A022693 -id:A022693 - 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

A022661 Expansion of Product_{m>=1} (1-m*q^m).

Original entry on oeis.org

1, -1, -2, -1, -1, 5, 1, 13, 4, 0, 2, -8, -61, -31, 13, -156, 21, 11, 223, 92, 91, 426, 972, 165, 141, -1126, 440, 1294, -4684, -2755, -5748, -2414, -6679, 10511, -10048, -19369, 19635, 22629, 14027, 76969, -1990, 40193, -10678, 75795, 215767, -54322, -40882

Offset: 0

N. J. A. Sloane

sign

Is a(9) the only occurrence of 0 in this sequence? - Robert Israel, Jun 02 2015

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A006906, A022629, A022693, A266964.

Coefficients(&*[(1-m*x^m):m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 18 2018

P:= mul(1-m*q^m,m=1..100):
S:= series(P,q,101):
seq(coeff(S,q,j),j=0..100); # Robert Israel, Jun 02 2015
# 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);  # Sean A. Irvine (after Alois P. Heinz), May 19 2019

nmax = 40; CoefficientList[Series[Product[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 *)
(* More efficient program: *) nmax = 50; 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 07 2016 *)

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n))) \\ G. C. Greubel, Feb 18 2018

A297325 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, -1, 0, 1, -3, -1, -2, 0, 1, -4, 0, -2, 2, 0, 1, -5, 2, -1, 9, -1, 0, 1, -6, 5, 0, 18, -2, 4, 0, 1, -7, 9, 0, 27, -12, 10, -1, 0, 1, -8, 14, -2, 35, -36, 11, -16, 18, 0, 1, -9, 20, -7, 42, -76, 14, -54, 38, -22, 0, 1, -10, 27, -16, 49, -132, 35, -104, 84, -98, 12, 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, -1, -1,   0,   2,   5,  ...
  0, -2, -2,  -1,   0,   0,  ...
  0,  2,  9,  18,  27,  35,  ...
  0, -1, -2, -12, -36, -76,  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Columns k=0..32 give A000007, A022693, A022694, A022695, A022696, A022697, A022698, A022699, A022700, A022701, A022702, A022703, A022704, A022705, A022706, A022707, A022708, A022709, A022710, A022711, A022712, A022713, A022714, A022715, A022716, A022717, A022718, A022719, A022720, A022721, A022722, A022723, A022724.
Main diagonal gives A297326.
Antidiagonal sums give A299210.
Cf. A266971, A297321, A297323, A297328.

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, -k*add(add(
      (-d)^(1+j/d), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Apr 20 2018

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

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

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

A299210 Expansion of 1/(1 - xProduct_{k>=1} 1/(1 + kx^k)).

Original entry on oeis.org

1, 1, 0, -2, -5, -3, 5, 20, 27, 17, -53, -152, -192, 31, 576, 1110, 694, -1297, -4519, -6160, -1107, 13665, 31914, 30643, -19339, -119260, -196142, -103318, 289543, 859631, 1062684, 13710, -2690348, -5675946, -4940757, 4167527, 21343918, 33874107, 16524162, -51704908, -150454546

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

N. J. A. Sloane, Transforms

Antidiagonal sums of A297325.

Cf. A022693, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299211, A299212.

nmax = 40; 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} A022693(k-1)*a(n-k).

A303190 a(n) = [x^n] Product_{k=1..n} 1/(1 + (n - k + 1)*x^k).

Original entry on oeis.org

1, -1, 3, -22, 224, -2759, 41629, -743319, 15285861, -355719616, 9242332881, -265191971970, 8328195163545, -284124989856012, 10463788330880961, -413744821089831397, 17482192791456272614, -786119610413822514764, 37482612103603819839034, -1888918995730788198553380

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

sign

			a(0) = 1;
a(1) = [x^1] 1/(1 + x) = -1;
a(2) = [x^2] 1/((1 + 2*x)*(1 + x^2)) = 3;
a(3) = [x^3] 1/((1 + 3*x)*(1 + 2*x^2)*(1 + x^3)) = -22;
a(4) = [x^4] 1/((1 + 4*x)*(1 + 3*x^2)*(1 + 2*x^3)*(1 + x^4)) = 224;
a(5) = [x^5] 1/((1 + 5*x)*(1 + 4*x^2)*(1 + 3*x^3)*(1 + 2*x^4)*(1 + x^5)) = -2759, 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,  (3),   -6,   13,    -26,  ...
n = 3:  1,  -3,   7,  (-22),  70,   -208,  ...
n = 4:  1,  -4,  13,   -54, (224),  -890,  ...
n = 5:  1,  -5,  21,  -108,  554, (-2759), ...

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

Cf. A022693, A292134, A303174, A303175, A303188, A303189.

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

a(n) ~ (-1)^n * n^n * (1 - 1/n + 3/n^2 - 7/n^3 + 15/n^4 - 32/n^5 + 65/n^6 - 131/n^7 + 260/n^8 - 501/n^9 + 965/n^10 - 1825/n^11 + 3419/n^12 - 6326/n^13 + 11652/n^14 - 21230/n^15 + 38405/n^16 - 69015/n^17 + 123334/n^18 - 218980/n^19 + 386809/n^20 - 679757/n^21 + 1189360/n^22 - 2071761/n^23 + 3594325/n^24 - 6211826/n^25 + 10698409/n^26 - 18363038/n^27 + 31420994/n^28 - 53605525/n^29 + 91198970/n^30 - ...). - Vaclav Kotesovec, Aug 22 2018

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.

A344369 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(1-s)).

Original entry on oeis.org

1, -2, -3, 0, -5, 0, -7, -8, 0, 0, -11, 0, -13, 0, 0, 16, -17, 0, -19, 0, 0, 0, -23, 24, 0, 0, -27, 0, -29, 30, -31, -32, 0, 0, 0, 36, -37, 0, 0, 40, -41, 42, -43, 0, 0, 0, -47, 0, 0, 0, 0, 0, -53, 54, 0, 56, 0, 0, -59, 60, -61, 0, 0, 64, 0, 66, -67, 0, 0, 70

Offset: 1

Ilya Gutkovskiy, May 16 2021

sign

Cf. A022693, A224892, A316441, A328731, A344368, A344370.

facs[n_] := If[n <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[facs[n/d], Min @@ # >= d &]], {d, Rest[Divisors[n]]}]]; A316441[n_] := Sum[(-1)^Length[f], {f, facs[n]}]; Table[n A316441[n], {n, 70}]

a(n) = n * A316441(n).

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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A022661 Expansion of Product_{m>=1} (1-m*q^m).

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A297325 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.

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

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

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A299210 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)).

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

A299210 Expansion of 1/(1 - xProduct_{k>=1} 1/(1 + kx^k)).

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