A092484 -id:A092484 - cp's OEIS frontend

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

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

A077335 Sum of products of squares of parts in all partitions of n.

Original entry on oeis.org

1, 1, 5, 14, 46, 107, 352, 789, 2314, 5596, 14734, 34572, 92715, 210638, 531342, 1250635, 3042596, 6973974, 16973478, 38399806, 91301956, 207992892, 483244305, 1089029008, 2533640066, 5642905974, 12912848789, 28893132440, 65342580250, 144803524640

Offset: 0

Vladeta Jovovic, Nov 30 2002

nonn

			The partitions of 4 are 4, 1+3, 2+2, 2+1+1, 1+1+1+1, the corresponding products of squares of parts are 16,9,16,4,1 and their sum is a(4) = 46.

Seiichi Manyama, Table of n, a(n) for n = 0..3134 (terms 0..1000 from Alois P. Heinz)

Cf. A006906, A074141, A092484, A265844, A292164.

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:= n-> b(n$2):
seq(a(n), n=0..30);  # 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, 0, i^2*b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 02 2015, after Alois P. Heinz *)
Table[Total[Times@@(#^2)&/@IntegerPartitions[n]],{n,0,30}] (* Harvey P. Dale, Apr 29 2018 *)
Table[Total[Times@@@(IntegerPartitions[n]^2)],{n,0,30}] (* Harvey P. Dale, Sep 07 2023 *)

S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

N=22;q='q+O('q^N); Vec(1/prod(n=1,N,1-n^2*q^n)) \\ Joerg Arndt, Aug 31 2015

G.f.: 1/Product_{m>0} (1 - m^2*x^m).
Recurrence: a(n) = (1/n)*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d^(2*k/d + 1).
a(n) = S(n,1), where S(n,m) = n^2 + Sum_{k=m..n/2} k^2*S(n-k,k), S(n,n) = n^2, S(n,m) = 0 for m > n. - Vladimir Kruchinin, Sep 07 2014
From Vaclav Kotesovec, Mar 16 2015: (Start)
a(n) ~ c * 3^(2*n/3), where
c = 668.1486183948153029651700839617715291485899132694809388646986235... if n=3k
c = 667.8494657534167286226227360927068283390090685342574808235616845... if n=3k+1
c = 667.8481656987523944806949678900876994934226621916594805916358627... if n=3k+2
(End)
In closed form, a(n) ~ (Product_{k>=4}(1/(1 - k^2/3^(2*k/3))) / ((1 - 3^(-2/3)) * (1 - 4*3^(-4/3))) + Product_{k>=4}(1/(1 - (-1)^(2*k/3)*k^2/3^(2*k/3))) / ((-1)^(2*n/3) * (1 + 4/3*(-1/3)^(1/3)) * (1 - (-1/3)^(2/3))) + Product_{k>=4}(1/(1 - (-1)^(4*k/3)*k^2/3^(2*k/3))) / ((-1)^(4*n/3) * (1 + (-1)^(1/3)*3^(-2/3)) * (1 - 4*(-1)^(2/3)*3^(-4/3)))) * 3^(2*n/3 - 1). - Vaclav Kotesovec, Apr 25 2017
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

A292189 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j^k*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 13, 7, 3, 1, 1, 16, 35, 25, 15, 4, 1, 1, 32, 97, 91, 77, 25, 5, 1, 1, 64, 275, 337, 405, 161, 43, 6, 1, 1, 128, 793, 1267, 2177, 1069, 393, 64, 8, 1, 1, 256, 2315, 4825, 11925, 7313, 3799, 726, 120, 10

Offset: 0

Seiichi Manyama, Sep 11 2017

			Square array begins:
   1, 1,  1,  1,   1, ...
   1, 1,  1,  1,   1, ...
   1, 2,  4,  8,  16, ...
   2, 5, 13, 35,  97, ...
   2, 7, 25, 91, 337, ...

Alois P. Heinz, Rows n = 0..150, flattened

Columns k=0..5 give A000009, A022629, A092484, A265840, A265841, A265842.
Rows 0+1, 2, 3 give A000012, A000079, A007689.
Main diagonal gives A292190.
Cf. A292166.

b:= proc(n, i, k) option remember; (m->
      `if`(mn, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Sep 11 2017

m = 14;
col[k_] := col[k] = Product[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 *)

A265840 Expansion of Product_{k>=1} (1 + k^3*x^k).

Original entry on oeis.org

1, 1, 8, 35, 91, 405, 1069, 3799, 8686, 36744, 86310, 235776, 686329, 1605779, 5230579, 13191702, 30608501, 73907925, 206052723, 433747560, 1324608945, 2995740974, 6973434054, 15364943439, 35816669079, 86662644756, 184871083828, 502089539734, 1098571699830

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

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

Cf. A022629, A092484, A265837, A265841, A265842.

Column k=3 of A292189.

nmax = 40; CoefficientList[Series[Product[1 + k^3*x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(3*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

Conjecture: log(a(n)) ~ 3*sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

A265841 Expansion of Product_{k>=1} (1 + k^4*x^k).

Original entry on oeis.org

1, 1, 16, 97, 337, 2177, 7313, 38529, 108594, 717186, 2053522, 7527458, 30757155, 88042387, 448973459, 1390503396, 4087546309, 12699966117, 49599776261, 124699632310, 608410782855, 1651128186296, 4862631132392, 13170300313769, 39285370060347, 130999461143020

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

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

Cf. A022629, A092484, A265838, A265840, A265842.

Column k=4 of A292189.

nmax = 40; CoefficientList[Series[Product[1 + k^4*x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(4*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

Conjecture: log(a(n)) ~ 4*sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

A265842 Expansion of Product_{k>=1} (1 + k^5*x^k).

Original entry on oeis.org

1, 1, 32, 275, 1267, 11925, 51445, 406183, 1406614, 14690040, 51144366, 251885088, 1481359033, 5108404955, 42614629915, 158222158038, 588574803125, 2360755022421, 13255325882835, 39266011999104, 325719196861377, 1031732678138822, 3791401325667894

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

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

Cf. A022629, A092484, A265839, A265840, A265841.

Column k=5 of A292189.

nmax = 40; CoefficientList[Series[Product[1 + k^5*x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(5*k)*x^(j*k)/k). - Ilya Gutkovskiy, Oct 18 2018

Conjecture: log(a(n)) ~ 5*sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

A292304 a(n) = [x^n] Product_{k>=1} (1 + n^2*x^k).

Original entry on oeis.org

1, 1, 4, 90, 272, 1275, 49284, 124901, 536640, 1620648, 104040100, 223290012, 880969104, 2485978170, 7454471332, 592164776475, 1138401673472, 4109108002310, 10877348160900, 30962024560494, 72270337440400, 7523649856001916, 13202150810778116, 44577985082575400

Offset: 0

Vaclav Kotesovec, Sep 14 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A092484, A291698.

nmax = 30; Table[SeriesCoefficient[Product[(1+n^2*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
Table[SeriesCoefficient[QPochhammer[-n^2, x, 1 + n]/(1 + n^2), {x, 0, n}], {n, 0, 30}]

Conjecture: a(n) ~ exp(2*sqrt((Pi^2/6 + 2*log(n)^2)*n)) * (Pi^2/6 + 2*log(n)^2)^(1/4) / (2 * sqrt(Pi) * n^(7/4)).

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.

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

A292165 Expansion of Product_{k>=1} 1/(1 + k^2*x^k).

Original entry on oeis.org

1, -1, -3, -6, 6, 5, 40, 11, 226, -516, -186, -844, 3731, -3734, 814, -33819, 85660, -46022, 210342, -411678, 593996, -2980156, 2076721, -3445584, 40785410, -37503158, 98085, -271846888, 336918770, -295108832, 2178341296, -2404059340, 6127604258

Offset: 0

Seiichi Manyama, Sep 10 2017

sign

Alois P. Heinz, Table of n, a(n) for n = 0..3145

Cf. A077335, A092484.

b:= proc(n, i) option remember; (m->
      `if`(mn, 0, i^2*b(n-i, i-1)))))(i*(i+1)/2)
    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] = Function[m,
     If[m < n, 0, If[n == m, i!^2, b[n, i - 1] +
     If[i > n, 0, i^2*b[n - i, i - 1]]]]][i*(i + 1)/2];
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, Mar 21 2022, after Alois P. Heinz *)

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

Convolution inverse of A092484.
From Vaclav Kotesovec, Sep 10 2017: (Start)
a(n) ~ (-1)^n * c * 3^(2*n/3), where
c = 0.717271758899891528435966115495396784611147877234945... if mod(n,3)=0
c = 0.387695187106751505296020614217498222070185848125472... if mod(n,3)=1
c = 0.241939482775588594057384356004734639024152664456553... if mod(n,3)=2
(End)
G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

cp's OEIS Frontend

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

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A077335 Sum of products of squares of parts in all partitions of n.

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A292189 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j^k*x^j).

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A265840 Expansion of Product_{k>=1} (1 + k^3*x^k).

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A265841 Expansion of Product_{k>=1} (1 + k^4*x^k).

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A265842 Expansion of Product_{k>=1} (1 + k^5*x^k).

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A292304 a(n) = [x^n] Product_{k>=1} (1 + n^2*x^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).