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

Author

Alois P. Heinz, Jul 15 2021

Keywords

• nonn
• tabl

Comments

The sequence of column k>0 satisfies a linear recurrence with constant coefficients of order k+1.

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

Columns k=0-4 give: A000007, A000045, A014335, A014337, A014341.

T(n+j,n) for j=0-2 give: A000012, A000217, A000914.

Row sums give A256180.

Programs

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 *)