A307059 -id:A307059 - cp's OEIS frontend

A307058 Expansion of 1/(2 - Product_{k>=1} (1 + x^(2*k-1))).

1, 1, 1, 2, 4, 7, 12, 21, 38, 68, 120, 212, 377, 670, 1188, 2107, 3740, 6638, 11778, 20898, 37084, 65808, 116775, 207212, 367696, 652478, 1157815, 2054524, 3645730, 6469316, 11479734, 20370656, 36147506, 64143372, 113821732, 201975429, 358403220, 635982680, 1128544452, 2002589998

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

nonn

Invert transform of A000700.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A000700, A055887, A302017, A304969, A307059.

Row sums of A341279.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1 + x^(2*j-1): j in [1..m+2]])) )); // G. C. Greubel, Jan 24 2024

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*g(i), i=1..n))
    end:
seq(a(n), n=0..39);  # Alois P. Heinz, Feb 09 2021

nmax = 39; CoefficientList[Series[1/(2 - Product[(1 + x^(2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]

m=80;
def f(x): return 1/(2 - product(1+x^(2*j-1) for j in range(1,m+3)))
def A307058_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307058_list(m) # G. C. Greubel, Jan 24 2024

a(0) = 1; a(n) = Sum_{k=1..n} A000700(k)*a(n-k).
From G. C. Greubel, Jan 24 2024: (Start)
G.f.: (1+x)/(2*(1+x) - x*QPochhammer(-1/x; x^2)).
G.f.: 1/( 2 - x^(1/24)*etx(x^2)^2/(eta(x^4)*eta(x)) ), where eta(x) is the Dedekind eta function. (End)

A307057 Expansion of 1/(2 - Product_{k>=2} 1/(1 - x^k)).

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 10, 15, 33, 55, 111, 195, 380, 684, 1306, 2389, 4507, 8313, 15591, 28881, 53991, 100257, 187086, 347860, 648512, 1206656, 2248399, 4185087, 7796011, 14514195, 27033073, 50334299, 93741325, 174552379, 325067573, 605316388, 1127249250, 2099115548, 3909023438, 7279285948

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

nonn

Invert transform of A002865.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A002865, A055887, A304969, A307058, A307059, A307060, A307062, A307063.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - 1/(&*[1 - x^j: j in [2..m+2]])) )); // G. C. Greubel, Jan 24 2024

nmax = 50; CoefficientList[Series[1/(2 - Product[1/(1 - x^k), {k, 2, nmax}]), {x, 0, nmax}], x]
a[0]= 1; a[n_]:= a[n]= Sum[(PartitionsP[k] -PartitionsP[k-1]) a[n-k], {k,n}];
Table[a[n], {n,0,50}]
CoefficientList[Series[1/(2 -(1-x)/QPochhammer[x]), {x,0,80}], x] (* G. C. Greubel, Jan 24 2024 *)

m=80;
def f(x): return 1/( 2 - (1-x)/product(1 - x^j for j in range(1,m+3)) )
def A307057_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307057_list(m) # G. C. Greubel, Jan 24 2024

a(0) = 1; a(n) = Sum_{k=1..n} A002865(k)*a(n-k).
a(n) ~ c / r^n, where r = 0.53700045638650021831634004949965496126950171484122... is the root of the equation 1 - r = 2*QPochhammer[r] and c = 0.2143395760756683581919851351414497181589685708674442097498294834747517926...
From G. C. Greubel, Jan 24 2024: (Start)
G.f.: 1/( 2 - (1-x)/QPochhammer(x) ).
G.f.: 1/( 2 - x^(1/24)*(1-x)/eta(x) ), where eta(x) is the Dedekind eta function. (End)

A307060 Expansion of 1/(2 - Product_{k>=1} 1/(1 + x^k)).

Original entry on oeis.org

1, -1, 1, -2, 4, -7, 12, -21, 38, -68, 120, -212, 377, -670, 1188, -2107, 3740, -6638, 11778, -20898, 37084, -65808, 116775, -207212, 367696, -652478, 1157815, -2054524, 3645730, -6469316, 11479734, -20370656, 36147506, -64143372, 113821732, -201975429, 358403220, -635982680, 1128544452, -2002589998

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

sign

Invert transform of A081362.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A081362, A299208, A304969, A307058.

Cf. A307057, A307058, A307059, A307062, A307063.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1-x^(2*j-1): j in [1..m+2]])) )); // G. C. Greubel, Jan 24 2024

nmax = 39; CoefficientList[Series[1/(2 - Product[1/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

m=80;
def f(x): return 1/( 2 - product(1-x^(2*j-1) for j in range(1,m+3)) )
def A307060_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307060_list(m) # G. C. Greubel, Jan 24 2024

G.f.: 1/(2 - Product_{k>=1} (1 - x^(2*k-1))).
a(0) = 1; a(n) = Sum_{k=1..n} A081362(k)*a(n-k).
From G. C. Greubel, Jan 24 2024: (Start)
G.f.: 1/(2 - QPochhammer(x)/QPochhammer(x^2)).
G.f.: 1/(2 - x^(1/24)*eta(x)/eta(x^2)), where eta(x) is the Dedekind eta function. (End)

A047265 Triangle T(n,k), for n >= 1, 1 <= k <= n, read by rows, giving coefficient of x^n in expansion of (Product_{j>=1} (1-(-x)^j) - 1 )^k.

Original entry on oeis.org

1, -1, 1, 0, -2, 1, 0, 1, -3, 1, -1, 0, 3, -4, 1, 0, -2, -1, 6, -5, 1, -1, 2, -3, -4, 10, -6, 1, 0, -2, 6, -3, -10, 15, -7, 1, 0, 2, -6, 12, 0, -20, 21, -8, 1, 0, 1, 6, -16, 19, 9, -35, 28, -9, 1, 0, 0, 0, 16, -35, 24, 28, -56, 36, -10, 1, -1, 2, -3, -6, 40, -65, 21, 62, -84, 45, -11, 1

Offset: 1

N. J. A. Sloane

This is an ordinary convolution triangle. If a column k=0 starting at n=0 is added, then this is the Riordan triangle R(1, f(x)), with

f(x) = Product_{j>=1} (1 - (-x)^j) - 1, generating {0, {A121373(n)}{n>=1}}. - _Wolfdieter Lang, Feb 16 2021

			Triangle starts:
   1,
  -1,   1,
   0,  -2,   1,
   0,   1,  -3,   1,
  -1,   0,   3,  -4,   1,
   0,  -2,  -1,   6,  -5,   1,
  -1,   2,  -3,  -4,  10,  -6,   1,
   0,  -2,   6,  -3, -10,  15,  -7,   1,
   0,   2,  -6,  12,   0, -20,  21,  -8,   1,
   0,   1,   6, -16,  19,   9, -35,  28,  -9,   1,
   0,   0,   0,  16, -35,  24,  28, -56,  36, -10,   1,
  -1,   2,  -3,  -6,  40, ...

Alois P. Heinz, Rows n = 1..200, flattened
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Columns: A001482 - A001488, A001490, A006665, A010815, A047649, A047654, A047655, A047938 - A047648.
Cf. A341418 (differently signed).
Cf. A000027, A000041, A000217, A000292, A121373, A307059.

R:=PowerSeriesRing(Integers(), 40);
T:= func< n,k | Coefficient(R!( (-1)^n*(-1 + (&*[1 - x^j: j in [1..n]]) )^k ), n) >;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Sep 07 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
T:= proc(n, k) option remember;
     `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, g(n)),
         (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Feb 07 2021

T[n_, k_]:= SeriesCoefficient[(-1)^n*(Product[(1-x^j), {j,n}] - 1)^k, {x, 0, n}];
Table[T[n, k], {n,12}, {k,n}]//Flatten (* Jean-François Alcover, Dec 05 2013 *)

T(n,k) = polcoeff((-1)^n*(Ser(prod(i=1,n,1-x^i)-1)^k), n) \\ Ralf Stephan, Dec 08 2013

from sage.combinat.q_analogues import q_pochhammer
P. = PowerSeriesRing(ZZ, 50)
def T(n,k): return P( (-1)^n*(-1 + q_pochhammer(n,x,x) )^k ).list()[n]
flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Sep 07 2023

G.f. column k: (Product_{j>=1} (1 - (-x)^j) - 1)^k, for k >= 1. See the name and a Riordan triangle comment above. - Wolfdieter Lang, Feb 16 2021
From G. C. Greubel, Sep 07 2023: (Start)
T(n, n) = 1.
T(n, n-1) = -A000027(n-1).
T(n, n-2) = A000217(n-3).
T(n, n-3) = -A000292(n-5).
Sum_{k=1..n} T(n, k) = (-1)^n * A307059(n).
Sum_{k=1..n} (-1)^k * T(n, k) = (-1)^n * A000041(n). (End)

A307062 Expansion of 1/(2 - Product_{k>=1} (1 + x^k)^k).

Original entry on oeis.org

1, 1, 3, 10, 29, 88, 264, 790, 2366, 7086, 21216, 63523, 190201, 569485, 1705121, 5105383, 15286247, 45769238, 137039743, 410316854, 1228548190, 3678451550, 11013817655, 32976968175, 98737827756, 295635383297, 885175234817, 2650343093602, 7935511791620, 23760073760720, 71141108467679

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

nonn

Invert transform of A026007.

a(n) is the number of compositions of n where there are A026007(k) sorts of part k. - Joerg Arndt, Jan 24 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026007, A257674, A299167, A304969.

Cf. A307057, A307058, A307059, A307060, A307063.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[(1+x^j)^j: j in [1..m+2]])) )); // G. C. Greubel, Jan 24 2024

b:= proc(n) b(n):= add((-1)^(n/d+1)*d^2, d=numtheory[divisors](n)) end:
g:= proc(n) g(n):= `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n) end:
a:= proc(n) a(n):= `if`(n=0, 1, add(g(k)*a(n-k), k=1..n)) end:
seq(a(n), n=0..45);  # Alois P. Heinz, Jan 24 2024

nmax = 30; CoefficientList[Series[1/(2 - Product[(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

m=80;
def f(x): return 1/( 2 - product((1+x^j)^j for j in range(1,m+3)) )
def A307062_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307062_list(m) # G. C. Greubel, Jan 24 2024

a(0) = 1; a(n) = Sum_{k=1..n} A026007(k)*a(n-k).

A307063 Expansion of 1/(2 - Product_{k>=1} (1 + k*x^k)).

Original entry on oeis.org

1, 1, 3, 10, 28, 85, 252, 745, 2202, 6530, 19326, 57194, 169341, 501242, 1483816, 4392531, 13002772, 38491212, 113943278, 337298400, 998482338, 2955742400, 8749688247, 25901125616, 76673399424, 226971213462, 671887935923, 1988945626648, 5887744768722, 17429103155892, 51594226501776

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

nonn

Invert transform of A022629.

a(n) is the number of compositions of n where there are A022629(k) sorts of part k. - Joerg Arndt, Jan 24 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A022629, A299164, A304969, A320652.

Cf. A307057, A307058, A307059, A307060, A307062.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[(1+j*x^j): j in [1..m+2]])) ));

nmax = 30; CoefficientList[Series[1/(2 - Product[(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

m=80;
def f(x): return 1/( 2 - product(1+j*x^j for j in range(1,m+3)) )
def A307063_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307063_list(m) # G. C. Greubel, Jan 24 2024

a(0) = 1; a(n) = Sum_{k=1..n} A022629(k)*a(n-k).

A341418 Triangle read by rows: T(n, m) gives the sum of the weights of weighted compositions of n with m parts from generalized pentagonal numbers {A001318(k)}_{k>=1}.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 1, 3, 1, -1, 0, 3, 4, 1, 0, -2, 1, 6, 5, 1, -1, -2, -3, 4, 10, 6, 1, 0, -2, -6, -3, 10, 15, 7, 1, 0, -2, -6, -12, 0, 20, 21, 8, 1, 0, 1, -6, -16, -19, 9, 35, 28, 9, 1, 0, 0, 0, -16, -35, -24, 28, 56, 36, 10, 1, 1, 2, 3, -6, -40, -65, -21, 62, 84, 45, 11, 1

Offset: 1

Wolfdieter Lang, Feb 15 2021

The sums of row n are given in A000041(n), for n >= 1 (number of partitions).
A differently signed triangle is A047265.
One could add a column m = 0 starting at n = 0 with T(0, 0) = 1 and T(n, 0) = 0 otherwise, by including the empty partition with no parts.
For the weights w of positive integer numbers n see a comment in A339885. It is w(n) = -A010815(n), for n >= 0. Also w(n) = A257628(n), for n >= 1.
The weight of a composition is the one of the respective partition, obtained by the product of the weights of the parts.
That the row sums give the number of partitions follows from the pentagonal number theorem. See also the Apr 04 2013 conjecture in A000041 by Gary W. Adamson, and the hint for the proof by Joerg Arndt. The INVERT map of A = {1, 1, 0, 0, -5, -7, ...}, with offset 1, gives the A000041(n) numbers, for n >= 0.
If the above mentioned column for m = 0, starting at n = 0 is added this is an ordinary convolution triangle of the Riordan type R(1, f(x)), with f(x) = -(Product_{j>=1} (1 - x^j) - 1), generating {A257628(n)}{n>=0}. See the formulae below. - _Wolfdieter Lang, Feb 16 2021

			The triangle T(n, m) begins:
  n\m   1  2  3   4   5   6   7  8  9 10 11 12 ... A000041
  --------------------------------------------------------
  1:    1                                                1
  2:    1  1                                             2
  3:    0  2  1                                          3
  4:    0  1  3   1                                      5
  5:   -1  0  3   4   1                                  7
  6:    0 -2  1   6   5   1                             11
  7:   -1 -2 -3   4  10   6   1                         15
  8:    0 -2 -6  -3  10  15   7  1                      22
  9:    0 -2 -6 -12   0  20  21  8  1                   30
  10:   0  1 -6 -16 -19   9  35 28  9  1                42
  11:   0  0  0 -16 -35 -24  28 56 36 10  1             56
  12:   1  2  3  -6 -40 -65 -21 62 84 45 11  1          77
  ...
For instance the case n = 6: The relevant weighted partitions with parts from the pentagonal numbers and number of compositions are: m = 2: 2*(1,-5) = -2*(1,5), m = 3: 1*(2^3), m = 4: 3*(1^2,2^2), m = 5: 1*(1^4,2), m = 6: 1*(1^6). The other partitions have weight 0.

Wikipedia, Pentagonal number theorem.

Cf. A000041, A008284, A010815, A047265, A257628, -A307059 (alternating row sums), A339885 (for partitions).

# Using function PMatrix from A357368. Adds a row and a column for n, m = 0.
PMatrix(14, proc(n) 24*n+1; if issqr(%) then sqrt(%); -(-1)^irem(iquo(%+irem(%,6),6),2) else 0 fi end); # Peter Luschny, Oct 06 2022

nmax = 12;
col[m_] := col[m] = (-(Product[(1-x^j), {j, 1, nmax}]-1))^m // CoefficientList[#, x]&;
T[n_, m_] := col[m][[n+1]];
Table[T[n, m], {n, 1, nmax}, {m, 1, n}] // Flatten (* Jean-François Alcover, Oct 23 2023 *)

T(n, m) = Sum_{j=1..p(n,m)} w(Part(n, m, j))*M0(n, m, j), where p(n, m) = A008284(n, m), M0(n, m, j) are the multinomials from A048996, i.e., m!/Prod_{k=1..m} e(n,m,j,k)! with the exponents of the parts, and the ternary weight of the j-th partition of n with m parts Part(n,m,j), in Abramowitz-Stegun order, is defined as the product of the weights of the parts, using w(n) = -A010815(n), for n >= 1, and m = 1, 2, ..., n.
From Wolfdieter Lang, Feb 16 2021: (Start)
G.f. column m: G(m, x) =  ( -(Product_{j>=1} (1 - x^j) - 1) )^m, for  m >= 1.
G.f. of row polynomials R(n, x) = Sum_{m=1..n}, that is g. f. of the triangle:
  GfT(z, x) = 1/(1 - x*G(1, z)) - 1. Riordan triangle (without m = 0 column). (End)

cp's OEIS Frontend

A307058 Expansion of 1/(2 - Product_{k>=1} (1 + x^(2*k-1))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A307057 Expansion of 1/(2 - Product_{k>=2} 1/(1 - x^k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A307060 Expansion of 1/(2 - Product_{k>=1} 1/(1 + x^k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A047265 Triangle T(n,k), for n >= 1, 1 <= k <= n, read by rows, giving coefficient of x^n in expansion of (Product_{j>=1} (1-(-x)^j) - 1 )^k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A307062 Expansion of 1/(2 - Product_{k>=1} (1 + x^k)^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A307063 Expansion of 1/(2 - Product_{k>=1} (1 + k*x^k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A341418 Triangle read by rows: T(n, m) gives the sum of the weights of weighted compositions of n with m parts from generalized pentagonal numbers {A001318(k)}_{k>=1}.

Views