A006906 -id:A006906 - cp's OEIS frontend

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

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)

A302830 Expansion of (1/(1 - x))Product_{k>=1} 1/(1 - kx^k).

Original entry on oeis.org

1, 2, 5, 11, 25, 50, 106, 203, 401, 755, 1427, 2597, 4804, 8566, 15352, 27027, 47551, 82187, 142445, 243025, 414919, 700739, 1181236, 1972552, 3293898, 5450728, 9008081, 14791741, 24244399, 39494615, 64266141, 103979929, 167991853, 270190879, 433773933, 693518984

Offset: 0

Ilya Gutkovskiy, Apr 13 2018

nonn

Partial sums of A006906.

Vaclav Kotesovec, Table of n, a(n) for n = 0..6000
Index entries for sequences related to partitions

Cf. A000041, A000070, A006906, A091360, A302831.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+add(b(n-i*j, i-1)*(i^j), j=1..n/i)))
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 13 2018

nmax = 35; CoefficientList[Series[1/(1 - x) Product[1/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[1/(1 - x) Exp[Sum[Sum[k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]

G.f.: (1/(1 - x))*exp(Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j).
From Vaclav Kotesovec, Apr 14 2018: (Start)
a(n) ~ c * 3^(n/3), where
c = 319343.48587983201292657132469068725642363369445... if mod(n,3)=0
c = 319343.34569378454521307030620964478962032866022... if mod(n,3)=1
c = 319343.21458897980925594955657564398036486423380... if mod(n,3)=2
(End)

A303175 a(n) = [x^n] Product_{k=1..n} 1/(1 - (n - k + 1)*x^k).

Original entry on oeis.org

1, 1, 5, 34, 322, 3803, 55297, 953815, 19086057, 434477488, 11086102633, 313318606066, 9714265351819, 327788649292844, 11957321196905337, 468872400449456885, 19666225828334583690, 878560858388253803180, 41645712575272737701666, 2087686693048676581394052

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

			a(0) = 1;
a(1) = [x^1] 1/(1 - x) = 1;
a(2) = [x^2] 1/((1 - 2*x)*(1 - x^2)) = 5;
a(3) = [x^3] 1/((1 - 3*x)*(1 - 2*x^2)*(1 - x^3)) = 34;
a(4) = [x^4] 1/((1 - 4*x)*(1 - 3*x^2)*(1 - 2*x^3)*(1 - x^4)) = 322;
a(5) = [x^5] 1/((1 - 5*x)*(1 - 4*x^2)*(1 - 3*x^3)*(1 - 2*x^4)*(1 - x^5)) = 3803, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 - (n - k + 1)*x^k) begins:
n = 0: (1), 0,   0,    0,    0,     0,  ...
n = 1:  1, (1),  1,    1,    1,     1,  ...
n = 2:  1,  2,  (5),  10,   21,    42,  ...
n = 3:  1,  3,  11,  (34), 106,   320,  ...
n = 4:  1,  4,  19,   78, (322), 1294,  ...
n = 5:  1,  5,  29,  148,  758, (3803), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..384

Cf. A006906, A124577, A206228, A303188, A303189, A303190.

Table[SeriesCoefficient[Product[1/(1 - (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

a(n) ~ n^n * (1 + 1/n + 1/n^2 - 1/n^3 - 3/n^4 - 8/n^5 - 7/n^6 - 13/n^7 + 2/n^8 - 3/n^9 + 31/n^10 + 21/n^11 + 81/n^12 + 2/n^13 + 152/n^14 - 114/n^15 + 173/n^16 - 341/n^17 + 260/n^18 - 936/n^19 + 861/n^20 - 2187/n^21 + 2630/n^22 - 4551/n^23 + 6211/n^24 - 8866/n^25 + 14889/n^26 - 22374/n^27 + 38490/n^28 - 55911/n^29 + 87688/n^30 - ...). - Vaclav Kotesovec, Aug 21 2018

A353993 Expansion of e.g.f. ( Product_{k>0} 1/(1 - k * x^k) )^(1/(1-x)).

Original entry on oeis.org

1, 1, 8, 63, 668, 7850, 115914, 1847286, 34031024, 682177464, 15049816200, 357564279600, 9212847784392, 252552128708568, 7395084613746816, 229412209982127480, 7524339637608261120, 259675490280634374720, 9418707076419411194304, 357606237255136232451264

Offset: 0

Seiichi Manyama, Aug 06 2022

nonn

Cf. A006906, A353992, A356335, A356337, A356408.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-k*x^k)^(1/(1-x))))

a353992(n) = n!*sum(k=1, n, sumdiv(k, d, (k/d)^d/d));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a353992(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A353992(k) * binomial(n-1,k-1) * a(n-k).

A022726 Expansion of 1/Product_{m>=1} (1 - m*q^m)^2.

Original entry on oeis.org

1, 2, 7, 18, 49, 114, 282, 624, 1422, 3058, 6597, 13700, 28564, 57698, 116479, 230398, 453698, 879080, 1696732, 3230578, 6124326, 11486884, 21439480, 39659598, 73036175, 133445640, 242756058, 438680734, 789328034, 1411926186, 2515574329, 4458203590, 7871211452, 13831782146

Offset: 0

N. J. A. Sloane

nonn

Self-convolution of A006906. - Vaclav Kotesovec, Jan 06 2016

Vaclav Kotesovec, Table of n, a(n) for n = 0..6000

Cf. A006906, A022662.

Column k=2 of A297328.

n:=40; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^2:m in [1..n]])); // G. C. Greubel, Jul 25 2018

nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 06 2016 *)

N=66; q='q+O('q^N);
gf= 1/prod(n=1,N, (1-n*q^n)^2 );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

From Vaclav Kotesovec, Jan 07 2016: (Start)
a(n) ~ c * n * 3^(n/3), where
c = 9588921272.54120308291761424720457... = (c0^2 + 2*c1*c2)/3  if mod(n,3)=0
c = 9588921272.50566179874517327053929... = (c2^2 + 2*c0*c1)/3  if mod(n,3)=1
c = 9588921272.49785814355801212400055... = (c1^2 + 2*c0*c2)/3  if mod(n,3)=2
For the constants c0, c1, c2 see A006906.
(End)
G.f.: exp(2*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018

Added more terms, Joerg Arndt, Oct 06 2012

A085643 Refines sequence A078812 using permutations and products of exponents on least prime signatures.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 6, 4, 6, 1, 5, 8, 12, 9, 12, 8, 1, 6, 10, 16, 9, 12, 36, 8, 12, 24, 10, 1, 7, 12, 20, 24, 15, 48, 36, 27, 16, 72, 32, 15, 40, 12, 1, 8, 14, 24, 30, 16, 18, 60, 48, 72, 54, 20, 96, 144, 54, 16, 20, 120, 80, 18, 60, 14, 1

Offset: 1

Alford Arnold, Aug 15 2003

The shape sequence for this table is A000041. Row sums in the example are A000079, A006906 and A001906.

			When the signatures (A025487) are listed in A036035 order the number of permutations of the exponents begin
1, 1 1, 1 2 1, 1 2 1 3 1, 1 2 2 3 3 4 1, ... b(n) and the products begin
1, 2 1, 3 2 1, 4 3 4 2 1, 5 4 6 3 4 2 1, ... c(n) thus we can write
1, 2 1, 3 4 1, 4 6 4 6 1, 5 8 12 9 12 8 1, ... a(n) = b(n)*c(n)

Cf. A078812.

A265955 Expansion of Product_{k>=1} (1 + 2kx^k)/(1 - 2kx^k).

Original entry on oeis.org

1, 4, 16, 60, 192, 596, 1776, 5020, 13760, 36916, 96336, 246316, 619392, 1530548, 3729392, 8976364, 21337920, 50195268, 116977232, 270114764, 618712640, 1406843940, 3176387120, 7126185948, 15894370816, 35253947940, 77796242768, 170868178332, 373606888128

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3280

Cf. A006906, A022629, A032302, A032309, A070933, A265951.

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

a(n) ~ c * n * 2^n, where c = 2 * Product_{m>=3} (1 + 2/(2^(m-1)/m - 1)) = 193.4198278838721371054040810054045645734538119720773785523616944906739...

A301577 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - kx^kA(x)^k).

Original entry on oeis.org

1, 1, 4, 16, 75, 366, 1887, 10010, 54493, 302302, 1703599, 9723774, 56101292, 326640411, 1916800425, 11325242328, 67316128903, 402245682741, 2414978550718, 14560379165160, 88122911824659, 535188028077586, 3260549998701951, 19921639754064470, 122041156818328779

Offset: 0

Ilya Gutkovskiy, Mar 23 2018

nonn

			G.f. A(x) = 1 + x + 4*x^2 + 16*x^3 + 75*x^4 + 366*x^5 + 1887*x^6 + 10010*x^7 + 54493*x^8 + 302302*x^9 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - 2*x^2*A(x)^2) * (1 - 3*x^3*A(x)^3) * ...).
log(A(x)) = x + 7*x^2/2 + 37*x^3/3 + 219*x^4/4 + 1276*x^5/5 + 7687*x^6/6 + 46551*x^7/7 + 285043*x^8/8 + ... + A297329(n)*x^n/n + ...

Cf. A006906, A109085, A297329, A301455, A301578.

A326885 E.g.f.: Product_{k>=1} 1/(1 - k*(exp(x)-1)^k).

Original entry on oeis.org

1, 1, 7, 55, 595, 7351, 110587, 1884415, 36154195, 771983911, 18141124267, 463345240975, 12792709110595, 379854657215671, 12057296962232347, 407072488594360735, 14565548824196479795, 550582832110097346631, 21917855760706255154827, 916261422041320023467695

Offset: 0

Vaclav Kotesovec, Jul 31 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A006906, A167137, A305986, A326884.

nmax = 20; CoefficientList[Series[Product[1/(1-k*(Exp[x]-1)^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = Sum_{k=0..n} A006906(k)*Stirling2(n,k)*k!.

a(n) ~ c * n! / ((3^(2/3) - 2) * (3^(2/3) - 1) * log(1 + 3^(-1/3))^(n+1)), where c = Product_{k>=4} 1/(1 - k/3^(k/3)) = 3468.14377687388560106742710672518465524...

A022739 Expansion of Product (1-m*q^m)^-15; m=1..infinity.

Original entry on oeis.org

1, 15, 150, 1175, 7875, 46953, 255745, 1293825, 6154530, 27772930, 119720802, 495673410, 1979777985, 7656005115, 28752006375, 105129718102, 375082371420, 1308260532180, 4468338006465

Offset: 0

N. J. A. Sloane

nonn

Robert Israel, Table of n, a(n) for n = 0..3000

Cf. A006906, A022610, A022643, A023013.

Column k=15 of A297328.

N:= 30: # for a(0)..a(N)
P:= mul(1-m*q^m,m=1..N):
S:= series(P,q,N+1):
S15:= series(S^(-15),q,N+1):
seq(coeff(S15,q,i),i=0..N); # Robert Israel, Dec 22 2019

nmax = 20; CoefficientList[Series[Product[1/(1 - k*x^k)^15, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 23 2019 *)

cp's OEIS Frontend

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

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A302830 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - k*x^k).

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Maple

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A303175 a(n) = [x^n] Product_{k=1..n} 1/(1 - (n - k + 1)*x^k).

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A353993 Expansion of e.g.f. ( Product_{k>0} 1/(1 - k * x^k) )^(1/(1-x)).

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PARI

PARI

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A022726 Expansion of 1/Product_{m>=1} (1 - m*q^m)^2.

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Magma

Mathematica

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Extensions

A085643 Refines sequence A078812 using permutations and products of exponents on least prime signatures.

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A265955 Expansion of Product_{k>=1} (1 + 2*k*x^k)/(1 - 2*k*x^k).

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Mathematica

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A301577 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - k*x^k*A(x)^k).

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A326885 E.g.f.: Product_{k>=1} 1/(1 - k*(exp(x)-1)^k).

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

A302830 Expansion of (1/(1 - x))Product_{k>=1} 1/(1 - kx^k).

A265955 Expansion of Product_{k>=1} (1 + 2kx^k)/(1 - 2kx^k).

A301577 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - kx^kA(x)^k).