A255672 -id:A255672 - cp's OEIS frontend

A008485 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^n.

1, 1, 5, 22, 105, 506, 2492, 12405, 62337, 315445, 1605340, 8207563, 42124380, 216903064, 1119974875, 5796944357, 30068145905, 156250892610, 813310723925, 4239676354650, 22130265931900, 115654632452535, 605081974091875, 3168828466966388, 16610409114771900

Offset: 0

T. Forbes (anthony.d.forbes(AT)googlemail.com)

nonn

Number of partitions of n into parts of n kinds. - Vladeta Jovovic, Sep 08 2002
Main diagonal of A144064. - Omar E. Pol, Jun 27 2012
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 supercongruence a(p) == p + 1 (mod p^2) holds for all primes p >= 3. Cf. A270913. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A000712, A000716, A023003-A023021, A005758, A006922.

Cf. A109085, A192435, A252782, A255526, A255672, A270913, A270915, A270919.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:= n-> etr(j->n)(n): seq(a(n), n=0..30); # Alois P. Heinz, Sep 09 2008

a[n_] := SeriesCoefficient[ Product[1/(1-x^k)^n, {k, 1, n}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Feb 24 2015 *)
Table[SeriesCoefficient[1/QPochhammer[x, x]^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 25 2016 *)
Table[SeriesCoefficient[Exp[n*Sum[x^j/(j*(1-x^j)), {j, 1, n}]], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 19 2018 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k +x*O(x^n))^n),n)}

{a(n)=n*polcoeff(log(1/x*serreverse(x*eta(x+x*O(x^n)))), n)} /* Paul D. Hanna, Apr 05 2012 */

a(n) = Sum_{pi} Product_{i=1..n} binomial(k_i+n-1, k_i) where pi runs through all nonnegative solutions of k_1+2*k_2+...+n*k_n=n. a(n) = b(n, n) where b(n, m)= m/n*Sum_{i=1..n} sigma(i)*b(n-i, m) is recurrence for number of partitions of n into parts of m kinds. - Vladeta Jovovic, Sep 08 2002
Equals the logarithmic derivative of A109085, the g.f. of which is (1/x)*Series_Reversion(x*eta(x)). - Paul D. Hanna, Apr 05 2012
Let G(x) = exp( Sum_{n>=1} a(n)*x^n/n ), then G(x) = 1/Product_{n>=1} (1-x^n*G(x)^n) is the g.f. of A109085. - Paul D. Hanna, Apr 05 2012
a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.352701333486642687772415814165..., c = A327279 = 0.26801521271073331568695383828... . - Vaclav Kotesovec, Sep 10 2014

a(0)=1 prepended by Alois P. Heinz, Mar 30 2015

A255961 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j*k).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 6, 0, 1, 4, 12, 18, 13, 0, 1, 5, 18, 37, 47, 24, 0, 1, 6, 25, 64, 111, 110, 48, 0, 1, 7, 33, 100, 215, 303, 258, 86, 0, 1, 8, 42, 146, 370, 660, 804, 568, 160, 0, 1, 9, 52, 203, 588, 1251, 1938, 2022, 1237, 282, 0

Offset: 0

Alois P. Heinz, Mar 11 2015

A(n,k) is the number of partitions of n when parts i are of k*i kinds. A(2,2) = 7: [2a], [2b], [2c], [2d], [1a,1a], [1a,1b], [1b,1b].

			Square array A(n,k) begins:
  1,  1,   1,    1,    1,     1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6,     7, ...
  0,  3,   7,   12,   18,    25,    33,    42, ...
  0,  6,  18,   37,   64,   100,   146,   203, ...
  0, 13,  47,  111,  215,   370,   588,   882, ...
  0, 24, 110,  303,  660,  1251,  2160,  3486, ...
  0, 48, 258,  804, 1938,  4005,  7459, 12880, ...
  0, 86, 568, 2022, 5400, 12150, 24354, 44885, ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A000219, A161870, A255610, A255611, A255612, A255613, A255614, A193427, A316461, A316462.
Rows n=0-3 give: A000012, A001477, A055998, A101853.
Main diagonal gives A255672.
Antidiagonal sums give A299166.
Cf. A144064, A257673.

A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      A(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[2, j], {j, 1, n}]/n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 02 2016, after Alois P. Heinz *)

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

T(n,k) = Sum_{i=0..k} C(k,i) * A257673(n,k-i).

A252782 a(n) = n-th term of Euler transform of n-th powers.

Original entry on oeis.org

1, 1, 5, 36, 490, 12729, 689896, 70223666, 13803604854, 5567490203192, 4386006155453382, 6711625359213752077, 21048250447828058144403, 131214686495783317936950378, 1603891839732647136012816743764, 40296598014204065945778862754895836

Offset: 0

Alois P. Heinz, Dec 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..80
Vaclav Kotesovec, Graph - The asymptotic ratio

Main diagonal of A144048.

Cf. A008485, A073229, A255672, A270917.

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
       d*d^k, d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
a:= n-> A(n$2):
seq(a(n), n=0..20);

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) = [x^n] Product_{j>=1} 1/(1-x^j)^(j^n).

Conjecture: limit n->infinity a(n)^(1/n^2) = exp(exp(-1)) = 1.444667861... . - Vaclav Kotesovec, Mar 25 2016

A270922 Coefficient of x^n in Product_{k>=1} (1 + x^k)^(k*n).

Original entry on oeis.org

1, 1, 5, 28, 141, 751, 4064, 22198, 122381, 679375, 3792155, 21263331, 119679000, 675763232, 3826165838, 21715370653, 123502583565, 703694143160, 4016079632039, 22953901314649, 131366012754691, 752709483123304, 4317601694413683, 24790635783551008

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, A270913, A270917, A270924, A380291.

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

a(n) ~ c * d^n / sqrt(n), where d = 5.86811560195778704624328861800917668... and c = 0.25351514412215050116013727161633502...

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

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

Original entry on oeis.org

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...

A380290 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} 1/(1 - x^k)^(k^2) is the g.f. of A023871.

Original entry on oeis.org

1, 1, 11, 73, 539, 3976, 30107, 229811, 1771803, 13749742, 107305836, 841211966, 6619647419, 52258136399, 413682035393, 3282569032273, 26101575743771, 207930807629248, 1659134361686186, 13258065574274885, 106084302933126364, 849845499077000534, 6815530442695480418, 54712839001004065090

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 supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for all primes p >= 7 and positive integers n and r. Some examples are given below.

			Examples of supercongruences:
a(7) - a(1) = 229811 - 1 = 2*5*(7^3)*67 == 0 (mod 7^3)
a(3*7) - a(3) = 849845499077000534 - 73 = (7^3)*29243*84727410689 == 0 (mod 7^3)
a(19) - a(1) = 13258065574274885 - 1 = (2^2)*11*(19^3)*29*26723*56687 == 0 (mod 19^3)

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
Peter Bala, Notes on A380290 and A380291

Cf. A023871, A001158, A023873, A283263, A380291, A380581, A380582, A380583.

Cf. A008485, A255672, A386720.

with(numtheory):
G(x) := series(exp(add(sigma[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/(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[DivisorSigma[3, 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} sigma_3(k)*x^k/k).

a(n) ~ c * d^n / sqrt(n), where d = 8.20432131153340331179513077696629277558952852444670658917204305357709... and c = 0.2513708881073263860977360125648021910598660424705749139651716452651... - Vaclav Kotesovec, Jul 30 2025

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...

A281266 Main diagonal of A279928.

Original entry on oeis.org

1, -1, -1, -1, 23, -51, 35, -197, 1367, -3889, 7649, -26258, 112739, -350676, 939623, -3063201, 11022167, -35276497, 106320311, -344831533, 1164544273, -3765456206, 11890410454, -38631658591, 127610160227, -414671018176, 1335126443260, -4348160271568

Offset: 0

Seiichi Manyama, Apr 13 2017

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..100 from Seiichi Manyama)

Cf. A255672, A279928.

nmax = 30; Table[SeriesCoefficient[Product[1/(1 + x^k)^(m*k), {k, 1, m}], {x, 0, m}], {m, 0, nmax}] (* Vaclav Kotesovec, Apr 17 2017 *)

a(n) ~ (-1)^n * c * d^n / sqrt(n), where d = 3.31585574856163070436... and c = 0.20250147602443379616... - Vaclav Kotesovec, Apr 17 2017

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

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

A301455 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - x^k*A(x)^k)^k.

Original entry on oeis.org

1, 1, 4, 16, 74, 360, 1840, 9698, 52409, 288697, 1615275, 9153850, 52434770, 303104532, 1765920785, 10358843904, 61129390652, 362650003202, 2161590275029, 12938838382316, 77745063802045, 468760264760369, 2835272729215565, 17198394229862818, 104598950726341920, 637709136315071504

Offset: 0

Ilya Gutkovskiy, Mar 21 2018

nonn

			G.f. A(x) = 1 + x + 4*x^2 + 16*x^3 + 74*x^4 + 360*x^5 + 1840*x^6 + 9698*x^7 + 52409*x^8 + 288697*x^9 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * ...).
log(A(x)) = x + 7*x^2/2 + 37*x^3/3 + 215*x^4/4 + 1251*x^5/5 + 7459*x^6/6 + 44885*x^7/7 + 272727*x^8/8 + ... + A255672(n)*x^n/n + ...

Cf. A000219, A001157, A109085, A145268, A255672, A301456.

G.f. A(x) satisfies: A(x) = exp(Sum_{k>=1} sigma_2(k)*x^k*A(x)^k/k).

cp's OEIS Frontend

A008485 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^n.

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A255961 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j*k).

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A252782 a(n) = n-th term of Euler transform of n-th powers.

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A270922 Coefficient of x^n in Product_{k>=1} (1 + x^k)^(k*n).

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

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A380290 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} 1/(1 - x^k)^(k^2) is the g.f. of A023871.

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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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