A266891 -id:A266891 - cp's OEIS frontend

A304210 Indices of primes in A266891.

3, 4, 25, 36, 54, 116, 122, 303, 559, 649, 720, 1294, 1359, 1757, 2727, 2938, 3618, 3661, 4217, 10143, 12204, 13382, 14136, 15398, 23661, 25880, 32345, 37859, 44955, 50119, 51712, 59989, 64645, 65715, 72135, 72250, 73702, 74939, 103364, 109425, 128847, 146987, 158189, 189186, 195836

Offset: 1

Vaclav Kotesovec, May 08 2018

nonn

No other terms below 200000.

			25 is in the sequence because A266891(25) = 222817961 is prime.

Cf. A285222, A266891, A304211.

A304211 Numbers k such that A266891(k) is divisible by k.

Original entry on oeis.org

1, 2, 252, 481, 497, 585, 619, 19141, 49508, 119096, 125897, 155036, 171507

Offset: 1

Vaclav Kotesovec, May 08 2018

No other terms below 200000.

			252 is in the sequence because A266891(252) = 31553893207174225739699170241560483640785962613789452 = 125213861933231054522615754926827316034864931007101 * 252.

Cf. A266891, A304043, A304210.

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

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)

A022629 Expansion of Product_{m>=1} (1 + m*q^m).

Original entry on oeis.org

1, 1, 2, 5, 7, 15, 25, 43, 64, 120, 186, 288, 463, 695, 1105, 1728, 2525, 3741, 5775, 8244, 12447, 18302, 26424, 37827, 54729, 78330, 111184, 159538, 225624, 315415, 444708, 618666, 858165, 1199701, 1646076, 2288961, 3150951, 4303995, 5870539, 8032571, 10881794, 14749051, 19992626

Offset: 0

N. J. A. Sloane

nonn

Sum of products of terms in all partitions of n into distinct parts. - Vladeta Jovovic, Jan 19 2002

Number of partitions of n into distinct parts, when there are j sorts of part j. a(4) = 7: 4, 4', 4'', 4''', 31, 3'1, 3''1. - Alois P. Heinz, Aug 24 2015

			The partitions of 6 into distinct parts are 6, 1+5, 2+4, 1+2+3, the corresponding products are 6,5,8,6 and their sum is a(6) = 25.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

Cf. A000009, A006906, A022661, A022693, A265758, A266891, A285222, A304043.
Cf. A092484, A265840, A265841, A265842, A322380, A322381.
Column k=1 of A292189.

Coefficients(&*[(1+m*x^m):m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 16 2018

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 1] elif i<1 then [0, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i-1));
         [f[1]+g[1], f[2]+g[2]*i]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 02 2012
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, i*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015

nn=20;CoefficientList[Series[Product[1+i x^i,{i,1,nn}],{x,0,nn}],x]  (* Geoffrey Critzer, Nov 02 2012 *)
nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^(j+1)*PolyLog[-j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2015 *)
(* More efficient program: 10000 terms, 4 minutes, 100000 terms, 6 hours *) nmax = 40; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j+1]] += k*poly[[j-k+1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 06 2016 *)

N=66; q='q+O('q^N); Vec(prod(n=1,N, (1+n*q^n) )) \\ Joerg Arndt, Oct 06 2012

Conjecture: log(a(n)) ~ sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, May 08 2018

A297321 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j*x^j)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 7, 0, 1, 5, 14, 28, 28, 15, 0, 1, 6, 20, 48, 69, 64, 25, 0, 1, 7, 27, 75, 137, 174, 133, 43, 0, 1, 8, 35, 110, 240, 380, 413, 266, 64, 0, 1, 9, 44, 154, 387, 726, 998, 933, 513, 120, 0, 1, 10, 54, 208, 588, 1266, 2075, 2488, 2046, 1000, 186, 0

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

			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,   2,   5,    9,   14,   20,  ...
0,   5,  14,   28,   48,   75,  ...
0,   7,  28,   69,  137,  240,  ...
0,  15,  64,  174,  380,  726,  ...

G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened

Columns k=0..32 give A000007, A022629, A022630, A022631, A022632, A022633, A022634, A022635, A022636, A022637, A022638, A022639, A022640, A022641, A022642, A022643, A022644, A022645, A022646, A022647, A022648, A022649, A022650, A022651, A022652, A022653, A022654, A022655, A022656, A022657, A022658, A022659, A022660.
Main diagonal gives A297322.
Antidiagonal sums give A299164.
Cf. A266891, A297323, A297325, A297328.

Table[Function[k, SeriesCoefficient[Product[(1 + i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

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

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

Vaclav Kotesovec, Jan 06 2016

nonn

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..6086 (terms 0..1000 from Vaclav Kotesovec)

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

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 *)

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

A261562 Expansion of Product_{k>=1} (1 + 2*x^k)^k.

Original entry on oeis.org

1, 2, 4, 14, 24, 58, 124, 238, 480, 922, 1764, 3238, 6008, 10794, 19292, 34166, 59504, 103042, 176452, 299958, 505240, 845570, 1403324, 2315118, 3794640, 6180370, 10009540, 16121374, 25829512, 41171690, 65320956, 103140062, 162149488, 253823178, 395698276

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 2001 terms from Vaclav Kotesovec)

Cf. A026007, A032302, A261561, A261563, A266857, A266891.

b:= proc(n, i) option remember;  `if`(n=0, 1, `if`(i<1, 0,
      add(2^j*binomial(i, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 21 2018

nmax = 50; CoefficientList[Series[Product[(1 + 2*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*2^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 50; s = 1+2*x; Do[s*=Sum[Binomial[k, j]*2^j*x^(j*k), {j, 0, nmax/k}]; s = Take[Expand[s], Min[nmax + 1, Exponent[s, x] + 1]];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Jan 08 2016 *)

{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, -(-2)^d * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 30 2015

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} -(-2)^d * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015

a(n) ~ c^(1/6) * exp(3^(2/3)*c^(1/3)*n^(2/3)/2) / (3^(3/4)*sqrt(2*Pi)*n^(2/3)), where c = Pi^2*log(2) + log(2)^3 - 6*polylog(3, -1/2) = 10.00970018379942727227807189532511265744588249928680712064... . - Vaclav Kotesovec, Jan 04 2016

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

A281790 Expansion of Product_{k>=1} (1+x^(k^2))^k.

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 0, 1, 4, 3, 0, 0, 6, 6, 0, 4, 7, 6, 3, 8, 8, 6, 6, 4, 21, 20, 4, 1, 34, 34, 2, 8, 23, 44, 28, 19, 18, 54, 54, 18, 56, 65, 46, 25, 100, 94, 38, 42, 85, 169, 107, 56, 69, 226, 194, 62, 111, 194, 241, 125, 215, 246, 258, 207, 283, 437, 292

Offset: 0

Vaclav Kotesovec, Apr 14 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Cf. A000219, A026007, A027998, A033461, A255528, A266891, A284896, A285047.

nmax = 100; CoefficientList[Series[Product[(1+x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; s = 1 + x; Do[s*=Sum[Binomial[k, j]*x^(j*k^2), {j, 0, Floor[nmax/k^2] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x]

a(n) ~ exp(sqrt(n/6)*Pi) / (2^(11/6) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 15 2017

A298987 a(n) = [x^n] Product_{k>=1} (1 + n*x^k)^k.

Original entry on oeis.org

1, 1, 4, 27, 80, 400, 1908, 6223, 31296, 116478, 450100, 1828915, 7360848, 26906828, 95776772, 403908975, 1421758720, 5072014447, 18481180644, 68350964211, 246180936400, 827642046294, 2958748580084, 10294629775620, 36607347335232, 120800714172500, 407951731319860, 1405943613730899

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

nonn

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

Cf. A026007, A261562, A266857, A266891, A291698, A297322, A298985, A298986, A298988.

Table[SeriesCoefficient[Product[(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

cp's OEIS Frontend

A304210 Indices of primes in A266891.

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A304211 Numbers k such that A266891(k) is divisible by k.

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

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A022629 Expansion of Product_{m>=1} (1 + m*q^m).

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A297321 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j*x^j)^k.

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

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A261562 Expansion of Product_{k>=1} (1 + 2*x^k)^k.

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

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