A270922 -id:A270922 - cp's OEIS frontend

A255672 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(k*n).

1, 1, 7, 37, 215, 1251, 7459, 44885, 272727, 1668313, 10263057, 63423482, 393440867, 2448542136, 15280435191, 95588065737, 599213418327, 3763242239317, 23673166664695, 149138199543613, 940796936557265, 5941862248557566, 37568309060087582, 237767215209245583

Offset: 0

Vaclav Kotesovec, Mar 01 2015

nonn

Number of partitions of n when parts i are of n*i kinds. - Alois P. Heinz, Nov 23 2018
From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 3 and all positive integers n and k. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from Vaclav Kotesovec)

Cf. A008485, A252782, A270913, A270922, A380290.

Main diagonal of A255961.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 11 2015

Table[SeriesCoefficient[Product[1/(1-x^k)^(k*n),{k,1,n}],{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 01 2015 *)

a(n) ~ c * d^n / sqrt(n), where d = 6.468409145117839606941857350154192468889057616577..., c = 0.25864792865819067933968646380369970564... . - Vaclav Kotesovec, Mar 01 2015

a(n) = [x^n] exp(n*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018

A270924 Coefficient of x^n in Product_{k>=1} ((1 + x^k) / (1 - x^k))^(k*n).

Original entry on oeis.org

1, 2, 16, 128, 1056, 8952, 77200, 673948, 5937792, 52689170, 470210016, 4215834328, 37945215552, 342650763392, 3102866408560, 28166168335128, 256220106742272, 2335126111557564, 21317113277158336, 194890649121580880, 1784158030393621056, 16353089279998330456

Offset: 0

Vaclav Kotesovec, Mar 26 2016

nonn

From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 3 and all positive integers n and k. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A255672, A270922, A270919, A270923.

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

a(n) ~ c * d^n / sqrt(n), where d = 9.38812912875337022533876219516002188057967... and c = 0.2845468763296311652189248055322905919858...

A301456 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + x^k*A(x)^k)^k.

Original entry on oeis.org

1, 1, 3, 12, 49, 217, 1006, 4810, 23576, 117812, 597937, 3073874, 15972678, 83758809, 442681653, 2355678968, 12610759255, 67868269712, 366979432955, 1992755590086, 10862329206524, 59414599714958, 326009477088080, 1793977307978268, 9898072238695390, 54744525395860053, 303463833091357785

Offset: 0

Ilya Gutkovskiy, Mar 21 2018

nonn

			G.f. A(x) = 1 + x + 3*x^2 + 12*x^3 + 49*x^4 + 217*x^5 + 1006*x^6 + 4810*x^7 + 23576*x^8 + 117812*x^9 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x)^2)^2 * (1 + x^3*A(x)^3)^3 * (1 + x^4*A(x)^4)^4 * ...
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 141*x^4/4 + 751*x^5/5 + 4064*x^6/6 + 22198*x^7/7 + 122381*x^8/8 + ... + A270922(n)*x^n/n + ...

Cf. A026007, A145267, A181315, A270922, A301455.

G.f. A(x) satisfies: A(x) = exp(Sum_{k>=1} (-1)^(k+1)*x^k*A(x)^k/(k*(1 - x^k*A(x)^k)^2)).

A277938 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j*k) in powers of x.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 8, 0, 1, 5, 14, 28, 30, 16, 0, 1, 6, 20, 48, 72, 68, 28, 0, 1, 7, 27, 75, 141, 183, 145, 49, 0, 1, 8, 35, 110, 245, 396, 443, 298, 83, 0, 1, 9, 44, 154, 393, 751, 1058, 1026, 600, 142, 0, 1, 10, 54, 208

Offset: 0

Seiichi Manyama, Apr 11 2017

			Square array begins:
   1, 1,  1,  1,   1, ...
   0, 1,  2,  3,   4, ...
   0, 2,  5,  9,  14, ...
   0, 5, 14, 28,  48, ...
   0, 8, 30, 72, 141, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-4 give: A000007, A026007, A026011, A027346, A027906.
Rows n=0-3 give: A000012, A001477, A000096, A005586.
Main diagonal gives A270922.
Antidiagonal sums give A299167.
Cf. A276554, A279928.

G.f. of column k: Product_{j>=1} (1+x^j)^(j*k).

A281267 Main diagonal of A276554.

Original entry on oeis.org

1, -1, -3, 8, 13, -51, -120, 538, 781, -5419, -3053, 47673, 5080, -427740, 136462, 3922383, -3278067, -34819588, 48561567, 299316651, -603368637, -2509708844, 6948730643, 20210062532, -76150197416, -152569240051, 801154765564, 1039352472008, -8158396721266

Offset: 0

Seiichi Manyama, Apr 13 2017

sign

From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 3 and all positive integers n and k. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A255672, A270922, A276554.

nmax = 40; Table[SeriesCoefficient[Product[(1 - x^k)^(n*k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Apr 17 2017 *)

a(n) = [x^n] exp(-n*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018

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

A300187 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k)^(n/k).

Original entry on oeis.org

1, 1, 4, 39, 488, 7615, 147024, 3371137, 89079808, 2665537713, 89142430400, 3295096700071, 133399600068096, 5870116973678191, 278971698167158528, 14239859507270510625, 776985219329347518464, 45130494178637796970273, 2780224621391401396134912, 181059775626543107582734183

Offset: 0

Ilya Gutkovskiy, Feb 28 2018

nonn

			The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} (1 + x^k)^(n/k) begins:
n = 0: (1), 0,   0,    0,     0,      0,       0,  ...
n = 1:  1, (1),  1,    5,    11,     59,     439,  ...
n = 2:  1,  2,  (4),  16,    68,    328,    2416,  ...
n = 3:  1,  3,   9,  (39),  207,   1197,    8811,  ...
n = 4:  1,  4,  16,   80,  (488),  3296,   25984,  ...
n = 5:  1,  5,  25,  145,   995,  (7615),  65575,  ...
n = 6:  1,  6,  36,  240,  1836,  15624, (147024), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A048272, A168243, A270922, A299033, A299034, A300188.

Table[n! SeriesCoefficient[Product[(1 + x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

a(n) = n! * [x^n] exp(n*Sum_{k>=1} A048272(k)*x^k/k).

a(n) ~ c * d^n * n^n, where d = 1.294982800733109500251... and c = 0.6755467963500480915... - Vaclav Kotesovec, Sep 07 2018

A300457 a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(n^k).

Original entry on oeis.org

1, -1, -3, -1, 25, 624, 9871, 170470, 3027249, 55077245, 979330606, 15079702923, 94670678245, -7958168036625, -626145997536240, -34564907982551791, -1733699815491494303, -84294315853736719077, -4067859614343931897505, -196552300464314521511610, -9519733465269825759734169

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

sign

			The table of coefficients of x^k in expansion of Product_{k>=1} (1 - x^k)^(n^k) begins:
n = 0: (1),  0,    0,    0,   0,     0,  ...
n = 1:  1, (-1),  -1,    0,   0,     1,  ...
n = 2:  1,  -2,  (-3),   0,   2,    12,  ...
n = 3:  1,  -3,   -6,  (-1),  9,    63,  ...
n = 4:  1,  -4,  -10,   -4, (25),  224,  ...
n = 5:  1,  -5,  -15,  -10,  55,  (624), ...

Cf. A010815, A008705, A252654, A252782, A255672, A270917, A270922, A281266, A281267, A281268, A283333, A292805, A300456, A300458.

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

A300458 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n^k).

Original entry on oeis.org

1, -1, -1, -10, 11, 374, 9792, 183847, 3469427, 65038049, 1195396233, 19667738452, 189089161562, -6219720781782, -606316892131934, -35104997710496175, -1795953382595105853, -88223902016631657740, -4283800987347611165184, -207864171877269042498096, -10102590396625592962089500

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

sign

			The table of coefficients of x^k in expansion of Product_{k>=1} 1/(1 + x^k)^(n^k) begins:
n = 0: (1),  0,    0,    0,   0,     0,  ...
n = 1:  1, (-1),   0,   -1,   1,    -1,  ...
n = 2:  1,  -2,  (-1),  -4,   3,    -2,  ...
n = 3:  1,  -3,   -3, (-10),  6,    15,  ...
n = 4:  1,  -4,   -6,  -20, (11),  104,  ...
n = 5:  1,  -5,  -10,  -35,  20,  (374), ...

Cf. A081362, A252654, A255526, A252782, A255672, A270917, A270922, A281266, A281267, A281268, A283333, A292805, A300456, A300457.

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

A386729 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} (1 + x^k)^(k^3) is the g.f. of A248882.

Original entry on oeis.org

1, 1, 17, 154, 1377, 13276, 127862, 1249746, 12321121, 122287798, 1220492192, 12235940113, 123133325382, 1243080020352, 12583773308102, 127688996851804, 1298370095026017, 13226355435367992, 134955405683954234, 1379032238329708409, 14110075394718902752, 144544237021110644340

Offset: 0

Vaclav Kotesovec, Jul 31 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..975

Cf. A248882, A284900, A386720.

Cf. A270913, A270922, A380291.

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

a(n) = [x^n] exp(n*Sum_{k >= 1} s_4(k)*x^k/k), where s_4(n) = Sum_{d divides n} (-1)^(n/d+1)*d^4 = A284900(n).

a(n) ~ c * d^n / sqrt(n), where d = 10.49088673566991578441632677715184699104285539252671173854512548234581416... and c = 0.2449508761900081824436717230940007974244164508939377916825513986093942...

cp's OEIS Frontend

A255672 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(k*n).

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A270924 Coefficient of x^n in Product_{k>=1} ((1 + x^k) / (1 - x^k))^(k*n).

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A301456 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + x^k*A(x)^k)^k.

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A277938 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j*k) in powers of x.

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A281267 Main diagonal of A276554.

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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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A300187 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k)^(n/k).

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A300457 a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(n^k).

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A300458 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n^k).

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