A174465 -id:A174465 - cp's OEIS frontend

A280473 G.f.: Product_{i>=1, j>=1, k>=1} (1 + x^(ijk)).

1, 1, 3, 6, 12, 21, 43, 70, 127, 215, 364, 591, 989, 1562, 2515, 3954, 6194, 9538, 14754, 22349, 33926, 50910, 76102, 112721, 166747, 244205, 356984, 518344, 749924, 1078711, 1547668, 2207418, 3140135, 4446572, 6276657, 8823776, 12371487, 17275879, 24061878

Offset: 0

Vaclav Kotesovec, Jan 04 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A007425, A107742, A174465, A219560, A280486.

nmax = 50; CoefficientList[Series[Product[(1+x^(i*j*k)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}], {x, 0, nmax}], x]
nmax = 50; A007425 = Table[Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 + x; Do[s *= Sum[Binomial[A007425[[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}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Aug 30 2018 *)

G.f.: Product_{k>=1} (1 + x^k)^tau_3(k), where tau_3() = A007425. - Ilya Gutkovskiy, May 22 2018

A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(ijk*l)).

Original entry on oeis.org

1, 1, 5, 9, 29, 49, 135, 235, 565, 995, 2177, 3821, 7900, 13728, 26974, 46606, 88128, 150644, 276283, 467647, 835708, 1400874, 2448818, 4065230, 6975307, 11470265, 19359345, 31552473, 52488142, 84808548, 139274675, 223191639, 362297234, 576064732, 925295844

Offset: 0

Vaclav Kotesovec, Jan 04 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000041, A006171, A007426, A174465, A280486.

nmax=50; CoefficientList[Series[Product[1/(1-x^(i*j*k*l)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}, {l, 1, nmax/i/j/k}], {x, 0, nmax}], x]
nmax = 50; tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#] * DivisorSigma[0, #] &], {n, 1, nmax}]; s = 1 - x; Do[s *= Sum[Binomial[tau4[[k]], j]*(-1)^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, Sep 08 2018 *)

G.f.: Product_{k>=1} 1/(1 - x^k)^tau_4(k), where tau_4() = A007426. - Ilya Gutkovskiy, May 22 2018

A305050 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))/(1 - x^(ijk)).

Original entry on oeis.org

1, 2, 8, 20, 56, 128, 316, 684, 1532, 3192, 6704, 13436, 26984, 52352, 101316, 191320, 359334, 662292, 1213360, 2189380, 3925432, 6951592, 12231332, 21298452, 36856840, 63211164, 107765896, 182295468, 306625208, 512190992, 851011960, 1405199028, 2308629300, 3771593392

Offset: 0

Ilya Gutkovskiy, May 24 2018

nonn

Convolution of the sequences A174465 and A280473.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms
Index entries for sequences related to partitions

Cf. A007425, A174465, A280473, A301554.

nmax = 33; CoefficientList[Series[Product[(1 + x^(i j k))/(1 - x^(i j k)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^Sum[DivisorSigma[0, d], {d, Divisors[k]}], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A007425(k).

A318413 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - x^(ijk))^(ijk).

Original entry on oeis.org

1, 1, 7, 16, 61, 130, 429, 945, 2684, 5990, 15530, 34313, 83995, 183070, 427046, 919480, 2067589, 4384678, 9577536, 20019243, 42664087, 87954522, 183573639, 373430131, 765524808, 1537737243, 3102614407, 6159028445, 12252086879, 24051526041, 47239506797, 91765428710, 178156003047

Offset: 0

Ilya Gutkovskiy, Aug 26 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000005, A007425, A034718, A174465, A174467, A280540, A318414.

a:=series(mul(mul(mul(1/(1-x^(i*j*k))^(i*j*k),k=1..55),j=1..55),i=1..55),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 02 2019

nmax = 32; CoefficientList[Series[Product[Product[Product[1/(1 - x^(i j k))^(i j k), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Product[1/(1 - x^k)^(k Sum[DivisorSigma[0, d], {d, Divisors[k]}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Exp[Sum[Sum[d^2 Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}]  x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]
nmax = 50; A034718 = Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 - x; Do[s *= Sum[Binomial[A034718[[k]], j]*(-1)^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 31 2018 *)

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*tau_3(k)), where tau_3() = A007425.
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d^2 * Sum_{j|d} tau(j) ) * x^k/k), where tau() = A000005.
Conjecture: log(a(n)) ~ (3*Zeta(3))^(1/3) * log(n)^(2/3) * n^(2/3) / 2. - Vaclav Kotesovec, Sep 02 2018

A321360 Expansion of Product_{1 <= i <= j <= k} 1/(1 - x^(ijk)).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 14, 19, 32, 44, 67, 91, 139, 186, 269, 362, 517, 686, 958, 1264, 1741, 2286, 3092, 4033, 5416, 7018, 9296, 11998, 15769, 20228, 26356, 33648, 43539, 55343, 71079, 89942, 114909, 144775, 183819, 230746, 291557, 364544, 458371, 571084, 714971, 887798, 1106704

Offset: 0

Seiichi Manyama, Nov 07 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A034836, A174465, A182269, A321359, A321361.

Euler transform of A034836.

G.f.: Product_{k>0} 1/(1 - x^k)^A034836(k).

A174466 a(n) = Sum_{d|n} dsigma(n/d)tau(d).

Original entry on oeis.org

1, 7, 10, 31, 16, 70, 22, 111, 64, 112, 34, 310, 40, 154, 160, 351, 52, 448, 58, 496, 220, 238, 70, 1110, 166, 280, 334, 682, 88, 1120, 94, 1023, 340, 364, 352, 1984, 112, 406, 400, 1776, 124, 1540, 130, 1054, 1024, 490, 142, 3510, 316, 1162, 520, 1240

Offset: 1

Paul D. Hanna, Apr 04 2010

Compare to sigma_2(n) = Sum_{d|n} d*sigma(n/d)*phi(d) = sum of squares of divisors of n.
tau(n) = A000005(n) = the number of divisors of n,
and sigma(n) = A000203(n) = sum of divisors of n.
Dirichlet convolution of A038040 and A000203. - R. J. Mathar, Feb 06 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005 (tau), A000203 (sigma), A007425 (tau_3), A034718, A038040, A174465.

a174466 n = sum $ zipWith3 (((*) .) . (*))
                  divs (map a000203 $ reverse divs) (map a000005 divs)
                  where divs = a027750_row n
-- Reinhard Zumkeller, Jan 21 2014

[&+[d*DivisorSigma(1, n div d)*#Divisors(d):d in Divisors(n)]:n in [1..55]]; // Marius A. Burtea, Oct 18 2019

f[p_, e_] := ((e^2+3*e+2)*p^(e+3) - 2*(e^2+4*e+3)*p^(e+2) + (e^2+5*e+6)*p^(e+1) - 2)/(2*(p-1)^3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2025 *)

{a(n)=sumdiv(n,d,d*sigma(n/d)*sigma(d,0))}

Logarithmic derivative of A174465.
Dirichlet g.f.: zeta(s)*(zeta(s-1))^3. - R. J. Mathar, Feb 06 2011
a(n) = Sum_{d|n} tau_3(d)*d = Sum_{d|n} A007425(d)*d. - Enrique Pérez Herrero, Jan 17 2013
G.f.: Sum_{k>=1} k*tau_3(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 06 2018
Sum_{k=1..n} a(k) ~ Pi^2*n^2/24 * (log(n)^2 + ((6*g - 1) + 12*z1/Pi^2) * log(n) + (1 - 6*g + 12*g^2 - 12*sg1)/2 + 6*((6*g - 1)*z1 + z2)/Pi^2), where g is the Euler-Mascheroni constant A001620, sg1 is the first Stieltjes constant A082633, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994. - Vaclav Kotesovec, Feb 02 2019
Multiplicative with a(p^e) = ((e^2+3*e+2)*p^(e+3) - 2*(e^2+4*e+3)*p^(e+2) + (e^2+5*e+6)*p^(e+1) - 2)/(2*(p-1)^3). - Amiram Eldar, May 26 2025

A304965 Expansion of Product_{k>=1} 1/(1 - x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

Original entry on oeis.org

1, 1, 3, 6, 19, 30, 96, 152, 461, 775, 1883, 3271, 8751, 14370, 34004, 59491, 140450, 239746, 541817, 932681, 2089189, 3606641, 7719178, 13398411, 28848808, 49603982, 103047935, 179154858, 370200348, 639269735, 1295389370, 2241994088, 4511677298, 7798101800, 15408901600

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Euler transform of A163767.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to partitions

Cf. A000219, A001970, A006171, A129373, A129374, A163767, A174465, A280487.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      A(d$2), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

nmax = 34; CoefficientList[Series[Product[1/(1 - x^k)^Times@@(Binomial[# + k - 1, k - 1]&/@FactorInteger[k][[All, 2]]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Times@@(Binomial[# + d - 1, d - 1]&/@FactorInteger[d][[All, 2]]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A163767(k).

A174467 G.f.: exp( Sum_{n>=1} A174468(n)x^n/n ) where A174468(n) = Sum_{d|n} dsigma(n/d)*sigma(d).

Original entry on oeis.org

1, 1, 5, 10, 31, 58, 157, 299, 711, 1367, 2987, 5679, 11807, 22117, 44006, 81513, 156885, 286413, 537058, 967367, 1773882, 3155223, 5677183, 9976095, 17661695, 30682683, 53544796, 92037152, 158575796, 269850363, 459636546, 774851829

Offset: 0

Paul D. Hanna, Apr 04 2010

nonn

Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A174465, A000203 (sigma).

{a(n)=polcoeff(exp(sum(m=1,n,x^m/m*sumdiv(m,d,d*sigma(m/d)*sigma(d)))+x*O(x^n)),n)}

From Ricardo Gómez Aíza, Mar 08 2023: (Start)
E.g.f.: Product_{n>=1,m>=1,k>=1} 1 / (1 - x^(n * m * k))^n.
log(a(n) / n!) ~ (3/2) * (Zeta(3) * Pi^4 / 18)^(1/3) * n^(2/3). (End)

A304963 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1} x^(ijk)).

Original entry on oeis.org

1, 1, 4, 10, 31, 82, 241, 664, 1898, 5316, 15058, 42374, 119718, 337432, 952373, 2685906, 7578248, 21376331, 60306495, 170120330, 479922212, 1353855927, 3819280961, 10774233218, 30394408336, 85743168417, 241883489742, 682358211402, 1924947591447, 5430317571250, 15319043353639

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Invert transform of A007425.

N. J. A. Sloane, Transforms
Index entries for sequences related to compositions

Cf. A000005, A007425, A011782, A129921, A174465, A280473, A304964.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(A(j, 3)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, May 22 2018

nmax = 30; CoefficientList[Series[1/(1 - Sum[x^(i j k), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}]), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[1/(1 - Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]

G.f.: 1/(1 - Sum_{k>=1} A007425(k)*x^k).

A319359 Expansion of Product_{i>=1, j>=1, k>=1} (1 - x^(ijk)).

Original entry on oeis.org

1, -1, -3, 0, 0, 9, 5, 4, -1, -27, -2, -33, -17, 8, 43, 92, 36, 100, -8, -11, -136, -120, -296, -363, -13, -203, 286, 306, 1010, 667, 724, 790, 151, -258, -1207, -964, -3325, -2059, -2924, -1992, -2116, 1277, 3625, 4437, 7724, 7734, 11524, 5801, 9685, -855, -2799, -13409, -16423

Offset: 0

Ilya Gutkovskiy, Sep 17 2018

sign

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

Cf. A000005, A000203, A007425, A174465, A174466, A280473, A288098.

a:=series(mul(mul(mul(1-x^(i*j*k),k=1..55),j=1..55),i=1..55),x=0,53): seq(coeff(a,x,n),n=0..52); # Paolo P. Lava, Apr 02 2019

nmax = 52; CoefficientList[Series[Product[(1 - x^(i j k)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}], {x, 0, nmax}], x]
nmax = 52; CoefficientList[Series[Product[(1 - x^k)^Sum[DivisorSigma[0, d], {d, Divisors[k]}], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[d DivisorSigma[1, k/d] DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 52}]

G.f.: Product_{k>=1} (1 - x^k)^A007425(k).

G.f.: exp(-Sum_{k>=1} A174466(k)*x^k/k).

cp's OEIS Frontend

A280473 G.f.: Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k)).

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A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(i*j*k*l)).

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A305050 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))/(1 - x^(i*j*k)).

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A318413 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - x^(i*j*k))^(i*j*k).

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A321360 Expansion of Product_{1 <= i <= j <= k} 1/(1 - x^(i*j*k)).

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A174466 a(n) = Sum_{d|n} d*sigma(n/d)*tau(d).

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A304965 Expansion of Product_{k>=1} 1/(1 - x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

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A174467 G.f.: exp( Sum_{n>=1} A174468(n)*x^n/n ) where A174468(n) = Sum_{d|n} d*sigma(n/d)*sigma(d).

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A304963 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1} x^(i*j*k)).

A280473 G.f.: Product_{i>=1, j>=1, k>=1} (1 + x^(ijk)).

A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(ijk*l)).

A305050 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))/(1 - x^(ijk)).

A318413 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - x^(ijk))^(ijk).

A321360 Expansion of Product_{1 <= i <= j <= k} 1/(1 - x^(ijk)).

A174466 a(n) = Sum_{d|n} dsigma(n/d)tau(d).

A174467 G.f.: exp( Sum_{n>=1} A174468(n)x^n/n ) where A174468(n) = Sum_{d|n} dsigma(n/d)*sigma(d).

A304963 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1} x^(ijk)).

A319359 Expansion of Product_{i>=1, j>=1, k>=1} (1 - x^(ijk)).