cp's OEIS Frontend

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

%I A152537 #26 Feb 03 2025 02:22:44
%S A152537 1,1,1,2,4,9,18,37,74,148,296,592,1183,2366,4732,9463,18926,37852,
%T A152537 75704,151408,302816,605632,1211265,2422530,4845060,9690120,19380241,
%U A152537 38760482,77520964,155041928,310083856,620167712,1240335424,2480670848,4961341696,9922683391
%N A152537 Convolution sequence: this sequence convolved with A000041 gives powers of 2, (A000079).
%C A152537 Terms are very similar to those of A178841. - _Georg Fischer_, Mar 23 2019
%H A152537 Vaclav Kotesovec, <a href="/A152537/b152537.txt">Table of n, a(n) for n = 0..1000</a>
%F A152537 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,...).
%F A152537 A152537 = sums of antidiagonal terms of the array.
%F A152537 The sequence may be obtained directly from the following set of operations:
%F A152537 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).
%F A152537 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.
%F A152537 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.
%F A152537 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
%F A152537 a(n) ~ c * 2^n, where c = A048651 = 0.28878809508660242127889972192923078... - _Vaclav Kotesovec_, Jun 02 2018
%F A152537 a(n) = 2^n - Sum_{j=1..n} A000041(j)*a(n-j). - _Alois P. Heinz_, Feb 02 2025
%e A152537 a(5) = 9 = 32 - 23 = (32 - ((7,5,3,2,1) dot (1,1,1,2,4)))
%e A152537 (1,1,2,3) convolved with (1,1,1,2) = 8, where (1,1,2,3...) = the first four partition numbers.
%p A152537 a:= proc(n) option remember;
%p A152537       2^n-add(combinat[numbpart](j)*a(n-j), j=1..n)
%p A152537     end:
%p A152537 seq(a(n), n=0..35);  # _Alois P. Heinz_, Feb 02 2025
%t A152537 nmax = 40; CoefficientList[Series[Product[1-x^k, {k, 1, nmax}] / (1-2*x), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 02 2018 *)
%o A152537 (PARI) /* computation by definition (division of power series) */
%o A152537 N=55;
%o A152537 A000079=vector(N,n,2^(n-1));
%o A152537 S000079=Ser(A000079);
%o A152537 A000041=vector(N,n,numbpart(n-1));
%o A152537 S000041=Ser(A000041);
%o A152537 S152537=S000079/S000041;
%o A152537 A152537=Vec(S152537) /* show terms */  /* _Joerg Arndt_, Feb 06 2011 */
%o A152537 (PARI) /* computation using power series eta(x) and 1/(1-2*x) */
%o A152537 x='x+O('x^55);  S152537=eta(x)/(1-2*x);
%o A152537 A152537=Vec(S152537) /* show terms */  /* _Joerg Arndt_, Feb 06 2011 */
%Y A152537 Cf. A010815, A000041, A000079, A152538, A178841.
%K A152537 nonn
%O A152537 0,4
%A A152537 _Gary W. Adamson_, Dec 06 2008