A265758 -id:A265758 - cp's OEIS frontend

A006906 a(n) is the sum of products of terms in all partitions of n.

1, 1, 3, 6, 14, 25, 56, 97, 198, 354, 672, 1170, 2207, 3762, 6786, 11675, 20524, 34636, 60258, 100580, 171894, 285820, 480497, 791316, 1321346, 2156830, 3557353, 5783660, 9452658, 15250216, 24771526, 39713788, 64011924, 102199026, 163583054, 259745051

Offset: 0

Simon Plouffe

a(0) = 1 since the only partition of 0 is the empty partition. The product of its terms is the empty product, namely 1.

Same parity as A000009. - Jon Perry, Feb 12 2004

			Partitions of 0 are {()} whose products are {1} whose sum is 1.
Partitions of 1 are {(1)} whose products are {1} whose sum is 1.
Partitions of 2 are {(2),(1,1)} whose products are {2,1} whose sum is 3.
Partitions of 3 are 3 => {(3),(2,1),(1,1,1)} whose products are {3,2,1} whose sum is 6.
Partitions of 4 are {(4),(3,1),(2,2),(2,1,1),(1,1,1,1)} whose products are {4,3,4,2,1} whose sum is 14.

G. Labelle, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..6000 (first 1001 terms from T. D. Noe)
Atreya Chatterjee, Emergent gravity from patterns in natural numbers, arXiv:2006.01170 [gr-qc], 2020.
Dean Hickerson, Comments on A006906
Robert Schneider and Andrew V. Sills, The product of parts or 'norm' of a partition, #A13 INTEGERS 20A (2020), Theorem 7, p. 4.

Row sums of A118851.

Cf. A000041, A007870, A022629, A022661, A022693, A077335, A163318, A265758, A302830, A318127, A322364, A322365.

a006906 n = p 1 n 1 where
   p _ 0 s = s
   p k m s | mReinhard Zumkeller, Dec 07 2011

A006906 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        add( A078308(k)*procname(n-k),k=1..n)/n ;
    end if;
end proc: # R. J. Mathar, Dec 14 2011
# second Maple program:
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:= n-> b(n, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 25 2013

(* a[n,k]=sum of products of partitions of n into parts <= k *) a[0,0]=1; a[n_,0]:=0; a[n_,k_]:=If[k>n, a[n,n], a[n,k] = a[n,k-1] + k a[n-k,k] ]; a[n_]:=a[n,n] (* Dean Hickerson, Aug 19 2007 *)
Table[Total[Times@@@IntegerPartitions[n]],{n,0,35}] (* Harvey P. Dale, Jan 14 2013 *)
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[PolyLog[-j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 15 2015 *)

The limit of a(n+3)/a(n) is 3. However, the limit of a(n+1)/a(n) does not exist. In fact, the sequence {a(n+1)/a(n)} has three limit points, which are about 1.4422447, 1.4422491 and 1.4422549. (See the Links entry.) - Dean Hickerson, Aug 19 2007
a(n) ~ c(n mod 3) 3^(n/3), where c(0)=97923.26765718877..., c(1)=97922.93936857030... and c(2)=97922.90546334208... - Dean Hickerson, Aug 19 2007
G.f.: 1 / Product_{k>=1} (1-k*x^k).
G.f.: 1 + Sum_{n>=1} n*x^n / Product_{k=1..n} (1-k*x^k) = 1 + Sum_{n>=1} n*x^n / Product_{k>=n} (1-k*x^k). - Joerg Arndt, Mar 23 2011
a(n) = (1/n)*Sum_{k=1..n} A078308(k)*a(n-k). - Vladeta Jovovic, Nov 22 2002
O.g.f.: exp( Sum_{n>=1} Sum_{k>=1} k^n * x^(n*k) / n ). - Paul D. Hanna, Sep 18 2017
O.g.f.: exp( Sum_{n>=1} Sum_{k=1..n} A008292(n,k)*x^(n*k)/(n*(1-x^n)^(n+1)) ), where A008292 is the Eulerian numbers. - Paul D. Hanna, Sep 18 2017

More terms from Vladeta Jovovic, Oct 04 2001

Edited by N. J. A. Sloane, May 19 2007

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

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

A265844 Expansion of Product_{k>=1} (1 + k^2x^k)/(1 - k^2x^k).

Original entry on oeis.org

1, 2, 10, 36, 118, 376, 1188, 3456, 10054, 28814, 79280, 215844, 581748, 1528456, 3987384, 10295952, 26130982, 65874532, 164661622, 406787220, 998529752, 2434022304, 5879630196, 14124455856, 33734350692, 80000820426, 188787849968, 443372664504, 1035137265552

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

Convolution of A092484 and A077335.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A015128, A077335, A092484, A265758.

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

a(n) ~ c * 3^(2*n/3), where
c = 33024.782174678163138510272317... if mod(n,3) = 0
c = 33024.230416953709449028604542... if mod(n,3) = 1
c = 33024.292470246596667257649964... if mod(n,3) = 2.

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)

A292317 Expansion of Product_{k>=1} ((1 - kx^k)/(1 + kx^k)).

Original entry on oeis.org

1, -2, -2, 0, 6, 8, 0, 4, -10, -50, -36, 16, 12, 80, 44, 88, 390, 180, -94, -712, -1624, -312, -688, 1476, 4444, -6954, -5812, 3816, 7728, 36600, 25708, -13308, -53586, -127048, 10104, 120936, 73490, 157400, -395168, -529472, 833888, 265916, 19300, -1132576

Offset: 0

Seiichi Manyama, Sep 14 2017

sign

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

Cf. A022661, A022693, A265758.

nmax = 50; CoefficientList[Series[Product[(1 - k*x^k)/(1 + k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2018 *)

Convolution of A022661 and A022693.

Convolution inverse of A265758.

A292418 a(n) = [x^n] Product_{k>=1} (1 + n^2x^k) / (1 - n^2x^k).

Original entry on oeis.org

1, 2, 40, 1800, 149024, 21223800, 4609532520, 1414165715200, 581109518753920, 307788983933760954, 204081628466048180200, 165541724073121026987224, 161233041454793035411134240, 185663865439487951708529417080, 249499302292252719726304186789160

Offset: 0

Vaclav Kotesovec, Sep 16 2017

nonn

Convolution of A292304 and A292417.

G. C. Greubel, Table of n, a(n) for n = 0..214

Cf. A265758, A265844, A292304, A292417, A292419.

nmax = 20; Table[SeriesCoefficient[Product[(1+n^2*x^k)/(1-n^2*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]

{a(n)= polcoef(prod(k=1, n, ((1+n^2*x^k)/(1-n^2*x^k) +x*O(x^n))), n)};
for(n=0,20, print1(a(n), ", ")) \\ G. C. Greubel, Feb 02 2019

a(n) ~ 2 * n^(2*n) * (1 + 2/n^2 + 4/n^4 + 8/n^6 + 14/n^8 + 24/n^10), for coefficients see A015128.

A326886 E.g.f.: Product_{k>=1} (1 + k(exp(x)-1)^k) / (1 - k(exp(x)-1)^k).

Original entry on oeis.org

1, 2, 14, 134, 1574, 22262, 370694, 7008374, 147805574, 3447703862, 88047037574, 2438080410614, 72703788119174, 2321967591003062, 79030014919422854, 2854499200663284854, 109018338380110506374, 4388176453133542327862, 185612789014681549094534

Offset: 0

Vaclav Kotesovec, Jul 31 2019

nonn

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

Cf. A265758, A306045, A326884, A326885, A326887.

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

a(n) = Sum_{k=0..n} A265758(k)*Stirling2(n,k)*k!.

a(n) ~ c * 2 * (3^(2/3) + 2) * n! / (3*(3^(2/3) - 2) * (3^(1/3) - 1) * log(1 + 3^(-1/3))^(n+1)), where c = Product_{k>=4} (1 + k/3^(k/3)) / (1 - k/3^(k/3)) = 153073.83255100475812062139772279157814388739...

cp's OEIS Frontend

A006906 a(n) is the sum of products of terms in all partitions of n.

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

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

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A265844 Expansion of Product_{k>=1} (1 + k^2*x^k)/(1 - k^2*x^k).

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

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A292317 Expansion of Product_{k>=1} ((1 - k*x^k)/(1 + k*x^k)).

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A292418 a(n) = [x^n] Product_{k>=1} (1 + n^2*x^k) / (1 - n^2*x^k).

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A326886 E.g.f.: Product_{k>=1} (1 + k*(exp(x)-1)^k) / (1 - k*(exp(x)-1)^k).

A265844 Expansion of Product_{k>=1} (1 + k^2x^k)/(1 - k^2x^k).

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

A292317 Expansion of Product_{k>=1} ((1 - kx^k)/(1 + kx^k)).

A292418 a(n) = [x^n] Product_{k>=1} (1 + n^2x^k) / (1 - n^2x^k).

A326886 E.g.f.: Product_{k>=1} (1 + k(exp(x)-1)^k) / (1 - k(exp(x)-1)^k).