750, 322, 28, 1, 0, 34, 3891, 22049, 30282, 16275, 4158, 546, 36, 1, 0, 55, 12595, 107580, 202565, 144375, 49035, 8820, 870, 45, 1
Offset: 0
Examples
Triangle T(n,k) begins: 1; 0, 1; 0, 1, 1; 0, 2, 3, 1; 0, 3, 11, 6, 1; 0, 5, 35, 35, 10, 1; 0, 8, 115, 180, 85, 15, 1; 0, 13, 371, 910, 630, 175, 21, 1; 0, 21, 1203, 4494, 4445, 1750, 322, 28, 1; 0, 34, 3891, 22049, 30282, 16275, 4158, 546, 36, 1; 0, 55, 12595, 107580, 202565, 144375, 49035, 8820, 870, 45, 1; ...
Links
- Alois P. Heinz, Rows n = 0..140, flattened
Crossrefs
Programs
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Maple
b:= proc(n) option remember; `if`(n=0, 1, add(expand(x*b(n-j) *binomial(n-1, j-1)*(<<0|1>, <1|1>>^j)[1, 2]), j=1..n)) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)): seq(T(n), n=0..12); # second Maple program: b:= proc(n, k) option remember; `if`(k=0, 0^n, `if`(k=1, combinat[fibonacci](n), (q-> add(binomial(n, j)* b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))) end: T:= (n, k)-> b(n, k)/k!: seq(seq(T(n, k), k=0..n), n=0..12); -
Mathematica
b[n_, k_] := b[n, k] = If[k == 0, 0^n, If[k == 1, Fibonacci[n], With[{q = Quotient[k, 2]}, Sum[Binomial[n, j] b[j, q] b[n-j, k-q], {j, 0, n}]]]]; T[n_, k_] := b[n, k]/k!; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 06 2021, after 2nd Maple program *)
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