A078308 -id:A078308 - 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

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

A087909 a(n) = Sum_{d|n} (n/d)^(d-1).

Original entry on oeis.org

1, 2, 2, 4, 2, 9, 2, 14, 11, 23, 2, 83, 2, 73, 108, 202, 2, 546, 2, 905, 780, 1037, 2, 5553, 627, 4111, 6644, 12647, 2, 40605, 2, 49682, 59172, 65555, 18028, 382424, 2, 262165, 531612, 869675, 2, 2706581, 2, 3147083, 5180382, 4194329, 2, 27246533, 117651

Offset: 1

Vladeta Jovovic, Oct 15 2003

Seiichi Manyama, Table of n, a(n) for n = 1..6290

Cf. A055225, A078308.

a[n_]:= DivisorSum[n, (n/#)^(#-1) &]; Array[a, 30] (* G. C. Greubel, May 16 2018 *)

a(n)=sumdiv(n, d, d^(n/d-1) );  /* Joerg Arndt, Oct 07 2012 */

N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k*x^k)^(1/k^2))))) \\ Seiichi Manyama, Jun 17 2019

G.f.: Sum_{k>0} x^k/(1-k*x^k).
From Seiichi Manyama, Jun 17 2019: (Start)
L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^(1/k^2)) = Sum_{k>=1} a(k)*x^k/k.
a(p) = 2 for prime p. (End)

A294579 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(1 + k*n/d).

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 9, 10, 7, 1, 17, 28, 25, 6, 1, 33, 82, 97, 26, 12, 1, 65, 244, 385, 126, 80, 8, 1, 129, 730, 1537, 626, 588, 50, 15, 1, 257, 2188, 6145, 3126, 4508, 344, 161, 13, 1, 513, 6562, 24577, 15626, 35652, 2402, 2049, 163, 18

Offset: 1

Seiichi Manyama, Nov 02 2017

			Square array begins:
  1,  1,   1,   1,    1, ...
  3,  5,   9,  17,   33, ...
  4, 10,  28,  82,  244, ...
  7, 25,  97, 385, 1537, ...
  6, 26, 126, 626, 3126, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..2 give A000203, A078308, A294567.
Rows k=0..1 give A000012, A000051(n+1).
Cf. A292166, A292193.

L.g.f. of column k: -log(Product_{j>=1} (1 - j^k * x^j)). - Seiichi Manyama, Jun 02 2019

G.f. of column k: Sum_{j>0} j^(k+1) * x^j / (1 - j^k * x^j). - Seiichi Manyama, Jan 14 2023

A308502 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n/d + k).

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 9, 10, 9, 1, 17, 28, 25, 6, 1, 33, 82, 81, 26, 24, 1, 65, 244, 289, 126, 80, 8, 1, 129, 730, 1089, 626, 330, 50, 41, 1, 257, 2188, 4225, 3126, 1604, 344, 161, 37, 1, 513, 6562, 16641, 15626, 8634, 2402, 833, 163, 68, 1, 1025, 19684, 66049, 78126, 49100, 16808, 5249, 973, 290, 12

Offset: 1

Seiichi Manyama, Jun 02 2019

			Square array begins:
    1,  1,   1,    1,    1,     1, ...
    3,  5,   9,   17,   33,    65, ...
    4, 10,  28,   82,  244,   730, ...
    9, 25,  81,  289, 1089,  4225, ...
    6, 26, 126,  626, 3126, 15626, ...
   24, 80, 330, 1604, 8634, 49100, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..2 give A055225, A078308, A296601.

Cf. A279394, A308504.

T[n_, k_] := DivisorSum[n, #^(n/# + k) &]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

L.g.f. of column k: -log(Product_{j>=1} (1 - j*x^j)^(j^(k-1))).

A296601 L.g.f.: -log(Product_{k>=1} (1 - kx^k)^k) = Sum_{n>=1} a(n)x^n/n.

Original entry on oeis.org

1, 9, 28, 81, 126, 330, 344, 833, 973, 1754, 1332, 5034, 2198, 5658, 8688, 13313, 4914, 28779, 6860, 54106, 45752, 33482, 12168, 254954, 93751, 78906, 255880, 505698, 24390, 1510700, 29792, 1671169, 1791312, 647114, 2819544, 12637371, 50654, 2282346, 14779520, 34058298, 68922, 68084220

Offset: 1

Ilya Gutkovskiy, May 20 2018

nonn

			L.g.f.: L(x) = x + 9*x^2/2 + 28*x^3/3 + 81*x^4/4 + 126*x^5/5 + 330*x^6/6 + 344*x^7/7 + 833*x^8/8 + 973*x^9/9 + ...
exp(L(x)) = 1 + x + 5*x^2 + 14*x^3 + 42*x^4 + 103*x^5 + 289*x^6 + 690*x^7 + 1771*x^8 + 4206*x^9 + ... + A266941(n)*x^n + ...

Seiichi Manyama, Table of n, a(n) for n = 1..5000
Index entries for sequences related to sums of divisors

Column k=2 of A308502.

Cf. A001157, A078308, A266941, A266964.

nmax = 42; Rest[CoefficientList[Series[-Log[Product[(1 - k x^k)^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 42; Rest[CoefficientList[Series[Sum[k^3 x^k/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Sum[d^(n/d + 2), {d, Divisors[n]}]; Table[a[n], {n, 1, 42}]

N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k*x^k)^k)))) \\ Seiichi Manyama, Jun 02 2019

G.f.: Sum_{k>=1} k^3*x^k/(1 - k*x^k).
a(n) = Sum_{d|n} d^(n/d+2).
a(p) = p^3 + 1 where p is a prime.
From Seiichi Manyama, Jun 24 2019: (Start)
Suppose given two sequences f(n) and g(n), n>0, we define a new sequence a(n), n>0, by a(n) = Sum_{d|n} d*f(d)*g(d)^(n/d).
L.g.f.: -log(Product_{n>0} (1 - g(n)*x^n)^f(n)) = Sum_{n>0} a(n)*x^n/n. (See A266964.)
If we set f(n) = n and g(n) = n, we get this sequence. (End)

A353992 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} d^(k/d + 1) )/k.

Original entry on oeis.org

1, 7, 41, 314, 2194, 22764, 195348, 2374224, 27940176, 384636960, 4673720160, 95522440320, 1323221996160, 23481816503040, 489968947641600, 10853692580505600, 190580382936115200, 5408424680491929600, 105077728210820198400, 3399507785578641408000

Offset: 1

Seiichi Manyama, Aug 06 2022

nonn

Cf. A078308, A353993, A356010, A356298, A356406.

a[n_] := n! * Sum[DivisorSum[k, #^(k/# + 1) &]/k, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Aug 06 2022 *)

a(n) = n!*sum(k=1, n, sumdiv(k, d, d^(k/d+1))/k);

a(n) = n!*sum(k=1, n, sumdiv(k, d, (k/d)^d/d));

my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-k*x^k))/(1-x)))

a(n) = n! * Sum_{k=1..n} A078308(k)/k.
a(n) = n! * Sum_{k=1..n} Sum_{d|k} (k/d)^d / d.
E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - k * x^k).

A354848 a(n) = (n-1)! * Sum_{d|n} d^(n/d + 1).

Original entry on oeis.org

1, 5, 20, 150, 624, 9600, 36000, 811440, 6572160, 105235200, 442713600, 39437798400, 81430272000, 4956708556800, 137741700096000, 3014189418240000, 6067609067520000, 1977977787641856000, 2317659281473536000, 1297953221362237440000

Offset: 1

Seiichi Manyama, Jun 08 2022

nonn

Cf. A038048, A078308, A354849, A354851.

a[n_] := (n - 1)! * DivisorSum[n, #^(n/# + 1) &]; Array[a, 20] (* Amiram Eldar, Jun 08 2022 *)

PARI
```
a(n) = (n-1)!*sumdiv(n, d, d^(n/d+1));
```

my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-k*x^k))))

a(n) = (n-1)! * A078308(n).
E.g.f.: -Sum_{k>0} log(1 - k * x^k).
If p is prime, a(p) = (p-1)! + p * p!.

A022736 Expansion of 1/Product_{m>=1} (1 - m*q^m)^12.

Original entry on oeis.org

1, 12, 102, 688, 4029, 21156, 102246, 461448, 1967658, 7990996, 31110432, 116685288, 423366831, 1490904528, 5110173678, 17088259888, 55862240688, 178836472032, 561532752086, 1731639278904, 5250722230962

Offset: 0

N. J. A. Sloane

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 12, g(n) = n. - Seiichi Manyama, Aug 16 2023

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Column k=12 of A297328.

Cf. A078308.

n:=50; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^12:m in [1..n]])); // G. C. Greubel, Jul 25 2018

With[{nmax = 50}, CoefficientList[Series[Product[(1 - k*q^k)^-12, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Jul 25 2018 *)

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^-12)) \\ G. C. Greubel, Jul 25 2018

a(0) = 1; a(n) = (12/n) * Sum_{k=1..n} A078308(k) * a(n-k). - Seiichi Manyama, Aug 16 2023

A022741 Expansion of 1/Product_{m>=1} (1 - m*q^m)^17.

Original entry on oeis.org

1, 17, 187, 1598, 11594, 74477, 435268, 2355265, 11950456, 57389960, 262741086, 1153256370, 4875584955, 19928324460, 79000508546, 304554050669, 1144374191351, 4199521413019, 15076785650771

Offset: 0

N. J. A. Sloane

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 17, g(n) = n. - Seiichi Manyama, Aug 16 2023

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Column k=17 of A297328.

Cf. A078308.

a(0) = 1; a(n) = (17/n) * Sum_{k=1..n} A078308(k) * a(n-k). - Seiichi Manyama, Aug 16 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A087909 a(n) = Sum_{d|n} (n/d)^(d-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A294579 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(1 + k*n/d).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A308502 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n/d + k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A296601 L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^k) = Sum_{n>=1} a(n)*x^n/n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A353992 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} d^(k/d + 1) )/k.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A354848 a(n) = (n-1)! * Sum_{d|n} d^(n/d + 1).

Views

A296601 L.g.f.: -log(Product_{k>=1} (1 - kx^k)^k) = Sum_{n>=1} a(n)x^n/n.