A333047 -id:A333047 - cp's OEIS frontend

A073266 Triangle read by rows: T(n,k) is the number of compositions of n as the sum of k integral powers of 2.

1, 1, 1, 0, 2, 1, 1, 1, 3, 1, 0, 2, 3, 4, 1, 0, 2, 4, 6, 5, 1, 0, 0, 6, 8, 10, 6, 1, 1, 1, 3, 13, 15, 15, 7, 1, 0, 2, 3, 12, 25, 26, 21, 8, 1, 0, 2, 6, 10, 31, 45, 42, 28, 9, 1, 0, 0, 6, 16, 30, 66, 77, 64, 36, 10, 1, 0, 2, 4, 18, 40, 76, 126, 126, 93, 45, 11, 1, 0, 0, 6, 16, 50, 96, 168, 224, 198, 130, 55, 12, 1

Offset: 1

Antti Karttunen, Jun 25 2002

Upper triangular region of the table A073265 read by rows. - Emeric Deutsch, Feb 04 2005

Also the convolution triangle of A209229. - Peter Luschny, Oct 07 2022

			T(6,3) = 4 because there are four ordered partitions of 6 into 3 powers of 2, namely: 4+1+1, 1+4+1, 1+1+4 and 2+2+2.
Triangle begins:
  1;
  1, 1;
  0, 2, 1;
  1, 1, 3, 1;
  0, 2, 3, 4, 1;
  0, 2, 4, 6, 5, 1;

G. C. Greubel, Rows n = 1..100 of triangle, flattened
S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.

Cf. A048298, A073265, A023359 (row sums), A089052 (partitions of n).

T(2n,n) gives A333047.

b:= proc(n) option remember; expand(`if`(n=0, 1,
       add(b(n-2^j)*x, j=0..ilog2(n))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)):
seq(T(n), n=1..14);  # Alois P. Heinz, Mar 06 2020
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> if n = 2^ilog2(n) then 1 else 0 fi); # Peter Luschny, Oct 07 2022

m:= 10; T[n_, k_]:= T[n, k]= Coefficient[(Sum[x^(2^j), {j,0,m+1}])^k, x, n]; Table[T[n, k], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Mar 06 2020 *)

T(n, k) = coefficient of x^n in the formal power series (x + x^2 + x^4 + x^8 + x^16 + ...)^k. - Emeric Deutsch, Feb 04 2005
T(0, k) = T(n, 0) = 0, T(n, k) = 0 if k > n, T(n, 1) = 1 if n = 2^m, 0 otherwise and in other cases T(n, k) = Sum_{i=0..floor(log_2(n-1))} T(n-(2^i), k-1). - Emeric Deutsch, Feb 04 2005
Sum_{k=0..n} T(n,k) = A023359(n). - Philippe Deléham, Nov 04 2006

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

A073266 Triangle read by rows: T(n,k) is the number of compositions of n as the sum of k integral powers of 2.

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