A347968 -id:A347968 - cp's OEIS frontend

A067687 Expansion of 1/( 1 - x / Product_{n>=1} (1-x^n) ).

1, 1, 2, 5, 12, 29, 69, 165, 393, 937, 2233, 5322, 12683, 30227, 72037, 171680, 409151, 975097, 2323870, 5538294, 13198973, 31456058, 74966710, 178662171, 425791279, 1014754341, 2418382956, 5763538903, 13735781840, 32735391558, 78015643589

Offset: 0

Alford Arnold, Feb 05 2002

nonn

Previous name was: Invert transform of right-shifted partition function (A000041).
Sums of the antidiagonals of the array formed by sequences A000007, A000041, A000712, A000716, ... or its transpose A000012, A000027, A000096, A006503, A006504, ....
Row sums of triangle A143866 = (1, 2, 5, 12, 29, 69, 165, ...) and right border of A143866 = (1, 1, 2, 5, 12, ...). - Gary W. Adamson, Sep 04 2008
Starting with offset 1 = A137682 / A000041; i.e. (1, 3, 7, 17, 40, 96, ...) / (1, 2, 3, 5, 7, 11, ...). - Gary W. Adamson, May 01 2009
From L. Edson Jeffery, Mar 16 2011: (Start)
Another approach is the following. Let T be the infinite lower triangular matrix with columns C_k (k=0,1,2,...) such that C_0=A000041 and, for k > 0, such that C_k is the sequence giving the number of partitions of n into parts of k+1 kinds (successive self-convolutions of A000041 yielding A000712, A000716, ...) and shifted down by k rows. Then T begins (ignoring trailing zero entries in the rows)
  (1,  0, ...            )
  (1,  1, 0, ...         )
  (2,  2, 1, 0, ...      )
  (3,  5, 3, 1, 0, ...   )
  (5, 10, 9, 4, 1, 0, ...)
etc., and a(n) is the sum of entries in row n of T. (End)

			The array begins:
  1,  1,  1,   1,   1,  1,  1, 1, ...
  0,  1,  2,   3,   4,  5,  6, 7, ...
  0,  2,  5,   9,  14, 20, 27, ...
  0,  3, 10,  22,  40, 65, ...
  0,  5, 20,  51, 105, ...
  0,  7, 36, 108, ...
  0, 11, 65, ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
N. J. A. Sloane, Transforms

Cf. A000007, A000041, A000712, A000716, A000012, A000027, A000096, A006503, A006504.
Cf. table A060850.
Cf. A137682, A143866.
Antidiagonal sums of A144064.

N=66; x='x+O('x^N); et=eta(x); Vec( sum(n=0,N, x^n/et^n ) ) \\ Joerg Arndt, May 08 2009

a(n) = Sum_{k=1..n} A000041(k-1)*a(n-k). - Vladeta Jovovic, Apr 07 2003
O.g.f.: 1/(1-x*P(x)), P(x) - o.g.f. for number of partitions (A000041). - Vladimir Kruchinin, Aug 10 2010
a(n) ~ c / r^n, where r = A347968 = 0.419600352598356478498775753566700025318... is the root of the equation QPochhammer(r) = r and c = 0.3777957165566422058901624844315414446044096308877617181754... = Log[r]/(Log[(1 - r)*r] + QPolyGamma[1, r] - Log[r]*Derivative[0, 1][QPochhammer][r, r]). - Vaclav Kotesovec, Feb 16 2017, updated Mar 31 2018

More terms from Vladeta Jovovic, Apr 07 2003
More terms and better definition from Franklin T. Adams-Watters, Mar 14 2006
New name (using g.f. by Vladimir Kruchinin), Joerg Arndt, Feb 19 2014

A206119 G.f.: A(x) = Sum_{n>=0} x^n / Product_{k=1..n} (1-x^k)^(n-k+1).

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 46, 106, 247, 578, 1359, 3204, 7573, 17930, 42512, 100902, 239694, 569768, 1355083, 3224124, 7673612, 18268414, 43500301, 103599089, 246761629, 587822094, 1400398656, 3336473471, 7949650646, 18942098721, 45136103113, 107555568419, 256302098369

Offset: 0

Paul D. Hanna, Feb 03 2012

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 106*x^7 +...
where
A(x) = 1 + x/(1-x) + x^2/((1-x)^2*(1-x^2)) + x^3/((1-x)^3*(1-x^2)^2*(1-x^3)) + x^4/((1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A067687.

{a(n)=polcoeff(sum(m=0,n,x^m/prod(k=1,m,(1-x^k +x*O(x^n))^(m-k+1))),n)}

a(n) ~ c / r^n, where r = A347968 = 0.419600352598356478498775753566700025318089363120016078... is the root of the equation QPochhammer(r) = r and c = 0.21842597743526022597060618810878279... - Vaclav Kotesovec, Aug 21 2018

A353823 Decimal expansion of the solution to Product_{k>=1} (1 - (-x)^k) = x.

Original entry on oeis.org

7, 6, 1, 0, 1, 7, 3, 9, 5, 9, 2, 4, 1, 0, 8, 9, 3, 9, 2, 9, 3, 4, 0, 4, 8, 2, 8, 8, 5, 5, 0, 3, 9, 2, 9, 8, 0, 7, 4, 9, 4, 6, 8, 8, 5, 3, 2, 8, 8, 3, 5, 6, 3, 6, 4, 6, 8, 7, 8, 7, 9, 7, 9, 8, 5, 1, 2, 3, 1, 3, 0, 7, 7, 1, 8, 0, 0, 6, 6, 5, 6, 4, 8, 2, 4, 9, 5, 5, 0, 0, 9, 7, 7, 4, 9, 6, 1, 0, 8, 5, 8, 4, 5, 6, 0, 4, 7

Offset: 0

Ilya Gutkovskiy, May 08 2022

			0.761017395924108939293404828855039298...

Cf. A121373, A347968.

RealDigits[x /. FindRoot[QPochhammer[-x] == x, {x, 1/2}, WorkingPrecision -> 120]][[1]] (* Vaclav Kotesovec, May 09 2022 *)

Root of the equation Sum_{j>=1} 1/(j*(1 - 1/(-x)^j)) = log(x). - Vaclav Kotesovec, May 09 2022

cp's OEIS Frontend

A067687 Expansion of 1/( 1 - x / Product_{n>=1} (1-x^n) ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A206119 G.f.: A(x) = Sum_{n>=0} x^n / Product_{k=1..n} (1-x^k)^(n-k+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A353823 Decimal expansion of the solution to Product_{k>=1} (1 - (-x)^k) = x.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula