A152537 Convolution sequence: this sequence convolved with A000041 gives powers of 2, (A000079).

1, 1, 1, 2, 4, 9, 18, 37, 74, 148, 296, 592, 1183, 2366, 4732, 9463, 18926, 37852, 75704, 151408, 302816, 605632, 1211265, 2422530, 4845060, 9690120, 19380241, 38760482, 77520964, 155041928, 310083856, 620167712, 1240335424, 2480670848, 4961341696, 9922683391

Offset: 0

Author

Gary W. Adamson, Dec 06 2008

Keywords

• nonn

Comments

Terms are very similar to those of A178841. - Georg Fischer, Mar 23 2019

Examples

			a(5) = 9 = 32 - 23 = (32 - ((7,5,3,2,1) dot (1,1,1,2,4)))
(1,1,2,3) convolved with (1,1,1,2) = 8, where (1,1,2,3...) = the first four partition numbers.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A010815, A000041, A000079, A152538, A178841.

Programs

Maple

a:= proc(n) option remember;
      2^n-add(combinat[numbpart](j)*a(n-j), j=1..n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 02 2025

Mathematica

nmax = 40; CoefficientList[Series[Product[1-x^k, {k, 1, nmax}] / (1-2*x), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 02 2018 *)

PARI

/* computation by definition (division of power series) */
N=55;
A000079=vector(N,n,2^(n-1));
S000079=Ser(A000079);
A000041=vector(N,n,numbpart(n-1));
S000041=Ser(A000041);
S152537=S000079/S000041;
A152537=Vec(S152537) /* show terms */  /* Joerg Arndt, Feb 06 2011 */

PARI

/* computation using power series eta(x) and 1/(1-2*x) */
x='x+O('x^55);  S152537=eta(x)/(1-2*x);
A152537=Vec(S152537) /* show terms */  /* Joerg Arndt, Feb 06 2011 */

Formula

Construct an array of rows such that n-th row = partial sums of (n-1)-th row of A010815: (1, -1, -1, 0, 0, 1, 0, 1,...).

A152537 = sums of antidiagonal terms of the array.

The sequence may be obtained directly from the following set of operations:

Our given sequence = A000041: (1, 1, 2, 3, 5, 7, 11,...). Delete the first "1" then consider (1, 2, 3, 5, 7, 11,...) as an operator Q which we write in reverse with 1,2,3,...terms for each operation. Letting R = the target sequence (1,2,4,8,...); we begin a(0) = 1, a(1) = 1, then perform successive operations of: "next term in (1,2,4,...) - dot product of Q*R" where Q is written right to left and R (the ongoing result) written left to right).

Examples: Given 4 terms Q, R, we have: (5,3,2,1) dot (1,1,1,2) = (5+3+2+2) = 12, which we subtract from 16, = 4.

Given 5 terms of Q,R and A152537, we have (7,5,3,2,1) dot (1,1,1,2,4) = 23 which is subtracted from 32 giving 9. Continue with analogous operations to generate the series.

a(n) = Sum_{j=0..n} A010815(j)*2^(n-j). G.f.: A000079(x)/A000041(x) = A010815(x)/(1-2x), where A......(x) denotes the g.f. of the associated sequence. - R. J. Mathar, Dec 09 2008

a(n) ~ c * 2^n, where c = A048651 = 0.28878809508660242127889972192923078... - Vaclav Kotesovec, Jun 02 2018

a(n) = 2^n - Sum_{j=1..n} A000041(j)*a(n-j). - Alois P. Heinz, Feb 02 2025