A047265 -id:A047265 - cp's OEIS frontend

A047654 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^2 in powers of x.

1, -2, 1, 0, -2, 2, -2, 2, 1, 0, 2, -2, 3, 0, 2, 0, 0, 2, -2, 0, -2, 2, -1, 0, 0, -2, -2, -2, 1, -2, 0, -2, -2, 0, 2, 0, -2, 0, -2, 0, 0, 0, 1, 2, 0, 0, 2, 0, 2, 0, 1, 2, 0, -2, 2, 2, 0, 2, 0, 2, 0, 2, 2, 0, -4, 0, 0, 2, 1, -2, 0, -2, 0, 0, 0, 0, 2, -4, 1, 0, 0, -2, -2, -2, -2, 0, 0, -2, 0, 2, -2, 2, -2

Offset: 2

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 2..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.

Cf. A001482 - A001488, A001490, A047265, A047638 - A047649, A047655, A341243.

m:=120;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^2 )); // 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:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 2):
seq(a(n), n=2..94);  # Alois P. Heinz, Feb 07 2021

nmax=94; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^2, {x, 0, nmax}], x]//Drop[#, 2] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=2}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 125}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

seq(n)={Vec((prod(j=1, n, 1-(-x)^j + O(x^n)) - 1)^2)} \\ Andrew Howroyd, Feb 07 2021

from sage.modular.etaproducts import qexp_eta
m=125; k=2;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047654_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047654_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^2. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047655 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^3 in powers of x.

Original entry on oeis.org

1, -3, 3, -1, -3, 6, -6, 6, 0, -3, 6, -9, 8, -6, 0, 0, -6, 6, -13, 3, -6, 3, 0, -3, 6, -9, 6, -3, 6, 0, 6, 6, -3, 11, 0, 6, 0, 9, 0, 0, 0, -3, 13, 0, 0, -6, 0, -6, 3, -3, -6, 0, -15, -6, -3, 0, -6, 0, -6, 0, -6, -6, 0, -11, 0, 0, -6, 0, 6, 0, 6, 0, 0, 0, -3, 19, 12, -3, 0, 0, 6, 6, 6, 6, 0, 0, 6, 0, 21, 3

Offset: 3

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 3..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.

Cf. A001482 - A001488, A001490, A047265, A047638 - A047649, A047654, A341243.

m:=120;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^3 )); // 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:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..92);  # Alois P. Heinz, Feb 07 2021

nmax=92; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^3, {x,0,nmax}], x]//Drop[#, 3] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=3}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 125}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

my(x='x+O('x^99)); Vec((eta(-x)-1)^3) \\ Joerg Arndt, Sep 07 2023

from sage.modular.etaproducts import qexp_eta
m=125; k=3;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047655_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047655_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^3. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A341279 Triangle read by rows: T(n,k) = coefficient of x^n in expansion of (-1 + Product_{j>=1} 1 / (1 + (-x)^j))^k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 3, 3, 4, 0, 1, 0, 1, 4, 6, 4, 5, 0, 1, 0, 2, 5, 9, 10, 5, 6, 0, 1, 0, 2, 8, 13, 16, 15, 6, 7, 0, 1, 0, 2, 9, 21, 26, 25, 21, 7, 8, 0, 1, 0, 2, 12, 27, 44, 45, 36, 28, 8, 9, 0, 1, 0, 3, 15, 40, 63, 80, 71, 49, 36, 9, 10, 0, 1

Offset: 0

Ilya Gutkovskiy, Feb 08 2021

			Triangle T(n,k) begins:
  1;
  0,  1;
  0,  0,  1;
  0,  1,  0,  1;
  0,  1,  2,  0,  1;
  0,  1,  2,  3,  0,  1;
  0,  1,  3,  3,  4,  0,  1;
  0,  1,  4,  6,  4,  5,  0,  1;
  0,  2,  5,  9, 10,  5,  6,  0,  1;
  0,  2,  8, 13, 16, 15,  6,  7,  0,  1;
  0,  2,  9, 21, 26, 25, 21,  7,  8,  0,  1;
  0,  2, 12, 27, 44, 45, 36, 28,  8,  9,  0,  1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give A000007, A000700, A338463, A341241, A341243, A341244, A341245, A341246, A341247, A341251, A341253.
Main diagonal and lower diagonals give A000012, A000004, A001477, A000217, A000290.
Row sums give A307058.
T(2n,n) gives A341265.
Cf. A000009, A047265, A060642, A286352, A308680.

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:
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=0..n), n=0..12);  # Alois P. Heinz, Feb 09 2021

T[n_, k_] := SeriesCoefficient[(-1 + 2/QPochhammer[-1, -x])^k, {x, 0, n}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten

G.f. of column k: (-1 + Product_{j>=1} (1 + x^(2*j-1)))^k.

Sum_{k=0..n} (-1)^(n-k) * T(n,k) = A000009(n).

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

A047654 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^2 in powers of x.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A047655 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^3 in powers of x.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A341279 Triangle read by rows: T(n,k) = coefficient of x^n in expansion of (-1 + Product_{j>=1} 1 / (1 + (-x)^j))^k, n >= 0, 0 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula