A270919 -id:A270919 - 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

A270913 Coefficient of x^n in Product_{k>=1} (1+x^k)^n.

Original entry on oeis.org

1, 1, 3, 13, 51, 206, 855, 3585, 15155, 64525, 276278, 1188353, 5130999, 22226049, 96544003, 420368858, 1834203955, 8018057345, 35107961175, 153950675585, 675978772326, 2971700764941, 13078268135683, 57613905606273, 254038914924791, 1121081799217231

Offset: 0

Vaclav Kotesovec, Mar 25 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 supercongruence a(p) == p + 1 (mod p^2) holds for all primes p. (End)

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

Cf. A000009, A008485, A255526, A270914, A270917, A270919, A324595, A362408.

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),
       (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 31 2021

Table[SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[QPochhammer[-1, x]^n, {x, 0, n}]/2^n, {n, 0, 25}]
Table[SeriesCoefficient[Exp[n*Sum[(-1)^j*x^j/(j*(x^j - 1)), {j, 1, n}]], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, May 19 2018 *)

{a(n)=polcoeff(prod(k=1, n, (1 + x^k +x*O(x^n))^n), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 26 2019

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = A327280 = 0.260542233142438469433860832160...

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...

A270923 Coefficient of x^n in Product_{k>=1} ((1 + x^k) / (1 - x^k))^(k^n).

Original entry on oeis.org

1, 2, 10, 88, 1414, 46648, 3026028, 373615284, 92794268694, 46265940243794, 44694344296430280, 86689242777435107120, 340600515192402995860548, 2624923513793602103874986688, 40749869155795866122979193705136, 1290021269710020392957588463834452744

Offset: 0

Vaclav Kotesovec, Mar 26 2016

nonn

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

Cf. A252782, A270917, A270919, A270924.

Table[SeriesCoefficient[Product[((1+x^k)/(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...

a(n) = [x^n] exp(Sum_{k>=1} (sigma_(n+1)(2*k) - sigma_(n+1)(k))*x^k/(2^n*k)). - Ilya Gutkovskiy, Apr 26 2019

A291697 a(n) = [x^n] Product_{k>=0} ((1 + x^(2k+1))/(1 - x^(2k+1)))^n.

Original entry on oeis.org

1, 2, 8, 44, 256, 1512, 9056, 54896, 335872, 2069774, 12827888, 79875996, 499305472, 3131436856, 19694403520, 124165133424, 784478240768, 4965659813668, 31484486937512, 199923173603596, 1271192603065856, 8092551782518688, 51574780342740256, 329022223268286288, 2100934234342260736

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

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 supercongruence a(p) == 2*p + 2 (mod p^3) holds for all primes p >= 5. Cf. A270919. (End)

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

Main diagonal of A289522.

Cf. A080054, A008705, A270919, A361008, A362408.

Table[SeriesCoefficient[Product[((1 + x^(2 k + 1))/(1 - x^(2 k + 1)))^n, {k, 0, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^n, {x, 0, n}], {n, 0, 24}]
(* Calculation of constant d: *) 1/r /. FindRoot[{s == QPochhammer[-r*s, r^2*s^2] / QPochhammer[r*s, r^2*s^2], QPochhammer[r*s, r^2*s^2] + QPochhammer[r*s, r^2*s^2]*((QPolyGamma[0, Log[-r*s]/Log[r^2*s^2], r^2*s^2] - QPolyGamma[0, Log[r*s]/Log[r^2*s^2], r^2*s^2]) / Log[r^2*s^2]) + 2*r^2*s^2*Derivative[0, 1][QPochhammer][r*s, r^2*s^2] == 2*r^2*s*Derivative[0, 1][QPochhammer][-r*s, r^2*s^2]}, {r, 1/8}, {s, 1}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 04 2023 *)

a(n) = A289522(n,n).

a(n) ~ c * d^n / sqrt(n), where d = 6.52085730573545526010335599231748172235904... and c = 0.296494808714349908707366708893... - Vaclav Kotesovec, Aug 30 2017

A288515 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)/(1 - x^j))^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 4, 0, 1, 6, 12, 8, 0, 1, 8, 24, 32, 14, 0, 1, 10, 40, 80, 76, 24, 0, 1, 12, 60, 160, 234, 168, 40, 0, 1, 14, 84, 280, 552, 624, 352, 64, 0, 1, 16, 112, 448, 1110, 1712, 1552, 704, 100, 0, 1, 18, 144, 672, 2004, 3912, 4896, 3648, 1356, 154, 0, 1, 20, 180, 960, 3346, 7896, 12600, 13120, 8184, 2532, 232, 0

Offset: 0

Ilya Gutkovskiy, Jun 10 2017

			Square array begins:
1,   1,    1,    1,     1,     1,  ...
0,   2,    4,    6,     8,    10,  ...
0,   4,   12,   24,    40,    60,  ...
0,   8,   32,   80,   160,   280,  ...
0,  14,   76,  234,   552,  1110,  ...
0,  24,  168,  624,  1712,  3913,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions

Columns k=0-24 give: A000007, A015128, A001934, A004404 (alternating values), A284286, A004406-A004425 (alternating values).
Rows n=0-2 give: A000012, A005843, A046092.
Main diagonal gives A270919.
Antidiagonal sums give A299108.
Cf. A122141, A286815.

# JacobiTheta4 is defined in A002448.
A288515Column(k, len) = JacobiTheta4(len, -k)
for k in 0:8 A288515Column(k, 8) |> println end # Peter Luschny, Mar 12 2018

Table[Function[k, SeriesCoefficient[Product[((1 + x^i)/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/EllipticTheta[4, 0, x]^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.
G.f. of column k: 1/theta_4(x)^k, where theta_4() is the Jacobi theta function.
For asymptotics of column k see comment from Vaclav Kotesovec in A001934.

A309986 Convolution of A270913 and A008485.

Original entry on oeis.org

1, 2, 9, 43, 206, 999, 4915, 24372, 121698, 611244, 3085612, 15645347, 79639602, 406809249, 2084567381, 10712007629, 55187254157, 284981396231, 1474729519719, 7646180479889, 39713643612380, 206600871071930, 1076372569004514, 5615363541987786, 29331204404385053

Offset: 0

Vaclav Kotesovec, Aug 26 2019

nonn

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

Cf. A008485, A270913, A270919, A327214, A327215.

A270913[n_] := SeriesCoefficient[Product[(1 + x^k)^n, {k, 1, n}], {x, 0, n}];
A008485[n_] := SeriesCoefficient[Product[1/(1 - x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A008485[k]*A270913[n-k], {k, 0, n}], {n, 0, 20}]

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.352701333486642687... and c = 0.446705640528056029457240607298917821281915554...

A300456 a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n^k).

Original entry on oeis.org

1, 2, 16, 200, 3264, 65752, 1565744, 42878432, 1324344832, 45464289482, 1715228012048, 70471268834936, 3129746696619072, 149318596196238328, 7612660420021177200, 412865831480749700928, 23725813528034949148672, 1439701175150489313314864, 91967625580609006328344400, 6167733266497532499924699672

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

nonn

			The table of coefficients of x^k in expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(n^k) begins:
n = 0: (1),  0,    0,    0,     0,       0,  ...
n = 1:  1,  (2),   4,    8,    14,      24,  ...
n = 2:  1,   4,  (16),  60,   208,     692,  ...
n = 3:  1,   6,   36, (200), 1038,    5160   ...
n = 4:  1,   8,   64,  472, (3264),  21608,  ...
n = 5:  1,  10,  100,  920,  7950,  (65752), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..377

Cf. A015128, A252654, A261519, A261520, A270919, A270923, A270924, A292805, A300457, A300458.

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

a(n) ~ exp(2*sqrt(2*n) - 1) * n^(n - 3/4) / (2^(3/4)*sqrt(Pi)). - Vaclav Kotesovec, Aug 26 2019

A304448 Coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(n^2).

Original entry on oeis.org

1, 2, 40, 1320, 61984, 3797560, 287566368, 25957422400, 2721948311680, 325260627848442, 43635601119149040, 6494550360714973304, 1062063969900788407680, 189301256401392643093560, 36526821128512112807216192, 7585918627122817713267856320

Offset: 0

Vaclav Kotesovec, May 12 2018

nonn

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

Cf. A206622, A270919, A304445, A304446, A304447.

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

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

A304447 Coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(2*n).

Original entry on oeis.org

1, 4, 40, 448, 5264, 63624, 783328, 9770240, 123040288, 1561033348, 19922193200, 255472920256, 3289122824000, 42488488508808, 550435283089088, 7148519205631488, 93038785849116736, 1213215382135324680, 15846906866928513736, 207302985358274247104

Offset: 0

Vaclav Kotesovec, May 12 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..880

Cf. A261386, A270919, A304443, A304444, A304448.

nmax = 20; Table[SeriesCoefficient[Product[((1+x^k)/(1-x^k))^(2*n), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2/QPochhammer[x])^(2*n), {x, 0, n}], {n, 0, nmax}]
(* Calculation of constants {d,c}: *) eq = FindRoot[{QPochhammer[-1, r*s] == 2*Sqrt[s]*QPochhammer[r*s], (QPochhammer[ r*s]*(Log[r*s] - 2*Log[1 - r*s] - 2*QPolyGamma[0, 1, r*s])) / Log[r*s] - r*Sqrt[s]*Derivative[0, 1][QPochhammer][-1, r*s] + 2*r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == 0}, {r, 1/12}, {s, 2}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[((1 - r*s)*Log[r*s]^2*QPochhammer[r*s]) / (Pi*(2*r*s*(-1 + r*s) * Log[r*s]*(2*(Log[r*s] - 2*Log[1 - r*s] - 2*QPolyGamma[0, 1, r*s]) * Derivative[0, 1][QPochhammer][r*s, r*s] + r*Sqrt[s]*Log[r*s] * (-Derivative[0, 2][QPochhammer][-1, r*s] + 2*Sqrt[s]*Derivative[0, 2][QPochhammer][r*s, r*s])) + QPochhammer[ r*s]*(16*r*s*ArcTanh[1 - 2*r*s] + (1 - r*s)*Log[r*s]^2 - 8*Log[1 - r*s] + 4*(-1 + r*s)*Log[1 - r*s]^2 + 8*(-1 + r*s)*(1 + Log[1 - r*s])* QPolyGamma[0, 1, r*s] + 4*(-1 + r*s)*QPolyGamma[0, 1, r*s]^2 + 4*(-1 + r*s)*(QPolyGamma[1, 1, r*s] - 2*r*s*Log[r*s]*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)

a(n) ~ c * d^n / sqrt(n), where d = 13.43567525239504624062504283058713960962824709850658926621911428148173077464... and c = 0.3323527904383991069791889982282236666403568774227549868882810268779...

cp's OEIS Frontend

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

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A270913 Coefficient of x^n in Product_{k>=1} (1+x^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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A270923 Coefficient of x^n in Product_{k>=1} ((1 + x^k) / (1 - x^k))^(k^n).

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A291697 a(n) = [x^n] Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^n.

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A288515 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)/(1 - x^j))^k.

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A309986 Convolution of A270913 and A008485.

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A300456 a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n^k).

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A291697 a(n) = [x^n] Product_{k>=0} ((1 + x^(2k+1))/(1 - x^(2k+1)))^n.