A270913 -id:A270913 - cp's OEIS frontend

A270914 Decimal expansion of a constant related to the asymptotics of A270913.

4, 5, 0, 2, 4, 7, 6, 7, 4, 7, 6, 1, 7, 3, 5, 4, 4, 8, 7, 7, 3, 8, 5, 9, 3, 9, 3, 2, 7, 0, 0, 7, 8, 4, 4, 0, 6, 7, 6, 3, 1, 2, 8, 7, 5, 6, 0, 9, 1, 6, 2, 1, 6, 3, 3, 4, 6, 4, 5, 4, 0, 4, 2, 4, 0, 8, 8, 8, 4, 0, 3, 2, 7, 9, 0, 6, 7, 7, 3, 2, 0, 2, 2, 1, 9, 2, 0, 6, 9, 6, 2, 5, 2, 5, 5, 1, 1, 4, 5, 3, 7, 2, 9, 6

Offset: 1

Vaclav Kotesovec, Mar 25 2016

This constant is very close to exp(5*Pi/(6*sqrt(2))) / sqrt(2) = 4.502476748630924546525119125234175537729... - Vaclav Kotesovec, May 17 2018

			4.502476747617354487738593932700784406763128756091621633464540424...

Cf. A181315, A206229, A270913, A270915, A303071, A304626, A327280.

RealDigits[1/r /. FindRoot[{2*s == QPochhammer[-1, r*s], r*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/2}, {s, 1/2}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Sep 26 2023 *)

Equals limit n->infinity A270913(n)^(1/n).

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...

A327214 Self-convolution of A270913.

Original entry on oeis.org

1, 2, 7, 32, 137, 592, 2597, 11442, 50567, 224112, 995392, 4428372, 19727877, 87983202, 392755207, 1754625632, 7844003907, 35086658052, 157023432677, 703037135122, 3148915010832, 14108913792342, 63235380631747, 283495965998772, 1271282293531077, 5702105357347602

Offset: 0

Vaclav Kotesovec, Aug 26 2019

nonn

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

Cf. A022567, A270913, A304443, A309986, A327215.

A270913[n_]:=SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A270913[k]*A270913[n-k], {k, 0, n}], {n, 0, 25}]

a(n) ~ c^2 * Pi * d^n, where d = A270914 = 4.5024767476173544877385939327... and c = A327280 = 0.260542233142438469433860832160... (see A270913).

A327280 Decimal expansion of a constant related to A270913 and A327214.

Original entry on oeis.org

2, 6, 0, 5, 4, 2, 2, 3, 3, 1, 4, 2, 4, 3, 8, 4, 6, 9, 4, 3, 3, 8, 6, 0, 8, 3, 2, 1, 6, 0, 4, 2, 2, 4, 8, 7, 6, 7, 6, 9, 0, 2, 0, 6, 0, 3, 9, 0, 8, 1, 6, 3, 0, 9, 2, 6, 3, 0, 2, 5, 9, 0, 1, 0, 2, 4, 1, 0, 3, 7, 5, 7, 2, 8, 9, 6, 2, 5, 4, 2, 2, 7, 8, 3, 5, 6, 5, 0, 8, 6, 0, 4, 0, 4, 4, 3, 1, 1, 4, 1, 0, 1, 0, 6, 2, 6

Offset: 0

Vaclav Kotesovec, Aug 28 2019

			0.26054223314243846943386083216042248767690206039081630926302590102410375728962...

Cf. A270913, A270914, A327214, A327279, A366018.

RealDigits[1/Sqrt[s*Pi*Derivative[0, 2][QPochhammer][-1, r*s]]/r /. FindRoot[{2*s == QPochhammer[-1, r*s], r*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/2}, {s, 1/2}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 02 2023 *)

Equals limit_{n->infinity} A270913(n) * sqrt(n) / A270914^n.

A286335 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)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 2, 0, 1, 5, 10, 13, 9, 3, 0, 1, 6, 15, 24, 24, 14, 4, 0, 1, 7, 21, 40, 51, 42, 22, 5, 0, 1, 8, 28, 62, 95, 100, 73, 32, 6, 0, 1, 9, 36, 91, 162, 206, 190, 120, 46, 8, 0, 1, 10, 45, 128, 259, 384, 425, 344, 192, 66, 10, 0

Offset: 0

Ilya Gutkovskiy, May 07 2017

A(n,k) is the number of partitions of n into distinct parts (or odd parts) with k types of each part.

			A(3,2) = 6 because we have [3], [3'], [2, 1], [2', 1], [2, 1'] and [2', 1'] (partitions of 3 into distinct parts with 2 types of each part).
Also A(3,2) = 6 because we have [3], [3'], [1, 1, 1], [1, 1, 1'], [1, 1', 1'] and [1', 1', 1'] (partitions of 3 into odd parts with 2 types of each part).
Square array begins:
  1,  1,  1,   1,   1,   1,  ...
  0,  1,  2,   3,   4,   5,  ...
  0,  1,  3,   6,  10,  15,  ...
  0,  2,  6,  13,  24,  40,  ...
  0,  2,  9,  24,  51,  95,  ...
  0,  3, 14,  42, 100, 206,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
N. J. A. Sloane, Transforms

Columns k=0-32 give: A000007, A000009, A022567-A022596.
Rows n=0-2 give: A000012, A001477, A000217.
Main diagonal gives A270913.
Antidiagonal sums give A299106.
Cf. A144064, A286352, A308680.

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..n/i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 29 2019

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

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

A(n,k) = Sum_{i=0..k} binomial(k,i) * A308680(n,k-i). - Alois P. Heinz, Aug 29 2019

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

Original entry on oeis.org

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

A270919 Coefficient of x^n in Product_{k>=1} ((1 + x^k) / (1 - x^k))^n.

Original entry on oeis.org

1, 2, 12, 80, 552, 3912, 28224, 206208, 1520784, 11297546, 84413912, 633713808, 4776117216, 36115518376, 273868321536, 2081866609920, 15859616674336, 121046064563376, 925411686479820, 7085465166635440, 54323193841192752, 416993869451825424, 3204447137019290944

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) == 2*p + 2 (mod p^2) holds for all primes p. Cf. A291697. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..1118 (terms 0..500 from Vaclav Kotesovec)

Main diagonal of A288515.

Cf. A008485, A008705, A156616, A270913, A270923, A270924, A291697, A309986, A362408.

Table[SeriesCoefficient[Product[((1+x^k)/(1-x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[(QPochhammer[-1, x]/QPochhammer[x, x])^n, {x, 0, n}]/2^n, {n, 0, 25}]
(* Calculation of constants {d,c}: *) eq = FindRoot[{2*s*QPochhammer[r*s] == QPochhammer[-1, r*s], (Log[1 - r*s] + QPolyGamma[0, 1, r*s])/Log[r*s] + r*((Derivative[0, 1][QPochhammer][-1, r*s] - 2*s*Derivative[0, 1][QPochhammer][r*s, r*s]) / (2*QPochhammer[r*s])) == 1}, {r, 1/8}, {s, 2}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[(1 - r*s)*Log[r*s]^2*(QPochhammer[r*s] / (Pi*(-r*s*(-1 + r*s) * Log[r*s]*(4*(2*ArcTanh[1 - 2*r*s] + QPolyGamma[0, 1, r*s])* Derivative[0, 1][QPochhammer][r*s, r*s] + r*Log[r*s]*(Derivative[0, 2][QPochhammer][-1, r*s] - 2*s*Derivative[0, 2][QPochhammer][r*s, r*s])) + 2*QPochhammer[r*s] * (4*r*s*ArcTanh[1 - 2*r*s] + 2*(-1 + (-1 + r*s)*ArcTanh[1 - 2*r*s])*Log[1 - r*s] - (-1 + r*s)*(-2 + Log[r*s] - 2*Log[1 - r*s])*QPolyGamma[0, 1, r*s] + (-1 + r*s) * QPolyGamma[0, 1, r*s]^2 + (-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 = 7.862983395705905261519347909953827161057584... and c = 0.299856802806668079413694689903953367699319...

a(n) = [x^n] 1/theta_4(x)^n, where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Nov 03 2017

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

Original entry on oeis.org

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

A255526 Coefficient of x^n in Product_{k>=1} 1/(1+x^k)^n.

Original entry on oeis.org

-1, 1, -4, 17, -56, 172, -547, 1809, -6061, 20316, -68135, 229244, -774372, 2624119, -8912759, 30328593, -103382254, 352975681, -1206921212, 4132159452, -14163858895, 48601267199, -166930975524, 573872089212, -1974472043081, 6798561779868, -23425506369715

Offset: 1

Vaclav Kotesovec, Feb 24 2015

sign

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Cf. A008485, A270913, A278428.

Table[SeriesCoefficient[Product[1/(1+x^k)^n,{k,1,n}],{x,0,n}],{n,1,30}]
(* Calculation of constant c: *) 1/Sqrt[(4 - r^2*s^3*Derivative[0, 2][QPochhammer][-1, r*s])*Pi] /. FindRoot[{QPochhammer[-1, r*s] == 2/s, 2/s + r*s*Derivative[0, 1][QPochhammer][-1, r*s] == 0}, {r, -1/3}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)

a(n) ~ (-1)^n * c * d^n / sqrt(n), where d = A318204 = 3.5097543279497033404372735..., c = 0.23322106096789389697797... .

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

cp's OEIS Frontend

A270914 Decimal expansion of a constant related to the asymptotics of A270913.

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A309986 Convolution of A270913 and A008485.

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A327214 Self-convolution of A270913.

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A327280 Decimal expansion of a constant related to A270913 and A327214.

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A286335 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)^k.

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

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A270919 Coefficient of x^n in Product_{k>=1} ((1 + x^k) / (1 - x^k))^n.

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

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