The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 5
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS The sequence of column k>0 satisfies a linear recurrence with constant coefficients of order k+1. LINKS Alois P. Heinz, Rows n = 0..140, flattened EXAMPLE 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; ... 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 *) 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. Sequence in context: A130717 A137396 A244213 * A352067 A178245 A338753 Adjacent sequences: A346412 A346413 A346414 * A346416 A346417 A346418 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 15 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 26 10:31 EDT 2023. Contains 361540 sequences. (Running on oeis4.)