A270913 -id:A270913 - cp's OEIS frontend

A270917 Coefficient of x^n in Product_{k>=1} (1 + x^k)^(k^n).

1, 1, 4, 35, 457, 12421, 678101, 69540142, 13730026114, 5551573311817, 4379029522335786, 6705866900012021577, 21038900445652125741759, 131183458646068931932668114, 1603688863449847489871671547959, 40294004792352613617780682256221711

Offset: 0

Vaclav Kotesovec, Mar 25 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..80

Cf. A252782, A255672, A270913.

Main diagonal of A284992.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i^k, j), j=0..n/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 16 2017

Table[SeriesCoefficient[Product[(1+x^k)^(k^n), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

Conjecture: limit n->infinity a(n)^(1/n^2) = exp(exp(-1)) = 1.444667861...

A324595 Number of colored integer partitions of 2n such that all colors from an n-set are used and parts differ by size or by color.

Original entry on oeis.org

1, 1, 5, 19, 85, 381, 1751, 8135, 38173, 180415, 857695, 4096830, 19645975, 94523729, 456079769, 2206005414, 10693086637, 51930129399, 252617434619, 1230714593340, 6003931991895, 29325290391416, 143393190367102, 701862880794183, 3438561265961263

Offset: 0

Alois P. Heinz, Sep 03 2019

nonn

			a(2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.

Alois P. Heinz, Table of n, a(n) for n = 0..1433
Wikipedia, Partition (number theory)

Cf. A000009, A270913, A308680, A340987.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
a:= n-> add(b(2*n$2, n-i)*(-1)^i*binomial(n, i), i=0..n):
seq(a(n), n=0..25);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n+1),
      (q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> g(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 29 2021

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k] Binomial[k, j]][n - i j], {j, 0, Min[k, n/i]}]]];
a[n_] := Sum[b[2n, 2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}];
a /@ Range[0, 25] (* Jean-François Alcover, May 06 2020, after Maple *)
Table[SeriesCoefficient[(-1 + QPochhammer[-1, Sqrt[x]]/2)^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jan 15 2024 *)
(* Calculation of constant d: *) 1/r /. FindRoot[{2 + 2*s == QPochhammer[-1, Sqrt[r*s]], Sqrt[r]*Derivative[0, 1][QPochhammer][-1, Sqrt[r*s]] == 4*Sqrt[s]}, {r, 1/5}, {s, 1}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jan 15 2024 *)

a(n) = A308680(2n,n).
a(n) ~ c * d^n / sqrt(n), where d = 5.0032778445310926321307990027... and c = 0.2798596129161126875318997... - Vaclav Kotesovec, Sep 14 2019
a(n) = [x^(2n)] (-1 + Product_{j>=1} (1 + x^j))^n. - Alois P. Heinz, Jan 29 2021

A296163 a(n) = [x^n] Product_{k>=1} ((1 - x^(5*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 1, 5, 22, 105, 501, 2456, 12160, 60801, 306130, 1550255, 7887034, 40281720, 206405967, 1060602800, 5463059772, 28199365873, 145832364580, 755420838614, 3918935839970, 20357605331355, 105878815699042, 551273881133750, 2873161931172668, 14988243880188600

Offset: 0

Ilya Gutkovskiy, Dec 06 2017

nonn

Eric Weisstein's World of Mathematics, Partition Function b_k

Cf. A008485, A035959, A121591, A263002, A270913, A285928, A296044, A296162.

Table[SeriesCoefficient[Product[((1 - x^(5 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k) + x^(3 k) + x^(4 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]
(* Calculation of constant d: *) With[{k = 5}, 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) + x^(3*k) + x^(4*k))^n.

a(n) ~ c * d^n / sqrt(n), where d = 5.3271035802753567624196808294779171420899175782347488197... and c = 0.2712048688090020853684153670711011713396954... - Vaclav Kotesovec, May 13 2018

A304443 Coefficient of x^n in Product_{k>=1} (1+x^k)^(2*n).

Original entry on oeis.org

1, 2, 10, 62, 394, 2562, 16966, 113794, 770442, 5254334, 36042250, 248403586, 1718732998, 11931569028, 83064794746, 579696375972, 4054279504266, 28408328186508, 199390547044342, 1401564307833908, 9865190079554954, 69522550703432476, 490484539061916794

Offset: 0

Vaclav Kotesovec, May 12 2018

nonn

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

Cf. A022567, A026011, A270913, A304444.

nmax = 25; Table[SeriesCoefficient[Product[(1+x^k)^(2*n), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 25; Table[SeriesCoefficient[(QPochhammer[-1, x]/2)^(2*n), {x, 0, n}], {n, 0, nmax}]
(* Calculation of constants {d,c}: *) {1/r, Sqrt[Derivative[0, 1][QPochhammer][-1, r*s] / (Pi*r*(Sqrt[s]*Derivative[0, 1][QPochhammer][-1, r*s]^2 + 2*s*Derivative[0, 2][QPochhammer][-1, r*s]))]} /. FindRoot[{4*s == QPochhammer[-1, r*s]^2, 2*r*Sqrt[s]*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)

a(n) ~ c * d^n / sqrt(n), where d = 7.21883059750200610514730564495768943165197819880185778427663522275469... and c = 0.300860732623379969554285615234449502629772950943717460278989499...

A304445 Coefficient of x^n in Product_{k>=1} (1+x^k)^(n^2).

Original entry on oeis.org

1, 1, 10, 174, 4132, 126905, 4802046, 216313349, 11313533064, 674172155790, 45102830448510, 3347925678474168, 273085613904116184, 24282144087974698445, 2337736453663291696624, 242272777074973285192935, 26892702305505020726198800, 3183326728470139531614913088

Offset: 0

Vaclav Kotesovec, May 12 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..330

Cf. A027998, A270913, A304446.

nmax = 20; Table[SeriesCoefficient[Product[(1+x^k)^(n^2), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2)^(n^2), {x, 0, n}], {n, 0, nmax}]

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

A380291 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} (1 + x^k)^(k^2) is the g.f. of A027998.

Original entry on oeis.org

1, 1, 9, 64, 425, 3026, 21672, 157095, 1149289, 8464240, 62683134, 466307865, 3482008904, 26083955002, 195932407939, 1475267031164, 11131100990825, 84140066313620, 637054366975740, 4830417047590165, 36674477204674750, 278779034863684377, 2121418004609211361, 16159262748227985561

Offset: 0

Peter Bala, Jan 19 2025

Given an integer sequence {f(n) : n >= 0} with f(0) = 1, there is a unique power series F(x) with rational coefficients, where F(0) = 1, such that f(n) = [x^n] F(x)^n. F(x) is given by F(x) = series_reversion(x/E(x)), where E(x) = exp(Sum_{n >= 1} f(n)*x^n/n). Furthermore, if the series E(x) has integer coefficients then the series F(x) also has integer coefficients and the sequence {f(n)} satisfies the Gauss congruences: f(n*p^r) == f(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r (by Stanley, Ch. 5, Ex. 5.2(a), p. 72 and the Lagrange inversion formula).
Thus the present sequence satisfies the Gauss congruences. In fact, stronger congruences appear to hold for the present sequence.
We conjecture that a(p) == 1 (mod p^3) for all primes p >= 7 (checked up to p = 61).
More generally, we conjecture that the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 7 and positive integers n and r. Some examples are given below.

			Examples of supercongruences:
a(7) - a(1) = 157095 - 1 = 2*(7^3)*229 == 0 (mod 7^3)
a(11) - a(1) = 466307865 - 1 = (2^3)*(11^3)*43793 == 0 (mod 11^3)
a(3*7) - a(3) = 278779034863684377 - 64 = (7^4)*43*26891*100413601 == 0 (mod 7^4)

R. P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.

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

Cf. A027998, A078307, A380290.

Cf. A270913, A270922.

with(numtheory):
s_3 := n-> add((-1)^(n/d+1)*d^3, d in divisors(n)):
G(x) := series(exp(add(s_3(k)*x^k/k, k = 1..23)), x, 24):
seq(coeftayl(G(x)^n, x = 0, n), n = 0..23);

Table[SeriesCoefficient[Product[(1 + x^k)^(n*k^2), {k, 1, n}], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)
(* or *)
Table[SeriesCoefficient[Exp[n*Sum[Sum[(-1)^(k/d + 1)*d^3, {d, Divisors[k]}]*x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

a(n) = [x^n] exp(n*Sum_{k >= 1} s_3(k)*x^k/k), where s_3(n) = Sum_{d divides n} (-1)^(n/d+1)*d^3 = A078307(n).

a(n) ~ c * d^n / sqrt(n), where d = 7.7846790125019502578773343468308844201627754275100035492213697757399421948... and c = 0.2484592487737716543953469621097743519172686743284742545545347906986158... - Vaclav Kotesovec, Jul 30 2025

A303071 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} (1 + x^k)^n.

Original entry on oeis.org

1, 2, 6, 23, 90, 362, 1491, 6225, 26242, 111479, 476466, 2046464, 8825559, 38191467, 165751529, 721177328, 3144703234, 13739010855, 60127642329, 263545670385, 1156732481150, 5083320593976, 22364017244278, 98491038664903, 434160710647831, 1915482295831037, 8457663096970431

Offset: 0

Ilya Gutkovskiy, Apr 18 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Index entries for sequences related to partitions

Cf. A036469, A270913, A286335, A303070.

Table[SeriesCoefficient[1/(1 - x) Product[(1 + x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x) Exp[n Sum[(-1)^(k + 1) x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

a(n) = [x^n] (1/(1 - x))*exp(n*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))).
a(n) = Sum_{j=0..n} A286335(j,n).
a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = 0.44252758868364961050787771300805... - Vaclav Kotesovec, May 19 2018

A304625 a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 0, 3, 19, 101, 501, 2486, 12398, 62329, 315436, 1605330, 8207552, 42124368, 216903051, 1119974861, 5796944342, 30068145889, 156250892593, 813310723907, 4239676354631, 22130265931880, 115654632452514, 605081974091853, 3168828466966365, 16610409114771876, 87141919856550506

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Number of partitions of n into 2 or more parts of n kinds. - Ilya Gutkovskiy, May 16 2018

Index entries for sequences related to partitions

Cf. A000065, A008485, A022567, A093160, A270913, A285927, A285928, A286653, A296044, A296162, A296163, A304626.

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

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.3527013334866426877724... and c = 0.268015212710733315686... - Vaclav Kotesovec, May 16 2018

A304626 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(n*k)))^n.

Original entry on oeis.org

1, 0, 1, 10, 47, 201, 849, 3578, 15147, 64516, 276268, 1188342, 5130987, 22226036, 96543989, 420368843, 1834203939, 8018057328, 35107961157, 153950675566, 675978772306, 2971700764920, 13078268135661, 57613905606250, 254038914924767, 1121081799217206, 4951199308679965

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Number of partitions of n into 2 or more distinct parts, with n types of each part. - Ilya Gutkovskiy, May 16 2018

Index entries for sequences related to partitions

Cf. A058576, A073252, A111133, A255526, A270913, A304625.

Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(n k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[(1 + x^k)^n, {k, 1, n - 1}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[(QPochhammer[-1, x, 1 + n]/QPochhammer[-1, x^n, 1 + n])^n, {x, 0, n}], {n, 0, 26}]

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.502476747617354487738... and c = 0.2605422331424384694... - Vaclav Kotesovec, May 16 2018

A319671 a(n) = [x^n] Product_{k>=2} (1 + x^k)^n.

Original entry on oeis.org

1, 0, 2, 3, 10, 30, 77, 252, 682, 2145, 6182, 18887, 56317, 170534, 515930, 1563843, 4759338, 14480073, 44203595, 134972504, 412984510, 1264601502, 3877302717, 11898761051, 36548512477, 112358685555, 345673541514, 1064250223230, 3278695047218, 10107173174013, 31174889414807

Offset: 0

Ilya Gutkovskiy, Sep 25 2018

nonn

Number of partitions of n into distinct parts > 1, with n types of each part.

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

Cf. A000593, A025147, A270913, A298598, A304626, A319670.

Table[SeriesCoefficient[Product[(1 + x^k)^n, {k, 2, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/((1 + x) QPochhammer[x, x^2])^n, {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[Exp[n Sum[(Sum[Mod[d, 2] d, {d, Divisors[k]}] + (-1)^k) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 30}]

a(n) = [x^n] exp(n*Sum_{k>=1} (A000593(k) + (-1)^k)*x^k/k).

a(n) ~ c * d^n / sqrt(n), where d = 3.136240011804974455586379053639831470878466... and c = 0.220695581251514154138820799337758703024... - Vaclav Kotesovec, Oct 06 2018

cp's OEIS Frontend

A270917 Coefficient of x^n in Product_{k>=1} (1 + x^k)^(k^n).

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A324595 Number of colored integer partitions of 2n such that all colors from an n-set are used and parts differ by size or by color.

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A296163 a(n) = [x^n] Product_{k>=1} ((1 - x^(5*k))/(1 - x^k))^n.

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A304443 Coefficient of x^n in Product_{k>=1} (1+x^k)^(2*n).

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A304445 Coefficient of x^n in Product_{k>=1} (1+x^k)^(n^2).

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A380291 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} (1 + x^k)^(k^2) is the g.f. of A027998.

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A303071 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} (1 + x^k)^n.

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A304625 a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.

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