cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A174065 Convolved with its aerated variant = A000041.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 19, 25, 31, 38, 48, 60, 73, 89, 109, 133, 161, 193, 232, 279, 333, 395, 470, 558, 658, 775, 912, 1071, 1254, 1464, 1708, 1991, 2313, 2681, 3107, 3595, 4149, 4782, 5506, 6331, 7268, 8330, 9538, 10912, 12462, 14213, 16199
Offset: 0

Views

Author

Gary W. Adamson, Mar 06 2010

Keywords

Comments

A000041 = (1, 1, 2, 3, 5, 7, ...) = (1, 1, 1, 2, 3, 4, 5, 7, ...) * (1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 7, 0, 9, 0, ...).
The sequence diverges from A100853 after 16 terms; and is a conjectured Euler transform of A035263: (1, 0, 1, 1, 1, 0, 1, 0, 1, ...).

Examples

			Heading at top, with triangle A174066 underneath (the generator for A174065):
1, 1, 1, 2, 3, 4,.... = heading
1;................... = 1
1;................... = 1
1, 1;................ = 2
2, 1;................ = 3
3, 1, 1;............. = 5
4, 2, 1;............. = 7
5, 3, 1, 2;.......... = 11
7, 4, 2, 2;.......... = 15
9, 5, 3, 2, 3;....... = 22
...
... where terms in the left column are the result of the two rules: multiply heading * left column, and row sums = partition numbers.
Thus leftmost term in column 8 must be 7 = 15 - (4 + 2 + 2). Then the 7 is placed in its spot in the left column and as the next heading term.

Links

Crossrefs

Cf. A000041, A327726, A373219, A373220, A373221.
Cf. A000009, A115362, A174066, A174067.

Programs

  • Maple
    p:= combinat[numbpart]:
a:= proc(n) option remember; `if`(n=0, 1, p(n)-add(a(j)*
      `if`(irem(n-j, 2, 'r')=1, 0, a(r)), j=0..n-1))
    end:
seq(a(n), n=0..61);  # Alois P. Heinz, Jul 27 2019
  • Mathematica
    nmax = 60; CoefficientList[Series[Product[QPochhammer[-1, x^(4^j)]/2, {j, 0, Log[nmax]/Log[4]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 24 2019 *)

Formula

Aerate and convolve sequences are generated by triangles (in this case A174066) in which ongoing terms are placed in the left column and at the top as a heading. Columns >1 are shifted down k times (k=2) in this case corresponding to (k-1) interpolated zeros. Next term in left column = n-th term in the "target sequence" S(n) (in this case A000041) minus (sum of terms in n-th row for columns >1). Place the latter term in the heading filling in missing terms.
G.f.: Product_{i>=1, j>=0} (1 + x^(i*4^j)). - Ilya Gutkovskiy, Sep 23 2019
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2^(11/8) * 3^(3/4) * n^(7/8)). - Vaclav Kotesovec, Sep 24 2019
From Seiichi Manyama, May 31 2024: (Start)
G.f.: Product_{k>=1} (1 + x^k)^(valuation(k,4) + 1).
Let A(x) be the g.f. of this sequence, and B(x) be the g.f. of A000009, then B(x) = A(x)/A(x^4). (End)

Extensions

More terms from R. J. Mathar, Mar 18 2010
Offset corrected by Alois P. Heinz, Jul 27 2019

A174066 Irregular triangle, row sums = A000041, left border = A174065.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 5, 3, 1, 2, 7, 4, 2, 2, 9, 5, 3, 2, 3, 12, 7, 4, 4, 3, 15, 9, 5, 6, 3, 4, 19, 12, 7, 8, 6, 4, 25, 15, 9, 10, 9, 4, 5, 31, 19, 12, 14, 12, 8, 5, 38, 25, 15, 18, 15, 12, 5, 7, 48, 31, 19, 24, 21, 16, 10, 7, 60, 38, 25, 30, 27, 20, 15, 7, 9, 73, 48, 31, 38, 36
Offset: 1

Views

Author

Gary W. Adamson, Mar 06 2010

Keywords

Comments

Left border = A174065: (1, 1, 1, 2, 3, 4, 5, 7, 9, 12,...) * its aerated variant (1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 7,...) = A000041, the partition sequence: (1, 1, 2, 3, 5, 7, 11, 15, 22, 30,...).

Examples

			Heading and first few rows of the triangle =
.1,...1,...1,...2,...3,...4,...5,...7,...9,... = A174065.
.1;........................................... = .. 1 (A000041)
.1;........................................... = .. 1
.1,...1;...................................... = .. 2
.2,...1;...................................... = .. 3
.3,...1,...1;................................. = .. 5
.4,...2,...1;................................. = .. 7
.5,...3,...1,...2;............................ = ..11
.7,...4,...2,...2;............................ = ..15
.9....5,...3,...2,...3;....................... = ..22
.12,..7,...4,...4,...3;....................... = ..30
.15,..9,...5,...6,...3;...4;.................. = ..42
.19,.12,...7,...8,...6,...4;.................. = ..56
.25,.15,...9,..10,...9,...4,...5;............. = ..77
.31,.19,..12,..14,..12,...8,...5;............. = .101
.38,.25,..15,..18,..15,..12,...5,...7;........ = .135
.48,.31,..19,..24,..21,..16,..10,...7;........ = .176
.60,.38,..25,..30,..27,..20,..15,...7,...9;... = .231
.73,.48,..31,..38,..36,..28,..20,..14,...9;... = .297
...
Example: leftmost term in 8th row has to be 7 = (15 - (4 + 2 + 2)); so we
place a 7 as next term in the heading, then multiply * leftmost column.
Finally, shift the columns down twice.

Crossrefs

Cf. A000041, A174065, A174067.

Formula

The triangle is the result of three rules after beginning (1, 1, 1, 1,...):
Columns >1 are shifted down twice from previous columns; column terms = left border * (left border placed as a heading row); and row sums = A000041, the partition numbers. The rules force the next missing term in the triangle to be the leftmost term in column 1. This is found by taking p(n) for row n, then subtracting the sum of row n terms (minus leftmost term).

A174068 Convolved with its aerated variant of two zeros between terms = A000041.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 9, 13, 17, 23, 29, 38, 48, 62, 77, 98, 121, 153, 187, 233, 283, 349, 422, 515, 620, 751, 900, 1083, 1291, 1544, 1832, 2180, 2576, 3050, 3590, 4234, 4965, 5830, 6813, 7971, 9286, 10824, 12572, 14608, 16921, 19600, 22640, 26150, 30130, 34709
Offset: 0

Views

Author

Gary W. Adamson, Mar 06 2010

Keywords

Comments

Considered k=3 in an infinite set of convolution sequences: (aerated with one zero, A174065; two zeros, A174068); such that A000041 =
(1, 1, 2, 3, 5, 7, 11,...) = (1, 1, 2, 2, 4, 5, 7, 9, 13, 17,...) *
(1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 4, 0, 0, 5, 0, 0, 7, 0, 0,...).

Examples

			The triangle heading and first few rows of the triangle =
1, 1, 2, 2, 4, 5, 7,...
1;
1;
2;
2, 1;
4, 1;
5, 2;
7, 2, 2;
...
G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 13*x^8 + 17*x^9 + ...

Links

Crossrefs

Cf. A000041, A174065, A174066, A174067.

Programs

  • Maple
    p:= combinat[numbpart]:
a:= proc(n) option remember; `if`(n=0, 1, p(n)-add(a(j)*
      `if`(irem(n-j, 3, 'r')>0, 0, a(r)), j=0..n-1))
    end:
seq(a(n), n=0..61);  # Alois P. Heinz, Jul 27 2019
  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, PartitionsP[n] - Sum[a[j]*If[Mod[n-j, 3] > 0, 0, a[(n-j)/3]], {j, 0, n-1}]];
a /@ Range[0, 61] (* Jean-François Alcover, May 17 2020, after Maple *)

Formula

Refer to A174065, and A174066, the case for k=3. The sequence = left border of a triangle generated from 3 rules: row sums = A000041; columns >1 are shifted down thrice from previous column; column terms are derived from self-convolution of left border, (with the left border placed at top as a heading).
A(x)*A(x^3) = A000041(x) for the generating functions. - R. J. Mathar, Mar 18 2010
Expansion of f(-x^3)/f(-x) * f(-x^27)/f(-x^9) * f(-x^243)/f(-x^27) * ... where f(-x) is a Ramanujan theta function. - Michael Somos, Jun 07 2012
a(n) ~ exp(Pi*sqrt(n/2)) / (2^(19/8) * 3^(1/8) * n^(7/8)). - Vaclav Kotesovec, Sep 24 2019

Extensions

More terms from R. J. Mathar, Mar 18 2010
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.