A303666 -id:A303666 - cp's OEIS frontend

A304908 Expansion of x * (d/dx) 1/(1 - Sum_{k>=0} x^(2^k)).

0, 1, 4, 9, 24, 50, 108, 217, 448, 882, 1740, 3366, 6504, 12428, 23660, 44745, 84352, 158270, 296064, 551950, 1026360, 1903524, 3522596, 6504998, 11990160, 22061700, 40528748, 74343096, 136183488, 249145148, 455265420, 830985473, 1515201792, 2760087990, 5023154832, 9133857670

Offset: 0

Ilya Gutkovskiy, May 20 2018

nonn

Sum of all parts of all compositions (ordered partitions) of n into powers of 2.

Index entries for sequences related to compositions

Cf. A000079, A001787, A023359, A036987, A281811, A304797, A303666, A304909.

nmax = 35; CoefficientList[Series[x D[1/(1 - Sum[x^2^k, {k, 0, Floor[Log[nmax]/Log[2]] + 1}]), x], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[k == 2^IntegerExponent[k, 2]] a[n - k], {k, 1, n}]; Table[n a[n], {n, 0, 35}]

a(n) = n*A023359(n).

A368299 a(n) is the number of permutations pi of [n] that avoid {231,321} so that pi^4 avoids 132.

Original entry on oeis.org

0, 1, 2, 4, 7, 13, 23, 41, 72, 127, 223, 392, 688, 1208, 2120, 3721, 6530, 11460, 20111, 35293, 61935, 108689, 190736, 334719, 587391, 1030800, 1808928, 3174448, 5570768, 9776017, 17155714, 30106180, 52832663, 92714861, 162703239, 285524281, 501060184, 879299327

Offset: 0

Kassie Archer, Dec 20 2023

Number of compositions of n of the form d_1+d_2+...+d_k=n where d_i is in {1,2,4} if i>1 and d_1 is any positive integer.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Kassie Archer and Aaron Geary, Powers of permutations that avoid chains of patterns, arXiv:2312.14351 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,1,-1).

Cf. A000071 (d_i in {1,2}), A077868 (d_i in {1,3}), A274110, A303666.

Partial sums of A181532.

a:= proc(n) option remember;
     `if`(n<1, 0, 1+add(a(n-j), j=[1, 2, 4]))
    end:
seq(a(n), n=0..37);  # Alois P. Heinz, Dec 20 2023

LinearRecurrence[{2,0,-1,1,-1},{0,1,2,4,7},38] (* Stefano Spezia, Dec 21 2023 *)

G.f.: x/((1-x)*(1-x-x^2-x^4)).
a(n) = Sum_{m=0..n-1} Sum_{r=0..floor(m/4)} Sum_{j=0..floor((m-4*r)/2)} binomial(m-3*r-j,r)*binomial(m-4*r-j,j).
a(n) = 1+a(n-1)+a(n-2)+a(n-4) where a(0)=0, a(1)=1, a(2)=2, a(3)=4.
a(n) = A274110(n+1) - 1.

cp's OEIS Frontend

A304908 Expansion of x * (d/dx) 1/(1 - Sum_{k>=0} x^(2^k)).

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