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 1-6 of 6 results.

A266964 Expansion of Product_{k>=1} (1 - k*x^k)^k.

Original entry on oeis.org

1, -1, -4, -5, -3, 23, 44, 104, 70, -93, -465, -1155, -1882, -1904, 804, 6195, 18755, 33296, 47327, 35198, -28493, -176199, -453792, -805453, -1126396, -1028297, -18994, 2946491, 8248080, 16444480, 25436984, 30736635, 22263981, -16098311, -102681575
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 07 2016

Keywords

Links

Crossrefs

Cf. A022661, A026007, A266891, A266941, A266971, A296601.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1-k*q^k)^k: k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
  • Maple
    seq(coeff(series(mul((1-k*x^k)^k,k=1..n),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 31 2018
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1-k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
(* More efficient program: *) nmax = 40; s = 1-x; Do[s*=Sum[Binomial[k, j]*(-1)^j*k^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax]
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-k*x^k)^k)) \\ Seiichi Manyama, Nov 18 2017
  • Ruby
    def s(f_ary, g_ary, n)
  s = 0
  (1..n).each{|i| s += i * f_ary[i] * g_ary[i] ** (n / i) if n % i == 0}
  s
end
def A(f_ary, g_ary, n)
  ary = [1]
  a = [0] + (1..n).map{|i| s(f_ary, g_ary, i)}
  (1..n).each{|i| ary << (1..i).inject(0){|s, j| s + a[j] * ary[-j]} / i}
  ary
end
def A266964(n)
  A((0..n).map{|i| -i}, (0..n).to_a, n)
end
p A266964(50) # Seiichi Manyama, Nov 18 2017

Formula

a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} (Sum_{d|k} d^(2+k/d)) * a(n-k) for n > 0. - Seiichi Manyama, Nov 02 2017
From Seiichi Manyama, Nov 14 2017: (Start)
A generalized Euler transform.
Suppose given two sequences f(n) and g(n), n>0, we define a new sequence a(n), n>=0, by Product_{n>0} (1 - g(n)*x^n)^(-f(n)) = a(0) + a(1)*x + a(2)*x^2 + ...
Since Product_{n>0} (1 - g(n)*x^n)^(-f(n)) = exp(Sum_{n>0} (Sum_{d|n} d*f(d)*g(d)^(n/d))*x^n/n), we see that a(n) is given explicitly by a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d*f(d)*g(d)^(n/d).
Examples:
1. If we set g(n) = 1, we get the usual Euler transform.
2. If we set f(n) = -h(n) and g(n) = -1, we get the weighout transform (cf. A026007).
3. If we set f(n) = -n and g(n) = n, we get this sequence.
(End)

A266941 Expansion of Product_{k>=1} 1 / (1 - k*x^k)^k.

Original entry on oeis.org

1, 1, 5, 14, 42, 103, 289, 690, 1771, 4206, 10142, 23449, 54786, 123528, 279480, 619206, 1366405, 2969071, 6425534, 13727775, 29187555, 61439660, 128620370, 267044222, 551527679, 1130806020, 2306746335, 4676096006, 9432394144, 18920266428, 37776372312
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 06 2016

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n, g(n) = n. - Seiichi Manyama, Nov 18 2017

Links

Crossrefs

Cf. A006906, A265758, A266891, A266942, A266964, A266971.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; s = 1 - x; Do[s *= Sum[Binomial[k, j]*(-1)^j*k^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2018 *)

Formula

From Vaclav Kotesovec, Jan 08 2016: (Start)
a(n) ~ c * n^2 * 3^(n/3), where
c = 3278974684037157122864203.021982619109776972432419491714093... if mod(n,3)=0
c = 3278974684037157122864202.999526122508793149896683112820555... if mod(n,3)=1
c = 3278974684037157122864203.001231135511323719311281438384212... if mod(n,3)=2
(End)
In closed form, a(n) ~ (Product_{k>=4}((1 - k/3^(k/3))^(-k)) / ((1 - 2/3^(2/3))^2 * (1 - 1/3^(1/3))) + Product_{k>=4}((1 - (-1)^(2*k/3)*k/3^(k/3))^(-k)) / ((-1)^(2*n/3) * ((1 + 2*(-1)^(1/3)/3^(2/3))^2 * (1 - (-1)^(2/3)/3^(1/3)))) + Product_{k>=4}((1 - (-1)^(4*k/3)*k/3^(k/3))^(-k)) / ((-1)^(4*n/3) * ((1 + (-1/3)^(1/3)) * (1 - 2*(-1/3)^(2/3))^2))) * 3^(n/3) * n^2 / 54. - Vaclav Kotesovec, Apr 24 2017
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (Sum_{d|k} d^(2+k/d)) * a(n-k) for n > 0. - Seiichi Manyama, Nov 02 2017

A297325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + j*x^j)^k.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -1, -2, 0, 1, -4, 0, -2, 2, 0, 1, -5, 2, -1, 9, -1, 0, 1, -6, 5, 0, 18, -2, 4, 0, 1, -7, 9, 0, 27, -12, 10, -1, 0, 1, -8, 14, -2, 35, -36, 11, -16, 18, 0, 1, -9, 20, -7, 42, -76, 14, -54, 38, -22, 0, 1, -10, 27, -16, 49, -132, 35, -104, 84, -98, 12, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2017

Keywords

Examples

			G.f. of column k: A_k(x) = 1 - k*x + (1/2)*k*(k - 3)*x^2 - (1/6)*k*(k^2 - 9*k + 20)*x^3 + (1/24)*k*(k^3 - 18*k^2 + 107*k - 42)*x^4 - (1/120)*k*(k^4 - 30*k^3 + 335*k^2 - 810*k + 624)*x^5 + ...
Square array begins:
  1,  1,  1,   1,   1,   1,  ...
  0, -1, -2,  -3,  -4,  -5,  ...
  0, -1, -1,   0,   2,   5,  ...
  0, -2, -2,  -1,   0,   0,  ...
  0,  2,  9,  18,  27,  35,  ...
  0, -1, -2, -12, -36, -76,  ...

Links

Crossrefs

Columns k=0..32 give A000007, A022693, A022694, A022695, A022696, A022697, A022698, A022699, A022700, A022701, A022702, A022703, A022704, A022705, A022706, A022707, A022708, A022709, A022710, A022711, A022712, A022713, A022714, A022715, A022716, A022717, A022718, A022719, A022720, A022721, A022722, A022723, A022724.
Main diagonal gives A297326.
Antidiagonal sums give A299210.
Cf. A266971, A297321, A297323, A297328.

Programs

  • Maple
    with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, -k*add(add(
      (-d)^(1+j/d), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Apr 20 2018
  • Mathematica
    Table[Function[k, SeriesCoefficient[Product[1/(1 + i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

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

A022693 Expansion of Product_{m>=1} 1/(1 + m*q^m).

Original entry on oeis.org

1, -1, -1, -2, 2, -1, 4, -1, 18, -22, 12, -26, 67, -86, 42, -235, 432, -364, 506, -868, 1434, -2396, 2225, -3348, 10842, -11822, 8049, -24468, 36662, -40024, 69766, -96052, 171976, -278242, 251886, -419723, 885806, -998468, 1103660, -2381042, 4009539, -4478416, 6372514, -9913690
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A006906, A022629, A022661, A266971.

Programs

  • Magma
    Coefficients(&*[1/(1+m*x^m):m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 + k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 15 2015 *)
nmax = 40; CoefficientList[Series[Exp[-Sum[(-1)^(j+1)*PolyLog[-j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 15 2015 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,1/(1+n*q^n))) \\ G. C. Greubel, Feb 25 2018

Formula

From Vaclav Kotesovec, Dec 15 2015: (Start)
a(n) ~ (-1)^n * c * 3^(n/3), where
c = 2.0319526534291644237634198503666896166412... if mod(n,3) = 0
c = 1.8420902462379331740718256785549611496880... if mod(n,3) = 1
c = 1.6677871810486313099783673373643842640151... if mod(n,3) = 2.
(End)
From Benedict W. J. Irwin, Mar 19 2017: (Start)
Conjecture: a(n) = Sum_{i_1,i_2,i_3,...}[(-1)^(i_1+i_2+i_3+...)*Product_{n>0} n^i_n], where the sum is over all valid sequences of positive i_k such that i_1+2*i_2+3*i_3+4*i_4+...= n.
Examples: Setting i_k=0 unless explicitly mentioned.
n=1, (i_1=1), a(1)= -1^1 = -1.
n=2, (i_1=2) or (i_2=1), a(2) = 1^2 - 2^1 = -1.
n=3, (i_1=3) or (i_1=1,i_2=1) or (i_3=1), a(3)=-1^3 + 1^1*2^1 - 3^1 = -2.
(End)

A298988 a(n) = [x^n] Product_{k>=1} 1/(1 + n*x^k)^k.

Original entry on oeis.org

1, -1, 0, -18, 208, -2400, 36504, -663754, 13808320, -324176418, 8487126400, -245122390601, 7741417124880, -265402847130421, 9816338228638872, -389618889514254225, 16518399076342421248, -745025763154442071130, 35619835529954597786208, -1799459812004380374518790, 95780758238408017088795600
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 31 2018

Keywords

Crossrefs

Cf. A255528, A261566, A261567, A266971, A292134, A297326, A298985, A298986, A298987.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}]

Formula

a(n) ~ (-1)^n * n^n * (1 - 2/n + 6/n^2 - 14/n^3 + 33/n^4 - 70/n^5 + 149/n^6 - 298/n^7 + 591/n^8 - 1132/n^9 + 2139/n^10 + ...), for coefficients, see A005380. - Vaclav Kotesovec, Aug 21 2018

A305745 Expansion of Product_{k>=1} ((1 - k*x^k) / (1 + k*x^k))^k.

Original entry on oeis.org

1, -2, -6, -4, 22, 72, 84, -32, -474, -1310, -1728, 60, 6420, 18712, 31080, 24992, -34074, -186468, -430138, -650612, -496296, 687120, 3599652, 8413968, 13374148, 12772246, -3910080, -50592280, -136089520, -244815336, -309079848, -176916784, 391358838
Offset: 0

Views

Author

Seiichi Manyama, Jun 09 2018

Keywords

Links

Crossrefs

Cf. A266942, A266964, A266971, A292317.

Programs

  • Maple
    N:= 50: # for a(0) .. a(N)
f:= mul(((1-k*x^k)/(1+k*x^k))^k,k=1..N):
S:= series(f,x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Mar 01 2024
  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-k*x^k)/(1+k*x^k))^k))

Formula

Convolution of A266964 and A266971.
Convolution inverse of A266942.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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