A107742 -id:A107742 - cp's OEIS frontend

A280451 G.f.: Product_{k>=1, j>=1} (1+x^(j*k^2)).

1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 16, 20, 27, 34, 42, 53, 67, 82, 102, 125, 153, 188, 227, 274, 332, 401, 478, 574, 686, 815, 969, 1147, 1356, 1600, 1884, 2210, 2597, 3040, 3547, 4141, 4824, 5607, 6508, 7546, 8732, 10100, 11656, 13431, 15473, 17793, 20429, 23436

Offset: 0

Vaclav Kotesovec, Jan 03 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A004101, A006171, A107742, A280663, A280664.

nmax = 100; CoefficientList[Series[Product[(1 + x^(j*k^2)), {k, 1, Floor[Sqrt[nmax]+1]}, {j, 1, Floor[nmax/k^2] + 1}], {x, 0, nmax}], x]

my(N=66, x='x+O('x^N)); Vec(prod(k=1, sqrt(N), eta(x^(2*k^2))/eta(x^(k^2)))) \\ Seiichi Manyama, Apr 29 2021

a(n) ~ exp(Pi^2*sqrt(n/2)/3 + sqrt(3) * (sqrt(2)-1) * Zeta(1/2) * Zeta(3/2) * n^(1/4) / (2^(3/4) * sqrt(Pi)) - 9*((sqrt(2)-1) * Zeta(1/2) * Zeta(3/2))^2 / (16*Pi^3)) * sqrt(Pi) / (2^(3/2) * sqrt(3) * n^(3/4)).

A288098 Convolution inverse of A006171.

Original entry on oeis.org

1, -1, -2, 0, 0, 4, 1, 3, 0, -5, 0, -7, -6, -4, 7, 0, 6, 9, 11, 10, -2, 13, -13, -10, -17, -20, -25, 0, -11, -11, -2, 11, 41, 27, 41, 17, 58, 12, 27, -21, -2, -36, -67, -52, -59, -95, -75, -20, -89, 35, 0, 62, 41, 142, 97, 172, 63, 154, 148, 85, 110, -36, -17, -156

Offset: 0

Seiichi Manyama, Jun 05 2017

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

Cf. A006171, A060640, A107742.

Product_{k>=1} (1 - x^k)^sigma_m(k): this sequence (m=0), A288385 (m=1), A288389 (m=2), A288392 (m=3).

nmax = 50; CoefficientList[Series[Product[(1 - x^(i*j)), {i, 1, nmax}, {j, 1, nmax/i}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2018 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]  (* Vaclav Kotesovec, Aug 28 2018 *)
nmax = 50; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[0, 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}]; Take[CoefficientList[s, x], nmax] (* Vaclav Kotesovec, Aug 28 2018 *)

G.f.: Product_{n>=1} E(q^n) where E(q) = Product_{n>=1} (1-q^n).
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A060640(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Aug 26 2018

A288415 Expansion of Product_{k>=1} (1 + x^k)^(sigma_3(k)).

Original entry on oeis.org

1, 1, 9, 37, 137, 487, 1749, 5901, 19695, 63832, 202905, 632689, 1941394, 5860868, 17448558, 51255292, 148726841, 426605755, 1210569740, 3400427281, 9460683203, 26083933370, 71300381025, 193313191005, 520057831035, 1388722752205, 3682100198763

Offset: 0

Seiichi Manyama, Jun 09 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..5402

Cf. A288391, A288392, A288420.

Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), A192065 (m=1), A288414 (m=2), this sequence (m=3).

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+q^k)^DivisorSigma(3,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

with(numtheory): seq(coeff(series(mul((1+x^k)^(sigma[3](k)),k=1..n),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 31 2018

nmax = 40; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

m=40; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^sigma(k,3))) \\ G. C. Greubel, Oct 30 2018

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288420(k)*a(n-k) for n > 0.
a(n) ~ exp(5*Pi^(4/5) * Zeta(5)^(1/5) * n^(4/5) / 2^(12/5)) * Zeta(5)^(1/10) / (2^(169/240) * sqrt(5) * Pi^(1/10) * n^(3/5)). - Vaclav Kotesovec, Mar 23 2018
G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^3). - Ilya Gutkovskiy, Aug 26 2018

A288007 Expansion of 1/Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

Original entry on oeis.org

1, -1, -1, -1, 1, 1, 0, 2, 2, -1, -2, 0, -1, -1, -4, -1, 2, 0, -1, 2, 2, 5, -1, 4, 8, -4, -5, 0, -1, -1, -6, -1, 3, -7, -9, -5, 1, 3, -3, 3, 17, 0, -6, 8, 12, 8, 0, 8, 17, -11, -9, -10, 0, -2, -20, 5, 14, -18, -25, -10, 1, -7, -21, 2, 29, -12, -17, 6, 17, 32, -4

Offset: 0

Seiichi Manyama, Jun 04 2017

sign

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

Cf. A107742, A109386.

Product_{k>=1} 1/(1 + x^k)^sigma_m(k): this sequence (m=0), A288421 (m=1), A288422 (m=2), A288423 (m=3).

m:=80; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(&*[(&*[1 + x^(j*k): j in [1..2*m]]): k in [1..2*m]]))); // G. C. Greubel, Oct 29 2018

with(numtheory): seq(coeff(series(exp(-add(sigma(k)*x^k/(k*(1-x^(2*k))),k=1..n)),x,n+1), x, n), n = 0 .. 70); # Muniru A Asiru, Jan 30 2019

A109386[n_] := DivisorSum[n, #*DivisorSum[#, Mod[#, 2] &] &]; a[0] = 1; a[n_] := a[n] = -(1/n) Sum[A109386[k] a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 04 2017 *)
CoefficientList[Series[1/Product[Product[1+x^(j*k), {j,1,100}], {k,1,100}], {x,0,80}], x] (* G. C. Greubel, Oct 29 2018 *)

m=80; x='x+O('x^m); Vec(1/(prod(k=1,2*m, prod(j=1,2*m, 1+x^(j*k) )))) \\ G. C. Greubel, Oct 29 2018

Convolution inverse of A107742.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 26 2018

A356939 MM-numbers of multisets of intervals. Products of primes indexed by members of A073485.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 30, 31, 32, 33, 34, 36, 39, 40, 41, 44, 45, 47, 48, 50, 51, 52, 54, 55, 59, 60, 62, 64, 65, 66, 67, 68, 72, 75, 78, 80, 81, 82, 83, 85, 88, 90, 93, 94, 96, 99, 100, 102, 104, 108

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The initial terms and corresponding multisets of multisets:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   6: {{},{1}}
   8: {{},{},{}}
   9: {{1},{1}}
  10: {{},{2}}
  11: {{3}}
  12: {{},{},{1}}
  13: {{1,2}}
  15: {{1},{2}}
  16: {{},{},{},{}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

The initial version is A356940.
Intervals are counted by A000012, A001227, ranked by A073485.
Other types: A107742, A356936, A356937, A356938.
Other conditions: A302478, A302492, A356930, A356935, A356944, A356955.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Cf. A000040, A000720, A003963, A055887, A076610, A270995, A302242, A304969.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
Select[Range[100],And@@chQ/@primeMS/@primeMS[#]&]

A280486 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(ijk*l)).

Original entry on oeis.org

1, 1, 4, 8, 20, 36, 86, 150, 314, 564, 1088, 1902, 3557, 6085, 10902, 18506, 32124, 53584, 91133, 149749, 249315, 405121, 662582, 1063152, 1714580, 2719842, 4327302, 6797316, 10686005, 16622003, 25861855, 39866017, 61422891, 93910783, 143406552, 217537696

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. A000009, A007426, A107742, A280473, A280487, A219561.

nmax = 50; CoefficientList[Series[Product[(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]*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, Sep 08 2018 *)

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

A321042 a(n) = [x^n] Product_{k>=1} (1 + x^k)^sigma_n(k).

Original entry on oeis.org

1, 1, 5, 37, 491, 12763, 690756, 70250881, 13805853214, 5567873958982, 4386114219458332, 6711687353310594027, 21048327399504558833175, 131214860796100022696745520, 1603892616451767287785208156624, 40296605442098101265893075903063822, 2031406440758379976992019043333960734724

Offset: 0

Ilya Gutkovskiy, Oct 26 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..80

Cf. A107742, A192065, A288414, A288415, A301548, A301549, A301550, A301551, A301552, A301553, A319647.

Table[SeriesCoefficient[Product[(1 + x^k)^DivisorSigma[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Product[Product[(1 + x^(i j))^(j^n), {j, 1, n}], {i, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^k/(k (1 - x^(2 k))), {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

a(n) = [x^n] Product_{i>=1, j>=1} (1 + x^(i*j))^(j^n).

a(n) = [x^n] exp(Sum_{k>=1} sigma_(n+1)(k)*x^k/(k*(1 - x^(2*k)))).

A301548 Expansion of Product_{k>=1} (1 + x^k)^(sigma_4(k)).

Original entry on oeis.org

1, 1, 17, 99, 491, 2429, 12056, 56618, 259074, 1155193, 5044288, 21585280, 90694483, 374661505, 1524090522, 6111565745, 24181962002, 94491963120, 364920615165, 1393789672170, 5268145436728, 19715988877445, 73096492576283, 268589397735778, 978533798885874

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..3000

Cf. A001159, A107742, A301542.

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

a(n) ~ exp(6^(2/3) * Pi * (31*Zeta(5)/7)^(1/6) * n^(5/6)/5 + Pi *(7/(31*Zeta(5)))^(1/6) * n^(1/6) / (240*6^(2/3))) * (31*Zeta(5)/7)^(1/12) / (2^(7/6) * 3^(2/3) * n^(7/12)).

G.f.: exp(Sum_{k>=1} sigma_5(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 26 2018

A301549 Expansion of Product_{k>=1} (1 + x^k)^(sigma_5(k)).

Original entry on oeis.org

1, 1, 33, 277, 1829, 12763, 89213, 584741, 3704421, 22964742, 139315315, 826585083, 4807922574, 27476514016, 154490531418, 855490577052, 4670177536402, 25157218161854, 133831334223869, 703601883107626, 3658023094714380, 18817745119097343, 95833879532504638

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..2000

Cf. A001160, A107742, A301543.

nmax = 30; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[5, k], {k, 1, nmax}], {x, 0, nmax}], x]

x='x+O('x^99); Vec(prod(i=1, 99, (1+x^i)^sigma(i, 5))) \\ Altug Alkan, Mar 26 2018

a(n) ~ exp(7 * Pi^(6/7) * Zeta(7)^(1/7) * n^(6/7) / (2^(9/7) * 3^(6/7))) * (3*Zeta(7)/Pi)^(1/14) / (2^(323/504) * sqrt(7) * n^(4/7)).

G.f.: exp(Sum_{k>=1} sigma_6(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 26 2018

A301550 Expansion of Product_{k>=1} (1 + x^k)^(sigma_6(k)).

Original entry on oeis.org

1, 1, 65, 795, 6971, 69317, 690756, 6316950, 55729130, 484275457, 4111328940, 34029153900, 275901508917, 2197552381491, 17207716281240, 132575879110175, 1006214596929014, 7531171360277228, 55632520744009711, 405876769498808480, 2926507055330036936

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A013954, A107742, A301544.

nmax = 30; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[6, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2^(5/2) * Pi * (127*Zeta(7)/15)^(1/8) * n^(7/8)/7 - Pi * (5/(127*Zeta(7)))^(1/8) * n^(1/8) / (504 * sqrt(2) * 3^(7/8))) * (127*Zeta(7)/15)^(1/16) / (2^(9/4) * n^(9/16)).

G.f.: exp(Sum_{k>=1} sigma_7(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 26 2018

cp's OEIS Frontend

A280451 G.f.: Product_{k>=1, j>=1} (1+x^(j*k^2)).

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A288098 Convolution inverse of A006171.

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A288415 Expansion of Product_{k>=1} (1 + x^k)^(sigma_3(k)).

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A288007 Expansion of 1/Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

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A356939 MM-numbers of multisets of intervals. Products of primes indexed by members of A073485.

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A280486 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(i*j*k*l)).

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A321042 a(n) = [x^n] Product_{k>=1} (1 + x^k)^sigma_n(k).

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A301548 Expansion of Product_{k>=1} (1 + x^k)^(sigma_4(k)).

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A280486 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(ijk*l)).