A137682 -id:A137682 - 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

A137680 Triangle read by rows, T(n,k) = T(n-1, k-1) - T(n-k, k-1); with leftmost term in each row = sum of all previous terms.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 7, 2, 0, 1, 17, 4, 1, 0, 1, 40, 10, 4, 1, 0, 1, 96, 23, 8, 3, 1, 0, 1, 228, 56, 19, 8, 3, 1, 0, 1, 544, 132, 46, 18, 7, 3, 1, 0, 1, 1296, 316, 109, 42, 18, 7, 3, 1, 0, 1, 3089, 752, 260, 101, 41, 17, 7, 3, 1, 0, 1, 7361, 1793, 620, 241, 98, 41, 17, 7, 3, 1, 0, 1, 17544, 4272, 1477, 574, 233, 97, 40, 17, 7, 3, 1, 0, 1

Offset: 1

Gary W. Adamson, Feb 05 2008

A variation of the same sequence = column 2 of the triangle: (1, 0, 2, 4, 10, 23, 56, 132, ...) = first difference row of column 1. Left border of the triangle = A137682.

Left column starting (1, 3, ...) = INVERT transform of A160096. - Gary W. Adamson, May 01 2009

			First few rows of the triangle:
     1;
     1,   1;
     3,   0,   1;
     7,   2,   0,   1;
    17,   4,   1,   0,  1;
    40,  10,   4,   1,  0,  1;
    96,  23,   8,   3,  1,  0, 1;
   228,  56,  19,   8,  3,  1, 0, 1;
   544, 132,  46,  18,  7,  3, 1, 0, 1;
  1296, 316, 109,  42, 18,  7, 3, 1, 0, 1;
  3089, 752, 260, 101, 41, 17, 7, 3, 1, 0, 1;
  ...

Cf. A137681 (row sums), A137682.

Cf. A160096. - Gary W. Adamson, May 01 2009

A137680 := proc(n,k)
    if k < 1 or k > n then
        0 ;
    elif n = 1 then
        1;
    elif k = 1 then
        add(add(procname(r,j),j=1..r),r=1..n-1) ;
    else
        procname(n-1,k-1)-procname(n-k,k-1) ;
    end if;
end proc: # R. J. Mathar, Aug 12 2012

T[n_, k_] := T[n, k] = Which[k < 1 || k > n, 0, n == 1, 1, k == 1, Sum[T[r, j], {r, 1, n-1}, {j, 1, r}], True, T[n-1, k-1] - T[n-k, k-1]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 02 2024, after R. J. Mathar *)

A137681 Row sums of triangle A137680.

Original entry on oeis.org

1, 2, 4, 10, 23, 56, 132, 316, 752, 1793, 4272, 10183, 24266, 57833, 137828, 328475, 782827, 1865651, 4446255, 10596406, 25253567, 60184809, 143433647, 341833954, 814665553, 1941527332, 4627086990, 11027366781, 26280642313, 62632555360, 149267165702

Offset: 1

Gary W. Adamson, Feb 05 2008

nonn

Partial sums of A137681 prefaced with a 1 = left border of triangle A137680: (1, 1, 3, 7, 17, 40, 96, 228, ...) = A137682. A variation of A137681 = column 2 of triangle A137680: (1, 0, 2, 4, 10, 23, 56, 132, 316, ...).

			a(4) = 10 = sum of row 4 terms of triangle A137680: (7 + 2 + 0 + 1).

Cf. A137680, A137682.

A160096 Partial sums of A010815 starting with offset 1, and signed (+ + - - + + ...).

Original entry on oeis.org

1, 2, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Gary W. Adamson, May 01 2009

nonn

INVERT transform of the sequence = A137682: (1, 3, 7, 17, 40, 96, 228, ...).
From Mats Granvik, Jan 01 2015: (Start)
(1) The positive integers are the row sums of the infinite lower triangular matrix "t" starting:
  1, 0, 0, 0, 0, 0, 0, ...
  1, 1, 0, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, 0, ...
  1, 1, 1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, 1, 1, ...
  ...
which satisfies the recurrence:
t(n, 1) = 1; t(n, k) = Sum_{i=1..n-1} t(n-i, k-1) - Sum_{i=1..n-1} t(n-i, k) if n >= k, otherwise 0;
(2) This sequence a(n), in turn, is the row sums of the infinite lower triangular matrix "t" starting:
  1, 0, 0, 0, 0, 0, 0, ...
  1, 1, 0, 0, 0, 0, 0, ...
  1, 0, 1, 0, 0, 0, 0, ...
  1, 0, 0, 1, 0, 0, 0, ...
  1, 0,-1, 0, 1, 0, 0, ...
  1, 0, 0,-1, 0, 1, 0, ...
  1, 0, 0,-1,-1, 0, 1, ...
  ...
which satisfies the recurrence:
t(n, 1) = 1; t(n, k) = Sum_{i=1..k-1} t(n-i, k-1) - Sum_{i=1..n-1} t(n-i, k) if n >= k, otherwise 0;
(3) The partition numbers are the row sums of the infinite lower triangular matrix "t" starting:
  1, 0, 0, 0, 0, 0, 0, ...
  1, 1, 0, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, 0, ...
  1, 2, 1, 1, 0, 0, 0, ...
  1, 2, 2, 1, 1, 0, 0, ...
  1, 3, 3, 2, 1, 1, 0, ...
  1, 3, 4, 3, 2, 1, 1, ...
  ...
which satisfies the recurrence:
t(n, 1) = 1; t(n, k) = Sum_{i=1..n-1} t(n-i, k-1) - Sum_{i=1..k-1} t(n-i, k) if n >= k, otherwise 0;
(4) The number of divisors of "n" is the row sums of the infinite lower triangular matrix "t" starting:
  1, 0, 0, 0, 0, 0, 0, ...
  1, 1, 0, 0, 0, 0, 0, ...
  1, 0, 1, 0, 0, 0, 0, ...
  1, 1, 0, 1, 0, 0, 0, ...
  1, 0, 0, 0, 1, 0, 0, ...
  1, 1, 1, 0, 0, 1, 0, ...
  1, 0, 0, 0, 0, 0, 1, ...
  ...
which satisfies the recurrence:
t(n, 1) = 1; t(n, k) = Sum_{i=1..k-1} t(n-i, k-1) - Sum_{i=1..k-1} t(n-i, k) if n >= k, otherwise 0.
In the four cases of recurrences only the summation indices within the sums change, from (1) "n-1" and "n-1" to (2) "k-1" and "n-1" to (3) "n-1" and "k-1" to (4) "k-1" and "k-1".
(End)

			The series begins (1, 2, 2, 2, 1, 1, 0, ...) since the signed q-series = (1, 1, 0, 0, -1, 0, ...).
G.f. = x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6 + x^12 + x^13 + x^14 + ...

Cf. A010815, A257628.

Cf. (1) A000027, (2) A160096, (3) A000041, (4) A000005.

(* A160096 as row sums of recursively defined table *)
Clear[t]; nn = 90; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}] - Sum[t[n - i, k], {i, 1, n - 1}], 0]; PartialSumsOfEulerqSeries = Table[Sum[t[n, k], {k, 1, n}], {n, 1, nn}] (* Mats Granvik, Jan 01 2015 *)
a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ x]) / (1 - x), {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
CoefficientList[Series[q*(1/(1 - q)^(2)*QHypergeometricPFQ[{q, q}, {q^2, q}, q, q^2]), {q, 0, 89}], q] (* Mats Granvik, Jan 09 2015 *)

{a(n) = if( n<0, 0, polcoeff( (1 - eta(x + x * O(x^n))) / (1 - x), n))}; /* Michael Somos, Jan 02 2015 */

Partial sums of Euler's q series (signed), starting from offset 1 = (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 0, 1, ...).
G.f.: (1 - f(-x)) / (1 - x) where f(-x) is the g.f. of A010815. - Michael Somos, Jan 02 2015
Partial sums of A257628. - Georg Fischer, May 29 2023

More terms from Mats Granvik, Jan 01 2015

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

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A137680 Triangle read by rows, T(n,k) = T(n-1, k-1) - T(n-k, k-1); with leftmost term in each row = sum of all previous terms.

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A137681 Row sums of triangle A137680.

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A160096 Partial sums of A010815 starting with offset 1, and signed (+ + - - + + ...).

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