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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.
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%I #19 Nov 06 2021 09:48:37

%S 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,

%T 13,371,910,630,175,21,1,0,21,1203,4494,4445,1750,322,28,1,0,34,3891,

%U 22049,30282,16275,4158,546,36,1,0,55,12595,107580,202565,144375,49035,8820,870,45,1

%N 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.

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

%H Alois P. Heinz, <a href="/A346415/b346415.txt">Rows n = 0..140, flattened</a>

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 2, 3, 1;

%e 0, 3, 11, 6, 1;

%e 0, 5, 35, 35, 10, 1;

%e 0, 8, 115, 180, 85, 15, 1;

%e 0, 13, 371, 910, 630, 175, 21, 1;

%e 0, 21, 1203, 4494, 4445, 1750, 322, 28, 1;

%e 0, 34, 3891, 22049, 30282, 16275, 4158, 546, 36, 1;

%e 0, 55, 12595, 107580, 202565, 144375, 49035, 8820, 870, 45, 1;

%e ...

%p b:= proc(n) option remember; `if`(n=0, 1, add(expand(x*b(n-j)

%p *binomial(n-1, j-1)*(<<0|1>, <1|1>>^j)[1, 2]), j=1..n))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):

%p seq(T(n), n=0..12);

%p # second Maple program:

%p b:= proc(n, k) option remember; `if`(k=0, 0^n, `if`(k=1,

%p combinat[fibonacci](n), (q-> add(binomial(n, j)*

%p b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))

%p end:

%p T:= (n, k)-> b(n, k)/k!:

%p seq(seq(T(n, k), k=0..n), n=0..12);

%t 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 T[n_, k_] := b[n, k]/k!;

%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Nov 06 2021, after 2nd Maple program *)

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

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

%Y Row sums give A256180.

%K nonn,tabl

%O 0,8

%A _Alois P. Heinz_, Jul 15 2021