A152537 Convolution sequence: convolved with A000041 = 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

0,4

Comments

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

Links

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

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

Example

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.

Mathematica

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

Prog

(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 */

Crossrefs

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

Sequence in context: A282986 A056185 A363262 * A182028 A081253 A118255

Adjacent sequences: A152534 A152535 A152536 * A152538 A152539 A152540

Keyword

nonn

Author

Gary W. Adamson, Dec 06 2008

Status

approved