A008485 -id:A008485 - cp's OEIS frontend

A309955 a(n) = [x^n] (1 + p(x))^n, where p(x) is the g.f. of A000040.

1, 2, 10, 59, 362, 2287, 14707, 95762, 629386, 4166627, 27743445, 185602188, 1246543559, 8399791922, 56762121398, 384513835219, 2610322687850, 17753944125159, 120954505004605, 825274753259894, 5638438272353597, 38569743775323134, 264127692090124488

Offset: 0

Alois P. Heinz, Aug 24 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1185

Cf. A000040, A008485, A008578, A025174, A292871, A309950, A340990.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, ithprime(n),
      (h-> add(b(j, h)*b(n-j, i-h), j=0..n))(iquo(i, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..31);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 1, Prime[n],
     Function[h, Sum[b[j, h]*b[n-j, i-h], {j, 0, n}]][Quotient[i, 2]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

A319670 a(n) = [x^n] Product_{k>=2} 1/(1 - x^k)^n.

Original entry on oeis.org

1, 0, 2, 3, 14, 30, 119, 301, 1078, 3036, 10242, 30624, 100451, 310128, 1004817, 3158343, 10182982, 32345186, 104145896, 332953929, 1072383374, 3442913407, 11100120528, 35742258497, 115377720235, 372326184555, 1203406838428, 3890040945078, 12588182588373, 40748118469180

Offset: 0

Ilya Gutkovskiy, Sep 25 2018

nonn

Number of partitions of n into parts > 1, if there are n kinds of parts.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000203, A002865, A008485, A147766, A191659, A304625, A319671.

Table[SeriesCoefficient[Product[1/(1 - x^k)^n , {k, 2, n}], {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[((1 - x)/QPochhammer[x])^n, {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, k] - 1) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 29}]

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

a(n) ~ c * d^n / sqrt(n), where d = 3.293558598422332665054219310876308... and c = 0.2154241499279313950113565475... - Vaclav Kotesovec, Oct 06 2018

A386720 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} 1/(1 - x^k)^(k^3) is the g.f. of A023872.

Original entry on oeis.org

1, 1, 19, 163, 1571, 15276, 152029, 1525420, 15460771, 157716235, 1617959044, 16672687769, 172459185341, 1789587777849, 18621317408384, 194222638392213, 2029985619026851, 21256104343844595, 222937740908641405, 2341629730618924374, 24627719497316157396, 259326672761381979574

Offset: 0

Vaclav Kotesovec, Jul 31 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..960

Cf. A023872.

Cf. A008485, A255672, A380290.

with(numtheory):
G(x) := series(exp(add(sigma[4](k)*x^k/k, k = 1..25)), x, 26):
seq(coeftayl(G(x)^n, x = 0, n), n = 0..25);

Table[SeriesCoefficient[Product[1/(1-x^k)^(n*k^3), {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Exp[n*Sum[DivisorSigma[4, k]*x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 25}]

a(n) = [x^n] exp(n*Sum_{k >= 1} sigma_4(k)*x^k/k).

a(n) ~ c * d^n / sqrt(n), where d = 10.783710654896500462544161711323081108292517438268962307143535279238... and c = 0.2464683956609371456774144752559018514863700235623819263696832303304...

A296162 a(n) = [x^n] Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 1, 5, 19, 89, 406, 1913, 9073, 43505, 209971, 1019390, 4971781, 24343037, 119579006, 589050663, 2908727839, 14393759457, 71360342129, 354372852011, 1762416036422, 8776797353574, 43761058841638, 218431753457637, 1091385769314272, 5458068218285909, 27319061357620056

Offset: 0

Ilya Gutkovskiy, Dec 06 2017

nonn

Eric Weisstein's World of Mathematics, Partition Function b_k

Cf. A000726, A008485, A121596, A128758, A270913, A285927, A296044, A296163.

Table[SeriesCoefficient[Product[((1 - x^(3 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
(* Calculation of constant d: *) With[{k = 3}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

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

a(n) ~ c * d^n / sqrt(n), where d = 5.1069752682291604222843644516987970799026747758649349... and c = 0.271879273312907861082536692355942116774864... - Vaclav Kotesovec, May 13 2018

A304460 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(n^3).

Original entry on oeis.org

1, 1, 44, 4410, 905840, 318906400, 172185088824, 132357574570221, 137406570363495360, 185242628827767255255, 314645673306845990409300, 657405676947400829561901359, 1656968286301847988118098735168, 4957425610652588047512198547937050

Offset: 0

Vaclav Kotesovec, May 13 2018

nonn

In general, for m>=3, coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(n^m) is asymptotic to n^(m*n)/n!.

Cf. A008485, A023872, A304446, A304459.

nmax = 20; Table[SeriesCoefficient[Product[1/(1-x^k)^(n^3), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[1/QPochhammer[x]^(n^3), {x, 0, n}], {n, 0, nmax}]

a(n) ~ exp(n) * n^(2*n - 1/2) / sqrt(2*Pi).

A254927 Coefficient of x^n in Product_{k=0..n} (1+k*x)^k.

Original entry on oeis.org

1, 8, 238, 15715, 1822678, 327061056, 83839010860, 29063729300694, 13090011332041111, 7428850394493811712, 5185703819680371737432, 4366227375438927437584444, 4363140133466727238167744916, 5104897162398639619205564019232, 6912594322573705179830176812524216

Offset: 1

Vaclav Kotesovec, Feb 10 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..200

Cf. A129256, A008485, A120295.

Table[Coefficient[Expand[Product[(1+k*x)^k,{k,0,n}]],x^n],{n,1,20}] (* or *)
p=1; Table[p=Expand[p*(1+n*x)^n]; Coefficient[p,x^n],{n,1,20}] (* faster *)

a(n) ~ exp(n+3/8) * n^(2*n-1/2) / (sqrt(2*Pi) * 3^n).

A301702 a(n) = [x^n] Product_{k>=0} 1/(1 - x^(2^k))^n.

Original entry on oeis.org

1, 1, 5, 19, 89, 401, 1877, 8821, 41969, 200899, 967605, 4681491, 22739705, 110816343, 541561333, 2653061819, 13024808161, 64063300481, 315624211781, 1557318893473, 7694243895289, 38060959885345, 188482408625373, 934323819631893, 4635781966972721, 23020536772620401

Offset: 0

Ilya Gutkovskiy, Mar 25 2018

nonn

Number of binary partitions of n into parts of n kinds.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions

Cf. A000123, A008485, A018819, A038712, A171238.

Table[SeriesCoefficient[Product[1/(1 - x^(2^k))^n, {k, 0, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[(1 + x^(2^k))^(n (k + 1)), {k, 0, n}], {x, 0, n}], {n, 0, 25}]

A301971 a(n) = [x^n] Product_{k>=1} 1/(1 - x^prime(k))^n.

Original entry on oeis.org

1, 0, 2, 3, 10, 30, 77, 252, 682, 2136, 6182, 18766, 56173, 169351, 512990, 1551828, 4720170, 14348289, 43751984, 133502873, 408029510, 1248460587, 3823949824, 11724787763, 35980251181, 110510334780, 339674840715, 1044812449722, 3215861978150, 9904301974294, 30521063942312, 94103983534015

Offset: 0

Ilya Gutkovskiy, Mar 29 2018

nonn

Number of partitions of n into prime parts of n kinds.

Index entries for sequences related to partitions

Cf. A000607, A008485, A030009, A117278, A219224, A259254, A291647, A298436, A299168.

Table[SeriesCoefficient[Product[1/(1 - x^Prime[k])^n, {k, 1, n}], {x, 0, n}], {n, 0, 31}]

A319457 a(n) = [x^n] Product_{k>=1} 1/((1 - x^k)(1 - x^(2k)))^n.

Original entry on oeis.org

1, 1, 7, 31, 175, 931, 5209, 29114, 165087, 940828, 5396777, 31090962, 179832625, 1043516371, 6072302726, 35420582431, 207051636799, 1212583329959, 7113193757656, 41788933655049, 245831162935825, 1447891754747672, 8537111315442222, 50387162650271055, 297664212003582753

Offset: 0

Ilya Gutkovskiy, Sep 19 2018

nonn

Cf. A002513, A008485, A270919, A296043, A296044, A319455, A319456.

Table[SeriesCoefficient[Product[1/((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[1/(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Exp[n Sum[(4 DivisorSigma[1, k] - DivisorSigma[1, 2 k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 24}]

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

a(n) = [x^n] exp(n*Sum_{k>=1} (4*sigma(k) - sigma(2*k))*x^k/k).

A356507 G.f.: Sum_{n>=0} x^(n(n+1)/2) P(x)^n, where P(x) is the partition function (A000041).

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 18, 34, 60, 109, 192, 339, 591, 1027, 1768, 3032, 5165, 8755, 14766, 24786, 41417, 68912, 114193, 188478, 309939, 507821, 829197, 1349437, 2189105, 3540253, 5708422, 9177939, 14715345, 23530180, 37527544, 59700283, 94741244, 149991677

Offset: 0

Paul D. Hanna, Aug 11 2022

nonn

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 10*x^5 + 18*x^6 + 34*x^7 + 60*x^8 + 109*x^9 + 192*x^10 + 339*x^11 + 591*x^12 + 1027*x^13 + 1768*x^14 + ...
such that
A(x) = 1 + x*P(x) + x^3*P(x)^2 + x^6*P(x)^3 + x^10*P(x)^4 + x^15*P(x)^5 + x^21*P(x)^6 + ... + x^(n*(n+1)/2) * P(x)^n + ...
where P(x) is the partition function and begins
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..820

Cf. A000041, A008485.

{a(n) = my(A = sum(m=0,n, x^(m*(m+1)/2) / prod(k=1,n,(1 - x^k +x*O(x^n))^m))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n equals the following expressions involving P(x), the partition function (A000041).
(1) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * P(x)^n.
(2) A(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k>=1} (1 - x^k)^n.
(3) A(x) = Sum_{n>=0} x^n * P(x)^n * Product_{k=1..n} (1 - x^(2*k-1)*P(x))/(1 - x^(2*k)*P(x)).
(4) A(x) = 1/(1 - x*P(x)/(1 + x*(1-x)*P(x)/(1 - x^3*P(x)/(1 + x^2*(1-x^2)*P(x)/(1 - x^5*P(x)/(1 + x^3*(1-x^3)*P(x)/(1 - x^7*P(x)/(1 + x^4*(1-x^4)*P(x)/(1 - ...))))))))), a continued fraction.

cp's OEIS Frontend

A309955 a(n) = [x^n] (1 + p(x))^n, where p(x) is the g.f. of A000040.

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A319670 a(n) = [x^n] Product_{k>=2} 1/(1 - x^k)^n.

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A386720 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} 1/(1 - x^k)^(k^3) is the g.f. of A023872.

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A296162 a(n) = [x^n] Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^n.

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A304460 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(n^3).

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A254927 Coefficient of x^n in Product_{k=0..n} (1+k*x)^k.

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A301702 a(n) = [x^n] Product_{k>=0} 1/(1 - x^(2^k))^n.

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A301971 a(n) = [x^n] Product_{k>=1} 1/(1 - x^prime(k))^n.

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A319457 a(n) = [x^n] Product_{k>=1} 1/((1 - x^k)*(1 - x^(2*k)))^n.

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A319457 a(n) = [x^n] Product_{k>=1} 1/((1 - x^k)(1 - x^(2k)))^n.

A356507 G.f.: Sum_{n>=0} x^(n(n+1)/2) P(x)^n, where P(x) is the partition function (A000041).