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 1-5 of 5 results.

A307059 Expansion of 1/(2 - Product_{k>=1} (1 - x^k)).

Original entry on oeis.org

1, -1, 0, 1, -1, 1, -1, 1, 0, -2, 4, -4, 1, 3, -5, 4, -3, 3, -1, -6, 13, -12, 2, 9, -13, 10, -6, 6, -4, -9, 28, -30, 5, 25, -28, 5, 9, 7, -27, 11, 32, -47, 2, 51, -27, -74, 128, -34, -131, 183, -78, -15, -37, 97, 89, -480, 649, -242, -498, 904, -663, 223, -140, 169, 488, -1818
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

Comments

Invert transform of A010815.
Alternating row sums of Riordan triangle (1, 1 - Product_{j>=1} (1-x^j) ), See A341418(n, m) without column {1, repeat(0)} for m = 0 and n >= 0. - Wolfdieter Lang, Feb 17 2021

Links

Crossrefs

Cf. A010815, A055887, A299105, A341418.
Cf. A307057, A307058, A307060, A307062, A307063.

Programs

  • Magma
    m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1 - x^j: j in [1..m+2]])) )); // G. C. Greubel, Sep 08 2023
  • Mathematica
    nmax=65; CoefficientList[Series[1/(2 - Product[(1 - x^k), {k, nmax}]), {x, 0, nmax}], x]
  • SageMath
    from sage.modular.etaproducts import qexp_eta
m=80;
def f(x): return 1/(2 - qexp_eta(QQ[['q']], m+2).subs(q=x) )
def A307059_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307059_list(m) # G. C. Greubel, Sep 08 2023

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A010815(k)*a(n-k).
G.f.: 1/(2 - QPochhammer(x)). - G. C. Greubel, Sep 08 2023

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

Views

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

Comments

Invert transform of A081362.

Links

Crossrefs

Cf. A081362, A299208, A304969, A307058.
Cf. A307057, A307058, A307059, A307062, A307063.

Programs

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

Formula

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)

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

Views

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

Comments

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

Links

Crossrefs

Cf. A026007, A257674, A299167, A304969.
Cf. A307057, A307058, A307059, A307060, A307063.

Programs

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

Formula

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

Views

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

Comments

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

Links

Crossrefs

Cf. A022629, A299164, A304969, A320652.
Cf. A307057, A307058, A307059, A307060, A307062.

Programs

  • Magma
    m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[(1+j*x^j): j in [1..m+2]])) ));
  • Mathematica
    nmax = 30; CoefficientList[Series[1/(2 - Product[(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
  • SageMath
    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

Formula

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

A307067 Expansion of 1/(2 - Product_{k>=2} (1 + x^k)).

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 6, 12, 19, 36, 60, 108, 187, 328, 576, 1005, 1765, 3084, 5408, 9461, 16575, 29017, 50812, 88977, 155792, 272813, 477684, 836466, 1464654, 2564685, 4490833, 7863610, 13769463, 24110774, 42218847, 73926591, 129448088, 226667986, 396903536, 694991728
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 22 2019

Keywords

Comments

Invert transform of A025147.

Links

Crossrefs

Cf. A025147, A055887, A299106, A304969, A307057, A307060, A317536.

Programs

  • Magma
    m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1+x^j: j in [2..m+2]])) )); // G. C. Greubel, Jan 24 2024
  • Maple
    a:=series(1/(2-mul((1+x^k),k=2..100)),x=0,40): seq(coeff(a,x,n),n=0..39); # Paolo P. Lava, Apr 03 2019
  • Mathematica
    nmax = 39; CoefficientList[Series[1/(2 - Product[(1 + x^k), {k, 2, nmax}]), {x, 0, nmax}], x]
  • SageMath
    m=80;
def f(x): return 1/( 2 - product(1+x^j for j in range(2, m+3)) )
def A307067_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307067_list(m) # G. C. Greubel, Jan 24 2024

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A025147(k)*a(n-k).
From G. C. Greubel, Jan 24 2024: (Start)
G.f.: (1+x)/(2*(1+x) - QP(x^2)/QP(x)), where QP(x) = QPochhammer(x).
G.f.: (1+x)/(2*(1+x) - x^(1/24)*eta(x^2)/eta(x)), where eta(x) is the Dedekind eta function. (End)
Showing 1-5 of 5 results.

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