A077335 -id:A077335 - 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

A092484 Expansion of Product_{m>=1} (1 + m^2*q^m).

Original entry on oeis.org

1, 1, 4, 13, 25, 77, 161, 393, 726, 2010, 3850, 7874, 16791, 31627, 69695, 139560, 255997, 482277, 986021, 1716430, 3544299, 6507128, 11887340, 21137849, 38636535, 70598032, 123697772, 233003286, 412142276, 711896765, 1252360770

Offset: 0

Jon Perry, Apr 04 2004

nonn

Sum of squares of products of terms in all partitions of n into distinct parts.

			The partitions of 6 into distinct parts are 6, 1+5, 2+4, 1+2+3, the corresponding squares of products are 36, 25, 64, 36 and their sum is a(6) = 161.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A022629, A077335, A265844, A285737, A292165.

Column k=2 of A292189.

b:= proc(n, i) option remember; (m->
      `if`(mn, 0, i^2*b(n-i, i-1)))))(i*(i+1)/2)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2017

Take[ CoefficientList[ Expand[ Product[1 + m^2*q^m, {m, 100}]], q], 31] (* Robert G. Wilson v, Apr 05 2005 *)

N=66; x='x+O('x^N); Vec(prod(n=1, N, 1+n^2*x^n)) \\ Seiichi Manyama, Sep 10 2017

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

Conjecture: log(a(n)) ~ sqrt(2*n) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

More terms from Robert G. Wilson v, Apr 05 2004

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

A265837 Expansion of Product_{k>=1} 1/(1 - k^3*x^k).

Original entry on oeis.org

1, 1, 9, 36, 164, 505, 2474, 7273, 31008, 103644, 379890, 1226802, 4747529, 14553648, 52167558, 171639695, 583371802, 1851395692, 6427705062, 19983302144, 67235043192, 214615427776, 697704303005, 2194982897304, 7262755260410, 22402942281766, 72461661415093

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

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

Cf. A006906, A077335, A265838, A265839, A265840.

Column k=3 of A292193.

nmax = 40; CoefficientList[Series[Product[1/(1 - k^3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^n, where
c = 86.60286320343345379122228784466307940393110978... if n mod 3 = 0
c = 86.27536745612304663727011387030370600864018892... if n mod 3 = 1
c = 86.29819842537784019895326532818285333403267092... if n mod 3 = 2.
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(3*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

A265838 Expansion of Product_{k>=1} 1/(1 - k^4*x^k).

Original entry on oeis.org

1, 1, 17, 98, 610, 2531, 18580, 72453, 449494, 2114440, 10753594, 48572844, 272867295, 1137441506, 5834448870, 27276382027, 129389072144, 576677550870, 2884567552542, 12401875640710, 59474089385344, 270438887909580, 1230979340265033, 5477371267093144

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1550

Cf. A006906, A077335, A265837, A265839, A265841.

Column k=4 of A292193.

nmax = 40; CoefficientList[Series[Product[1/(1 - k^4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^(4*n/3), where
c = 27.2472595510480930563087281042486261391960582835336715327... if n mod 3 = 0
c = 26.8841208067599453033952496040472485838861626762931432887... if n mod 3 = 1
c = 26.9277867007233095885556073185206409643421012262073908850... if n mod 3 = 2.
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(4*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

A265839 Expansion of Product_{k>=1} 1/(1 - k^5*x^k).

Original entry on oeis.org

1, 1, 33, 276, 2324, 13225, 145586, 760057, 6836328, 45996924, 322816122, 2064921330, 16881567137, 96217644312, 708147553326, 4769313137735, 31412238427954, 198869428043476, 1442034056253438, 8596120396405880, 58954590481229064, 387170921610808720

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1240

Cf. A006906, A077335, A265837, A265838, A265842.

Column k=5 of A292193.

nmax = 40; CoefficientList[Series[Product[1/(1 - k^5*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^(5*n/3), where
c = 12.8519823810391431573687005461910113782018563173082562291... if n mod 3 = 0
c = 12.4535903496941652158697054030067622653283880393322526099... if n mod 3 = 1
c = 12.5138855694494734654940524026530463555984202132997900068... if n mod 3 = 2.
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(5*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

A151954 Expansion of Product_{k>0} (1-k^2*x^k)^(-1/k).

Original entry on oeis.org

1, 1, 3, 6, 16, 27, 79, 126, 331, 632, 1436, 2509, 6800, 11218, 26044, 51958, 114941, 205183, 502228, 875545, 2027193, 3963938, 8389190, 15504996, 37555290, 66502859, 145809046, 292860564, 621638120, 1156065731, 2701045579

Offset: 0

Franklin T. Adams-Watters, Jul 03 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..3149

Cf. A028342, A073705, A077335.

nmax = 40; CoefficientList[Series[Product[(1-k^2*x^k)^(-1/k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 05 2017 *)

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} A073705(k)*a(n-k) for n > 0. - Seiichi Manyama, Nov 05 2017
From Vaclav Kotesovec, Nov 05 2017: (Start)
a(n) ~ c * 3^(2*n/3) / n^(2/3), where
c = 4.674336739118905298732313884863019... if mod(n,3)=0
c = 4.299861572054701010776554223312792... if mod(n,3)=1
c = 4.239106098573472870377481583112857... if mod(n,3)=2
(End)

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.

A285674 Expansion of Product_{k>=1} 1/(1 - k^2*x^k)^k.

Original entry on oeis.org

1, 1, 9, 36, 148, 489, 1959, 6326, 22741, 74072, 246436, 781189, 2523042, 7773342, 24200874, 73439472, 222247101, 660405663, 1958564056, 5715567301, 16623600991, 47780474694, 136623175876, 386983158080, 1090779014163, 3048348195528, 8478106666045

Offset: 0

Vaclav Kotesovec, Apr 24 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Cf. A077335, A266941, A285241, A285737.

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

a(n) ~ c * 3^(2*n/3) * n^2, where
c = 76631915822.1860553820452485980060616094557062528483009... if mod(n,3)=0
c = 76631915822.1819974623120987784506295282600132985390786... if mod(n,3)=1
c = 76631915822.1825610530012010285873110459423711856434442... if mod(n,3)=2
In closed form, a(n) ~ (Product_{k>=4}((1 - k^2/3^(2*k/3))^(-k)) / ((1 - 1/3^(2/3)) * (1 - 4/3^(4/3))^2) + Product_{k>=4}((1 - (-1)^(2*k/3)*k^2/3^(2*k/3))^(-k)) / ((-1)^(2*n/3) * ((1 + 4/3*(-1/3)^(1/3))^2 * (1 - (-1/3)^(2/3)))) + Product_{k>=4}((1 - (-1)^(4*k/3)*k^2/3^(2*k/3))^(-k)) / ((-1)^(4*n/3) * ((1 + (-1)^(1/3)/3^(2/3)) * (1 - 4*(-1)^(2/3) / 3^(4/3))^2))) * 3^(2*n/3) * n^2 / 54.

A292164 Expansion of Product_{k>=1} (1 - k^2*x^k).

Original entry on oeis.org

1, -1, -4, -5, -7, 27, 17, 167, 110, -42, 10, -706, -4001, -3915, 3079, -18640, 9869, 21403, 130565, 107250, -15661, 420664, 599540, -161785, -1232833, -5836888, -5129796, 6516714, -29068180, -14953045, -41490510, 20261320, 30395771, 441235155, 205289550

Offset: 0

Seiichi Manyama, Sep 10 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Column k=2 of A292166.

Cf. A022629, A022661, A077335, A092484.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, i^2*b(n-i, i))))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i$2)*a(i$2), i=0..n-1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2017

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
    b[n, i - 1] + If[i > n, 0, i^2*b[n - i, i]]]];
a[n_] := a[n] = If[n == 0, 1,
    -Sum[b[n - i, n - i]*a[i], {i, 0, n - 1}]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 04 2024, after Alois P. Heinz *)

N=66; x='x+O('x^N); Vec(prod(n=1, N, 1-n^2*x^n))

Convolution inverse of A077335.

G.f.: exp(-Sum_{k>=1} Sum_{j>=1} j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

cp's OEIS Frontend

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

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A092484 Expansion of Product_{m>=1} (1 + m^2*q^m).

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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).

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A265837 Expansion of Product_{k>=1} 1/(1 - k^3*x^k).

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A265838 Expansion of Product_{k>=1} 1/(1 - k^4*x^k).

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A265839 Expansion of Product_{k>=1} 1/(1 - k^5*x^k).

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A151954 Expansion of Product_{k>0} (1-k^2*x^k)^(-1/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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A265844 Expansion of Product_{k>=1} (1 + k^2x^k)/(1 - k^2x^k).