A341072 -id:A341072 - cp's OEIS frontend

A121548 Triangle read by rows: T(n,k) is the number of compositions of n into k Fibonacci numbers (1 <= k <= n; only one 1 is considered as a Fibonacci number).

1, 1, 1, 1, 2, 1, 0, 3, 3, 1, 1, 2, 6, 4, 1, 0, 3, 7, 10, 5, 1, 0, 2, 9, 16, 15, 6, 1, 1, 2, 9, 23, 30, 21, 7, 1, 0, 2, 10, 28, 50, 50, 28, 8, 1, 0, 3, 9, 34, 71, 96, 77, 36, 9, 1, 0, 2, 12, 36, 95, 156, 168, 112, 45, 10, 1, 0, 0, 12, 43, 115, 231, 308, 274, 156, 55, 11, 1, 1, 2, 9, 48, 140, 312, 504, 560, 423, 210, 66, 12, 1

Offset: 1

Emeric Deutsch, Aug 07 2006

			T(5,3)=6 because we have [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1] and [3,1,1].
Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  0,  3,  3,  1;
  1,  2,  6,  4,  1;
  0,  3,  7, 10,  5,  1;
  0,  2,  9, 16, 15,  6,  1;
  ...

Alois P. Heinz, Rows n = 0..150, flattened

Cf. A076739, A010056, A121549, A121550, A121551.

T(2n,n) gives A341072.

with(combinat): G:=1/(1-t*sum(z^fibonacci(i),i=2..40))-1: Gser:=simplify(series(G,z=0,25)): for n from 1 to 23 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 15 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form
# second Maple program:
g:= proc(n) g(n):= (t-> issqr(t+4) or issqr(t-4))(5*n^2) end:
T:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(g(j), T(n-j, t-1), 0), j=1..n)))
    end:
seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Oct 10 2022

nmax = 14;
T = Rest@CoefficientList[#, t]& /@ Rest@(1/(1 - t*Sum[z^Fibonacci[i],
     {i, 2, nmax}]) - 1 + O[z]^(nmax+1) // CoefficientList[#, z]&);
Table[T[[n, k]], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 02 2022 *)

G.f.: G(t,z) = 1 / (1 - t*Sum_{i>=2} z^Fibonacci(i)) - 1.
Sum of terms in row n = A076739(n).
T(n,1) = A010056(n) (the characteristic function of the Fibonacci numbers);
T(n,2) = A121549(n);
T(n,3) = A121550(n);
Sum_{k=1..n} k*T(n,k) = A121551(n).

A341071 Number of compositions of 2n into n squarefree parts.

Original entry on oeis.org

1, 1, 3, 7, 23, 76, 267, 932, 3263, 11410, 40028, 140955, 498467, 1769249, 6299282, 22485487, 80435919, 288275461, 1034862558, 3720499175, 13393720428, 48275699654, 174194970601, 629187941167, 2274710068067, 8230764250326, 29805342037721, 108009638665648

Offset: 0

Alois P. Heinz, Feb 04 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A005117, A333047, A338464, A338465, A341072.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..35);

b[n_, t_] := b[n, t] =
  If[n == 0, If[t == 0, 1, 0], If[t < 1, 0, Sum[
  If[SquareFreeQ[j], b[n - j, t - 1], 0], {j, 1, n}]]];
a[n_] := b[2n, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 26 2023, from Maple program *)

a(n) ~ c * d^n / sqrt(n), where d = 3.6921332109291502908930783402282652076342401140592914... and c = 0.27220098720735952890181498005903942865018992276116... - Vaclav Kotesovec, Feb 14 2021

cp's OEIS Frontend

A121548 Triangle read by rows: T(n,k) is the number of compositions of n into k Fibonacci numbers (1 <= k <= n; only one 1 is considered as a Fibonacci number).

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