A266941 -id:A266941 - cp's OEIS frontend

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

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

Vaclav Kotesovec, Jan 07 2016

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)

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

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1-k*q^k)^k: k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

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

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]

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-k*x^k)^k)) \\ Seiichi Manyama, Nov 18 2017

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

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)

A297328 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, 3, 0, 1, 3, 7, 6, 0, 1, 4, 12, 18, 14, 0, 1, 5, 18, 37, 49, 25, 0, 1, 6, 25, 64, 114, 114, 56, 0, 1, 7, 33, 100, 219, 312, 282, 97, 0, 1, 8, 42, 146, 375, 676, 855, 624, 198, 0, 1, 9, 52, 203, 594, 1276, 2030, 2178, 1422, 354, 0, 1, 10, 63, 272, 889, 2196, 4155, 5736, 5496, 3058, 672, 0

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

			G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k + 5)*x^2 + (1/6)*k*(k^2 + 15*k + 20)*x^3 + (1/24)*k*(k^3 + 30*k^2 + 155*k + 150)*x^4 + (1/120)*k*(k^4 + 50*k^3 + 575*k^2 + 1750*k + 624)*x^5 + ...
Square array begins:
  1,   1,    1,    1,    1,     1,  ...
  0,   1,    2,    3,    4,     5,  ...
  0,   3,    7,   12,   18,    25,  ...
  0,   6,   18,   37,   64,   100,  ...
  0,  14,   49,  114,  219,   375,  ...
  0,  25,  114,  312,  676,  1276,  ...

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

Columns k=0..32 give A000007, A006906, A022726, A022727, A022728, A022729, A022730, A022731, A022732, A022733, A022734, A022735, A022736, A022737, A022738, A022739, A022740, A022741, A022742, A022743, A022744, A022745, A022746, A022747, A022748, A022749, A022750, A022751, A022752, A022753, A022754, A022755, A022756.
Main diagonal gives A297329.
Antidiagonal sums give A299162.
Cf. A078308, A266941, A297321, A297323, A297325.

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

first(n, k) = my(res = matrix(n, k)); for(u=1, k, my(col = Vec(prod(j=1, n, 1/(1 - j*x^j)^(u-1)) + O(x^n))); for(v=1, n, res[v, u] = col[v])); res \\ Iain Fox, Dec 28 2017

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

A(0,k) = 1; A(n,k) = (k/n) * Sum_{j=1..n} A078308(j) * A(n-j,k). - Seiichi Manyama, Aug 16 2023

A285047 Expansion of Product_{k>=1} 1/(1-x^(k^2))^k.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 3, 6, 9, 9, 9, 13, 19, 19, 19, 28, 37, 43, 43, 57, 69, 81, 81, 100, 132, 150, 160, 184, 236, 260, 280, 319, 391, 460, 490, 565, 657, 771, 811, 922, 1084, 1243, 1363, 1510, 1781, 1985, 2185, 2388, 2775, 3159, 3439, 3832, 4335, 4963, 5323

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000219, A001156, A023871, A266941, A281790.

nmax=100; CoefficientList[Series[Product[1/(1-x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]

log(a(n)) ~ Pi*sqrt(n/3).

A266971 Expansion of Product_{k>=1} 1 / (1 + k*x^k)^k.

Original entry on oeis.org

1, -1, -3, -6, 2, 9, 41, 46, 91, -110, -210, -713, -574, -1152, 792, 1066, 9317, 8553, 21302, 745, 8051, -82940, -76750, -276022, -82369, -404100, 381095, -38110, 2427272, 1126260, 6527840, 198507, 9754305, -14320206, 2879362, -60271740, -5154261, -143468194

Offset: 0

Vaclav Kotesovec, Jan 07 2016

sign

For n > 36 is a(n) > 0 if n is even and a(n) < 0 if n is odd.

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

Seiichi Manyama, Table of n, a(n) for n = 0..6224 (terms 0..1000 from Vaclav Kotesovec)

Cf. A022629, A022693, A266891, A266941, A266964.

nmax=50; CoefficientList[Series[Product[1/(1+k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+k*x^k)^k)) \\ Seiichi Manyama, Nov 18 2017

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 A266971(n)
  A((0..n).to_a, (0..n).map{|i| -i}, n)
end
p A266971(50) # Seiichi Manyama, Nov 18 2017

a(n) ~ c * (-1)^n * n^2 * 3^(n/3), where
c = 50.5838262902886367070621... if mod(n,3)=0,
c = 50.5827771239052189170531... if mod(n,3)=1,
c = 50.5832885870455104598393... if mod(n,3)=2.
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(-d)^(n/d). - Seiichi Manyama, Nov 18 2017

A285241 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^2).

Original entry on oeis.org

1, 1, 9, 36, 140, 481, 1774, 5925, 20076, 64980, 208486, 652058, 2017023, 6117878, 18347256, 54222195, 158463794, 457570786, 1307951914, 3700153918, 10371860026, 28810051738, 79359812567, 216834266612, 587961817595, 1582612248239, 4230325722508

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A006906, A023871, A266941, A285240, A285243.

nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^(k^2), {k,1,nmax}], {x,0,nmax}], x]

a(n) ~ c * n^8 * 3^(n/3), where
if mod(n,3) = 0 then c = 3435237242728465092737309192093188152686332293\
03276380306112638865540880372901642880694943679256417087889777743957063\
209444405157397505005623042846150296486667845382334521513094023.8560142\
40331306860864399770618296475558098172993864629247911801570500913143642\
65158886200452165335605783203726486071335...
if mod(n,3) = 1 then c = 3435237242728465092737309192093188152686332293\
03276380306112638865540880372901642880694943679256417087889777743957063\
209444405157397505005623042846150296486667845382334521513094023.8560112\
77299895134841028015999951571187798033179513268954711586617617334007687\
07198348808962592621276659532114355538024...
if mod(n,3) = 2 then c = 3435237242728465092737309192093188152686332293\
03276380306112638865540880372901642880694943679256417087889777743957063\
209444405157397505005623042846150296486667845382334521513094023.8560117\
00278534968233203470801053870003971422069097966617636511346003845666735\
79293861331368526745743422198017148868212...
In closed form, a(n) ~ -(27*Product_{k>=4}((1 - k / 3^(k/3))^(-k^2)) / (13 + 128*3^(1/3) - 95*3^(2/3)) + 243*Product_{k>=4}((1 + (-1)^(1 + 2*k/3) * k / 3^(k/3))^(-k^2)) / ((-1)^(2*n/3) * ((3 + 2*(-3)^(1/3))^4 * (-3 + (-3)^(2/3)))) + (-1)^(1 - 4*n/3) * Product_{k>=4}((1 + (-1)^(1 + 4*k/3) * k / 3^(k/3))^(-k^2)) / ((1 + (-1/3)^(1/3)) * (1 - 2*(-1/3)^(2/3))^4)) / 793618560 * n^8 * 3^(n/3).

A298985 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k)^k.

Original entry on oeis.org

1, 1, 8, 54, 496, 5400, 73728, 1204322, 23167808, 512093178, 12781430600, 355128859129, 10863077554224, 362572265689777, 13107541496092960, 510105773344747725, 21258690342206888192, 944467894258279964254, 44555341678790400325512, 2224158766859058600584834, 117123916650423288611260400

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A000219, A124577, A261561, A261565, A266941, A297329, A298986, A298987, A298988.

b:= proc(n, i, k) option remember;   `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i+j-1, j)*b(n-i*j, i-1, k)*k^j, j=0..n/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 23 2018

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

a(n) ~ n^n. - Vaclav Kotesovec, Feb 02 2018

A266942 Expansion of Product_{k>=1} ((1 + kx^k) / (1 - kx^k))^k.

Original entry on oeis.org

1, 2, 10, 36, 118, 376, 1156, 3392, 9734, 27230, 74256, 198724, 522292, 1348968, 3432824, 8613856, 21330374, 52190692, 126262774, 302222388, 716247128, 1681575344, 3912919956, 9028823856, 20667406276, 46949343786, 105881451120, 237135574392, 527580701456

Offset: 0

Vaclav Kotesovec, Jan 06 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A006906, A022629, A265758, A266891, A266941.

nmax = 40; CoefficientList[Series[Product[((1+k*x^k)/(1-k*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

From Vaclav Kotesovec, Jan 08 2016: (Start)
a(n) ~ c * n^2 * 3^(n/3), where
c = 1122422673446372185062691708933615715850.583956830118389527... if mod(n,3)=0
c = 1122422673446372185062691708933615715849.484130848291097773... if mod(n,3)=1
c = 1122422673446372185062691708933615715849.782119252925454917... if mod(n,3)=2
(End)

A285243 Expansion of Product_{k>=1} 1/(1 - k*x^(k^2))^k.

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 5, 5, 17, 26, 26, 26, 58, 94, 94, 94, 190, 298, 352, 352, 608, 896, 1112, 1112, 1752, 2641, 3289, 3559, 5095, 7499, 9227, 10307, 14051, 20111, 25520, 28760, 38843, 53467, 68191, 76831, 102187, 138283, 175543, 202813, 263905, 355220, 445364

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A285047, A266941, A285241, A285242, A285245.

nmax = 100; CoefficientList[Series[Product[1/(1 - k*x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]

a(n) ~ c * n * 2^(n/4), where
c = 37.4093119651465404809069752821426852731608123... if mod(n,4)=0
c = 37.6275180026872367633343656570058911570800766... if mod(n,4)=1
c = 37.7650387085085950514850376086515488784106690... if mod(n,4)=2
c = 37.4702467422193571732026074780460498930830447... if mod(n,4)=3

A285674 Expansion of Product_{k>=1} 1/(1 - k^2*x^k)^k.

Original entry on oeis.org

1, 1, 9, 36, 148, 489, 1959, 6326, 22741, 74072, 246436, 781189, 2523042, 7773342, 24200874, 73439472, 222247101, 660405663, 1958564056, 5715567301, 16623600991, 47780474694, 136623175876, 386983158080, 1090779014163, 3048348195528, 8478106666045

Offset: 0

Vaclav Kotesovec, Apr 24 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Cf. A077335, A266941, A285241, A285737.

nmax=40; CoefficientList[Series[Product[1/(1-k^2*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^(2*n/3) * n^2, where
c = 76631915822.1860553820452485980060616094557062528483009... if mod(n,3)=0
c = 76631915822.1819974623120987784506295282600132985390786... if mod(n,3)=1
c = 76631915822.1825610530012010285873110459423711856434442... if mod(n,3)=2
In closed form, a(n) ~ (Product_{k>=4}((1 - k^2/3^(2*k/3))^(-k)) / ((1 - 1/3^(2/3)) * (1 - 4/3^(4/3))^2) + Product_{k>=4}((1 - (-1)^(2*k/3)*k^2/3^(2*k/3))^(-k)) / ((-1)^(2*n/3) * ((1 + 4/3*(-1/3)^(1/3))^2 * (1 - (-1/3)^(2/3)))) + Product_{k>=4}((1 - (-1)^(4*k/3)*k^2/3^(2*k/3))^(-k)) / ((-1)^(4*n/3) * ((1 + (-1)^(1/3)/3^(2/3)) * (1 - 4*(-1)^(2/3) / 3^(4/3))^2))) * 3^(2*n/3) * n^2 / 54.

A294585 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^k*x^j)^(j^k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 14, 5, 1, 1, 65, 98, 42, 7, 1, 1, 257, 794, 514, 103, 11, 1, 1, 1025, 6818, 7194, 2435, 289, 15, 1, 1, 4097, 60074, 107170, 69475, 12752, 690, 22, 1, 1, 16385, 535538, 1649322, 2177411, 715277, 58849, 1771, 30

Offset: 0

Seiichi Manyama, Nov 03 2017

			Square array begins:
   1,  1,   1,    1,      1, ...
   1,  1,   1,    1,      1, ...
   2,  5,  17,   65,    257, ...
   3, 14,  98,  794,   6818, ...
   5, 42, 514, 7194, 107170, ...

Seiichi Manyama, Antidiagonals n = 0..113, flattened

Columns k=0..2 give A000041, A266941, A294586.

Rows n=0-1 give A000012.

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(k+1+k*j/d)) * A(n-j,k) for n > 0.

cp's OEIS Frontend

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

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A297328 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.

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A285047 Expansion of Product_{k>=1} 1/(1-x^(k^2))^k.

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A266971 Expansion of Product_{k>=1} 1 / (1 + k*x^k)^k.

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A285241 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^2).

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A298985 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k)^k.

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A266942 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - k*x^k))^k.

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A285243 Expansion of Product_{k>=1} 1/(1 - k*x^(k^2))^k.

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A266942 Expansion of Product_{k>=1} ((1 + kx^k) / (1 - kx^k))^k.