A006906 -id:A006906 - cp's OEIS frontend

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

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

A007870 Determinant of character table of symmetric group S_n.

Original entry on oeis.org

1, 1, 2, 6, 96, 2880, 9953280, 100329062400, 10651768002183168000, 150283391703941024789299200000, 9263795272057860957392207640004657152000000000, 16027108137650009941734148595388542471170145479274004480000000000000

Offset: 0

Peter J. Cameron, Götz Pfeiffer [ goetz(AT)dcs.st-and.ac.uk ]

nonn

			1 + x + 2*x^2 + 6*x^3 + 96*x^4 + 2880*x^5 + 9953280*x^6 + 100329062400*x^7 + ...
The integer partitions of 4 are {(4), (3,1), (2,2), (2,1,1), (1,1,1,1)} with product 4*3*1*2*2*2*1*1*1*1*1*1 = 96. - _Gus Wiseman_, May 09 2019

Alois P. Heinz, Table of n, a(n) for n = 0..18
Amritanshu Prasad, Symmetric Functions, Chapter 5, Representation Theory: a Combinatorial Viewpoint, Cambridge Studies in Adv. Math. 147 (2014), p. 107.
F. W. Schmidt and R. Simion, On a partition identity, J. Combin. Theory, A 36 (1984), 249-252.
D. Vaintrob, A product identity for partitions, MathOverflow, June 2012.

Row-products of A302246 and A302247.
Cf. A000041, A000142, A006128, A006906, A066186, A066633, A086644, A325501, A325504, A325507, A325536.
Cf. A063073.

List(List([0..11],n->Flat(Partitions(n))),Product); # Muniru A Asiru, Dec 21 2018

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1$2], ((f, g)->
      [f[1]+g[1], f[2]*g[2]*i^g[1]])(b(n, i-1), b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 30 2013

Needs["Combinatorica`"]; Table[Times@@Flatten[Partitions[n]], {n, 10}]
a[ n_] := If[n < 0, 0, Times @@ Flatten @ IntegerPartitions @ n] (* Michael Somos, Jun 11 2012 *)
Table[Exp[Total[Map[Log, IntegerPartitions [n]], 2]], {n, 1, 25}] (* Richard R. Forberg, Dec 08 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 1}, Function[{f, g}, {f[[1]] + g[[1]], f[[2]]*g[[2]]*i^g[[1]]}][If[i < 2, {0, 1}, b[n, i - 1]], If[i > n, {0, 1}, b[n - i, i]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

from sympy import prod
from sympy.utilities.iterables import ordered_partitions
a = lambda n: prod(map(prod, ordered_partitions(n))) if n > 0 else 1
print([a(n) for n in range(0, 12)]) # Darío Clavijo, Feb 22 2024

Product of all parts of all partitions of n.
From Gus Wiseman, May 09 2019: (Start)
a(n) = A003963(A325501(n)).
A001222(a(n)) = A325536(n).
A001221(a(n)) = A000720(n).
(End)

A067553 Sum of products of terms in all partitions of n into odd parts.

Original entry on oeis.org

1, 1, 1, 4, 4, 9, 18, 25, 40, 76, 122, 178, 321, 472, 734, 1303, 1874, 2852, 4782, 6984, 10808, 17552, 25461, 38512, 61586, 90894, 135437, 213260, 312180, 463340, 728806, 1057468, 1562810, 2422394, 3511962, 5215671, 7985196, 11550542, 17022228, 25924746, 37638033

Offset: 0

Naohiro Nomoto, Jan 29 2002

a(0) = 1 as the empty product equals 1. [Joerg Arndt, Oct 06 2012]

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Alois P. Heinz)

Cf. A006906, A022629, A000009.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n or irem(i, 2)=0, 0, i*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 07 2014

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n || Mod[i, 2] == 0, 0, i*b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 02 2015, after Alois P. Heinz *)
nmax = 40; CoefficientList[Series[Product[1/(1-(2*k-1)*x^(2*k-1)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 15 2015 *)

g(n):= if n=0 then 1 else if oddp(n)=true  then n else 0;
P(m,n):=if n=m then g(n) else sum(g(k)*P(k,n-k),k,m,n/2)+g(n);
a(n):=P(1,n);
makelist(a(n),n,0,27); /* Vladimir Kruchinin, Sep 06 2014 */

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

G.f.: 1/(Product_{k>=0} (1-(2*k+1)*x^(2*k+1)) ). - Vladeta Jovovic, May 09 2003
From Vaclav Kotesovec, Dec 15 2015: (Start)
a(n) ~ c * 3^(n/3), where
c = 28.8343667894061364904068323836801301428320806272385991... if mod(n,3) = 0
c = 28.4762018725001067057188975211539643762050439184376103... if mod(n,3) = 1
c = 28.3618072960214990676207117911869616961300790076910101... if mod(n,3) = 2.
(End)

Corrected a(0) from 0 to 1, Joerg Arndt, Oct 06 2012

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

N. J. A. Sloane

sign

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

Cf. A006906, A022629, A022661, A266971.

Coefficients(&*[1/(1+m*x^m):m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018

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 *)

m=50; q='q+O('q^m); Vec(prod(n=1,m,1/(1+n*q^n))) \\ G. C. Greubel, Feb 25 2018

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)

A299162 Expansion of 1/(1 - xProduct_{k>=1} 1/(1 - kx^k)).

Original entry on oeis.org

1, 1, 2, 6, 17, 49, 135, 380, 1051, 2925, 8119, 22548, 62574, 173767, 482360, 1339126, 3717700, 10321163, 28653557, 79548612, 220843925, 613110573, 1702128034, 4725475979, 13118945083, 36421037100, 101112695940, 280710759278, 779313926949, 2163544401343, 6006468273440

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2256
N. J. A. Sloane, Transforms

Antidiagonal sums of A297328.

Cf. A006906, A067687, A299105, A299106, A299108, A299164, A299166, A299167.

nmax = 30; CoefficientList[Series[1/(1 - x Product[1/(1 - k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(0) = 1; a(n) = Sum_{k=1..n} A006906(k-1)*a(n-k).

A265758 Expansion of Product_{k>=1} ((1 + kx^k)/(1 - kx^k)).

Original entry on oeis.org

1, 2, 6, 16, 38, 88, 200, 428, 902, 1874, 3780, 7504, 14732, 28368, 54052, 101960, 189750, 349996, 640218, 1159624, 2084952, 3722008, 6593560, 11606268, 20308188, 35312170, 61065636, 105060200, 179795936, 306244136, 519291476, 876554860, 1473504846

Offset: 0

Vaclav Kotesovec, Dec 15 2015

nonn

Convolution of A022629 and A006906.

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

Cf. A006906, A015128, A022629, A022661, A022693, A261584.

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

a(n) ~ c * 3^(n/3), where
c = 28711548.45004804552683870974706458425598... if mod(n,3) = 0
c = 28711547.74098394497470795294574937283075... if mod(n,3) = 1
c = 28711547.58138731567204220029302329316039... if mod(n,3) = 2.

A002098 G.f.: 1/Product_{k>=1} (1-prime(k)*x^prime(k)).

Original entry on oeis.org

1, 0, 2, 3, 4, 11, 17, 29, 49, 85, 144, 226, 404, 603, 1025, 1679, 2558, 4201, 6677, 10190, 16599, 25681, 39643, 61830, 96771, 147114, 228338, 352725, 533291, 818624, 1263259, 1885918, 2900270, 4396577, 6595481, 10040029, 15166064, 22642064

Offset: 0

N. J. A. Sloane

nonn

a(n) is sum of all numbers k for which A001414(k), the sum of prime factors with repetition, equals n. See Havermann's link. - J. M. Bergot, Jun 14 2013

S.M. Kerawala, On a Pair of Arithmetic Functions Analogous to Chawla's Pair, J. Natural Sciences and Mathematics, 9 (1969), circa p. 103.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..6274 (terms 0..500 from T. D. Noe)
H. Havermann: Tables of sum-of-prime-factors sequences (overview with links to the first 50000 sums).

Cf. A002099, A006906.

Row sums of A064364, A116864.

b:= proc(n,i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=0 then 0
    else b(n, i-1) +b(n-ithprime(i), i) *ithprime(i)
      fi
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 20 2010

b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i==0, 0, True, b[n, i-1] + b[n - Prime[i], i]*Prime[i]]; a[n_] := b[n, PrimePi[n]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)
With[{nn=40},CoefficientList[Series[1/Product[1-Prime[k]x^Prime[k],{k,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Jun 20 2021 *)

my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-isprime(k)*k*x^k)) \\ Seiichi Manyama, Feb 27 2022

Better description and more terms from Vladeta Jovovic, May 09 2003

A265951 Expansion of Product_{k>=1} 1/(1 - 2kx^k).

Original entry on oeis.org

1, 2, 8, 22, 68, 170, 484, 1166, 3048, 7274, 18000, 41806, 100684, 229258, 535692, 1206230, 2758944, 6123650, 13798088, 30284894, 67272756, 146426002, 321513284, 693944814, 1510245960, 3236648578, 6985572672, 14885926182, 31904642348, 67618415690

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

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

Cf. A006906, A265974, A265975, A265976.

Cf. A032309, A070933, A265955.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      2^n, b(n, i-1) +i*2*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 23 2019

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

a(n) ~ c * n * 2^n, where c = 1/2 * Product_{m>=3} 1/(1 - m/2^(m-1)) = 9.281573281805057363737677116134642024212942973614535341005126953773818...

A292193 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).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 5, 1, 1, 9, 14, 14, 7, 1, 1, 17, 36, 46, 25, 11, 1, 1, 33, 98, 164, 107, 56, 15, 1, 1, 65, 276, 610, 505, 352, 97, 22, 1, 1, 129, 794, 2324, 2531, 2474, 789, 198, 30, 1, 1, 257, 2316, 8986, 13225, 18580, 7273, 2314, 354, 42

Offset: 0

Seiichi Manyama, Sep 11 2017

			Square array begins:
   1,  1,  1,   1,   1, ...
   1,  1,  1,   1,   1, ...
   2,  3,  5,   9,  17, ...
   3,  6, 14,  36,  98, ...
   5, 14, 46, 164, 610, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A000041, A006906, A077335, A265837, A265838, A265839.
Rows 0+1, 2 give A000012, A000051.
Main diagonal gives A292194.
Cf. A292166.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      `if`(i>n, 0, i^k*b(n-i, i, k))+b(n, i-1, k))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 11 2017

m = 12;
col[k_] := col[k] = Product[1/(1 - j^k*x^j), {j, 1, m}] + O[x]^(m+1) // CoefficientList[#, x]&;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*j/d)) * A(n-j,k) for n > 0. - Seiichi Manyama, Nov 02 2017

A322364 Numerator of the sum of inverse products of parts in all partitions of n.

Original entry on oeis.org

1, 1, 3, 11, 7, 27, 581, 4583, 2327, 69761, 775643, 147941, 30601201, 30679433, 10928023, 6516099439, 445868889691, 298288331489, 7327135996801, 1029216937671847, 14361631943741, 837902013393451, 2766939485246012129, 274082602410356881, 835547516381094139939

Offset: 0

Alois P. Heinz, Dec 04 2018

			1/1, 1/1, 3/2, 11/6, 7/3, 27/10, 581/180, 4583/1260, 2327/560, 69761/15120, 775643/151200, 147941/26400, 30601201/4989600, 30679433/4633200 ... = A322364/A322365

Alois P. Heinz, Table of n, a(n) for n = 0..505
A. Knopfmacher, J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
D. Zeilberger, N. Zeilberger, Fractional Counting of Integer Partitions, 2018.

Denominators: A322365.

Cf. A000041, A006906, A080130, A177208, A177209, A322380, A322381, A323290, A323291, A323339, A323340.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +b(n-i, min(i, n-i))/i)
    end:
a:= n-> numer(b(n$2)):
seq(a(n), n=0..30);

b[n_, i_] := b[n, i] = If[n==0||i==1, 1, b[n, i-1] + b[n-i, Min[i, n-i]]/i];
a[n_] := Numerator[b[n, n]];
a /@ Range[0, 30] (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)

a(n) = {my(s=0); forpart(p=n, s += 1/vecprod(Vec(p))); numerator(s);} \\ Michel Marcus, Apr 29 2020

Limit_{n-> infinity} a(n)/(n*A322365(n)) = exp(-gamma) = A080130.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A007870 Determinant of character table of symmetric group S_n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

Python

Formula

A067553 Sum of products of terms in all partitions of n into odd parts.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A299162 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 - k*x^k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A265758 Expansion of Product_{k>=1} ((1 + k*x^k)/(1 - k*x^k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A002098 G.f.: 1/Product_{k>=1} (1-prime(k)*x^prime(k)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A299162 Expansion of 1/(1 - xProduct_{k>=1} 1/(1 - kx^k)).

A265758 Expansion of Product_{k>=1} ((1 + kx^k)/(1 - kx^k)).

A265951 Expansion of Product_{k>=1} 1/(1 - 2kx^k).