A248884 -id:A248884 - cp's OEIS frontend

A026007 Expansion of Product_{m>=1} (1 + q^m)^m; number of partitions of n into distinct parts, where n different parts of size n are available.

1, 1, 2, 5, 8, 16, 28, 49, 83, 142, 235, 385, 627, 1004, 1599, 2521, 3940, 6111, 9421, 14409, 21916, 33134, 49808, 74484, 110837, 164132, 241960, 355169, 519158, 755894, 1096411, 1584519, 2281926, 3275276, 4685731, 6682699, 9501979, 13471239, 19044780, 26850921, 37756561, 52955699

Offset: 0

N. J. A. Sloane

In general, for t > 0, if g.f. = Product_{m>=1} (1 + t*q^m)^m then a(n) ~ c^(1/6) * exp(3^(2/3) * c^(1/3) * n^(2/3) / 2) / (3^(2/3) * (t+1)^(1/12) * sqrt(2*Pi) * n^(2/3)), where c = Pi^2*log(t) + log(t)^3 - 6*polylog(3, -1/t). - Vaclav Kotesovec, Jan 04 2016

			For n = 4, we have 8 partitions
  01: [4]
  02: [4']
  03: [4'']
  04: [4''']
  05: [3, 1]
  06: [3', 1]
  07: [3'', 1]
  08: [2, 2']

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
Vaclav Kotesovec, Graph - The asymptotic ratio
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 18.

Cf. A000009, A000027, A026741, A073592, A255528, A261562, A266857, A285223, A304040.
Cf. A000219. - Gary W. Adamson, Jun 13 2009
Cf. A027998, A248882, A248883, A248884.
Cf. A026011, A027346, A027906.
Column k=1 of A284992.

with(numtheory):
b:= proc(n) option remember;
      add((-1)^(n/d+1)*d^2, d=divisors(n))
    end:
a:= proc(n) option remember;
      `if`(n=0, 1, add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Aug 03 2013

a[n_] := a[n] = 1/n*Sum[Sum[(-1)^(k/d+1)*d^2, {d, Divisors[k]}]*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Apr 17 2014, after Vladeta Jovovic *)
nmax=50; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*x^k/(k*(1-x^k)^2),{k,1,nmax}]],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

N=66; q='q+O('q^N);
gf= prod(n=1,N, (1+q^n)^n );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

a(n) = (1/n)*Sum_{k=1..n} A078306(k)*a(n-k). - Vladeta Jovovic, Nov 22 2002
G.f.: Product_{m>=1} (1+x^m)^m. Weighout transform of natural numbers (A000027). Euler transform of A026741. - Franklin T. Adams-Watters, Mar 16 2006
a(n) ~ zeta(3)^(1/6) * exp((3/2)^(4/3) * zeta(3)^(1/3) * n^(2/3)) / (2^(3/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where zeta(3) = A002117. - Vaclav Kotesovec, Mar 05 2015

A013665 Decimal expansion of zeta(7).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

From Dimitris Valianatos, Apr 29 2020: (Start)
Let p_n = Product_{k >= 1, 4*k-1 is prime} (((4*k - 1)^n + 1) / ((4*k - 1)^n - 1)).
Then (2^(n + 1) / (2^n - 1)) * Sum_{k >= 1} 1 / (4*k - 3)^n = ((p_n + 1) / p_n) * Sum_{k >= 1} 1 / k^n = ((p_n + 1) / p_n) * zeta(n), n >= 3 odd number.
For n = 7, p_7 = 1.00091744947834007403796003463414...
The product (256 / 127) * Sum_{k >= 1} 1 / (4*k - 3)^7 = 2.01577429320860871987548541116538... is equal to the product ((p_7 + 1) / p_7) * Sum_{k >= 1} 1 / k^7 = 1.9990833914636834116748... * zeta(7) = 2.01577429320860871987548541116538... (End)

			1.0083492773819228268397975498497967595998635605652387064172831365716014...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

Jakob Ablinger, Proving two conjectural series for zeta(7) and discovering more series for zeta(7), arXiv:1908.06631 [math.CO], 2019.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. Borwein and D. Bradley, Empirically determined Apéry-like formulas for zeta(4n+3), arXiv:math/0505124 [math.CA], 2005.
Michael J. Dancs and Tian-Xiao He, An Euler-type formula for zeta(2k+1), Journal of Number Theory, Volume 118, Issue 2, June 2006, Pages 192-199.
Simon Plouffe, Plouffe's Inverter, Zeta(7) to 50000 digits.
Simon Plouffe, Zeta(7) to 512 places:sum(1/n^7, n=1..infinity).

Cf. A013661, A002117, A013662, A013663, A013664, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.

Cf. A023874, A023875, A248884.

RealDigits[Zeta[7],10,120][[1]] (* Harvey P. Dale, Oct 23 2012 *)

PARI
```
zeta(7) \\ Michel Marcus, Apr 17 2016
```

zeta(7) = Sum_{n >= 1} (A010052(n)/n^(7/2)) = Sum_{n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^(7/2) ). - Mikael Aaltonen, Feb 22 2015
zeta(7) = Product_{k>=1} 1/(1 - 1/prime(k)^7). - Vaclav Kotesovec, Apr 30 2020
From Artur Jasinski, Jun 27 2020: (Start)
zeta(7) = (-1/840)*Integral_{x=0..1} log(1-x^6)^7/x^7.
zeta(7) = (1/720)*Integral_{x=0..oo} x^6/(exp(x)-1).
zeta(7) = (4/2835)*Integral_{x=0..oo} x^6/(exp(x)+1).
zeta(7) = (1/(182880*Zeta(1/2)^7))*(-61*Pi^7*zeta(1/2)^7 + 2880* zeta'(1/2)^7 - 10080*zeta(1/2)*zeta'(1/2)^5*zeta''(1/2) + 10080* zeta(1/2)^2*zeta'(1/2)^3*zeta''(1/2)^2 - 2520*zeta(1/2)^3*zeta'(1/2)* zeta''(1/2)^3 + 3360*zeta(1/2)^2*zeta'(1/2)^4*zeta'''(1/2) - 5040 zeta(1/2)^3*zeta'(1/2)^2*zeta''(1/2)*zeta'''(1/2) + 840*zeta(1/2)^4* zeta''(1/2)^2*zeta'''(1/2) + 560*zeta(1/2)^4*zeta'(1/2)*zeta'''(1/2)^3  - 840*zeta(1/2)^3*zeta'(1/2)^3*zeta''''(1/2) + 840*zeta(1/2)^4*zeta'(1/2)* zeta''(1/2)*zeta''''(1/2) - 140*zeta(1/2)^5*zeta'''(1/2)*zeta''''(1/2) + 168*zeta(1/2)^4*zeta'(1/2)^2*zeta'''''(1/2) - 84*zeta(1/2)^5*zeta''(1/2)* zeta'''''(1/2) - 28*zeta(1/2)^5*zeta'(1/2)*zeta''''''(1/2) + 4* zeta(1/2)^6*zeta'''''''(1/2)). (End)
Equals 19*Pi^7/56700 - 2*Sum_{k>=1} 1/(k^7*(exp(2*Pi*k) - 1)) [Grosswald] (see Finch). - Stefano Spezia, Nov 01 2024
From Peter Bala, Apr 27 2025: (Start)
zeta(7) = 1/7! * Integral_{x >= 0} x^7 * exp(x)/(exp(x) - 1)^2 dx = 2^6/(2^6 - 1) * 1/7! * Integral_{x >= 0} x^7 * exp(x)/(exp(x) + 1)^2 dx.
zeta(7) = 1/8! * Integral_{x >= 0} x^8 * exp(x)*(exp(x) + 1) /(exp(x) - 1)^3 dx = 1/ (2*3*7*15*63) * Integral_{x >= 0} x^8 * exp(x)*(exp(x) - 1)/(exp(x) + 1)^3 dx. (End)

A027998 Expansion of Product_{m>=1} (1+q^m)^(m^2).

Original entry on oeis.org

1, 1, 4, 13, 31, 83, 201, 487, 1141, 2641, 5972, 13309, 29248, 63360, 135688, 287197, 601629, 1247909, 2565037, 5226816, 10565132, 21192569, 42202909, 83466925, 163999684, 320230999, 621579965, 1199659836, 2302765961, 4397132933, 8354234552, 15795913477

Offset: 0

N. J. A. Sloane

nonn

In general, if g.f. = Product_{k>=1} (1 + x^k)^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(2*Pi/3 * (14*c2/15)^(1/4) * n^(3/4) + 3*c1 * Zeta(3) / Pi^2 * sqrt(15*n/(14*c2)) + (Pi * c0 * (5/(14*c2))^(1/4) / (2*3^(3/4)) - 9*c1^2 * Zeta(3)^2 * (15/(14*c2))^(5/4) / Pi^5) * n^(1/4) + 2025 * c1^3 * Zeta(3)^3 / (49 * c2^2 * Pi^8) - 15*c0*c1*Zeta(3) / (28*c2 * Pi^2)) * ((7*c2)/15)^(1/8) / (2^(15/8 + c0/2 + c1/12) * n^(5/8)). - Vaclav Kotesovec, Nov 08 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 22.

Cf. A026007, A248882, A248883, A248884, A285224, A304049.

Column k=2 of A284992.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^2: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

with(numtheory):
b:= proc(n) option remember;
      add((-1)^(n/d+1)*d^3, d=divisors(n))
    end:
a:= proc(n) option remember;
      `if`(n=0, 1, add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 03 2013

a[0] = 1; a[n_] := a[n] = 1/n*Sum[Sum[(-1)^(k/d+1)*d^3, {d, Divisors[k]}]*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 31} ] (* Jean-François Alcover, Jan 17 2014, after Vladeta Jovovic *)
nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 05 2015 *)

x = 'x + O('x ^ 50); Vec(prod(k=1, 50, (1 + x^k)^(k^2))) \\ Indranil Ghosh, Apr 05 2017

a(n) = 1/n*Sum_{k=1..n} A078307(k)*a(n-k). - Vladeta Jovovic, Nov 22 2002
a(n) ~ 7^(1/8) * exp(2/3 * Pi * (14/15)^(1/4) * n^(3/4)) / (2^(15/8) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 05 2015
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + x^k)/(k*(1 - x^k)^3)). - Ilya Gutkovskiy, May 30 2018

A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).

Original entry on oeis.org

1, 1, 8, 35, 119, 433, 1476, 4962, 16128, 51367, 160105, 490219, 1476420, 4378430, 12805008, 36962779, 105417214, 297265597, 829429279, 2291305897, 6270497702, 17008094490, 45744921052, 122052000601, 323166712109, 849453194355, 2217289285055, 5749149331789

Offset: 0

Vaclav Kotesovec, Mar 05 2015

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..5510 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 22.

Cf. A023872, A026007, A027998, A248883, A248884, A309335.

Column k=3 of A284992.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^3: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

b:= proc(n) option remember; add(
      (-1)^(n/d+1)*d^4, d=numtheory[divisors](n))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 16 2017

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^3),{k,1,nmax}],{x,0,nmax}],x]

x = 'x + O('x^50); Vec(prod(k=1, 50, (1 + x^k)^(k^3))) \\ Indranil Ghosh, Apr 06 2017

a(n) ~ Zeta(5)^(1/10) * 3^(1/5) * exp(2^(-11/5) * 3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) * n^(4/5)) / (2^(71/120) * 5^(2/5)* sqrt(Pi) * n^(3/5)), where Zeta(5) = A013663.
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A284900(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 4*x^k + x^(2*k))/(k*(1 - x^k)^4)). - Ilya Gutkovskiy, May 30 2018
Euler transform of A309335. - Georg Fischer, Nov 10 2020

A248883 Expansion of Product_{k>=1} (1+x^k)^(k^4).

Original entry on oeis.org

1, 1, 16, 97, 457, 2297, 11113, 52049, 235334, 1039886, 4497930, 19074006, 79418883, 325184763, 1311252535, 5212704708, 20449320159, 79231806015, 303428397505, 1149325838320, 4308477305997, 15993198330782, 58815616643170, 214383601754107, 774837953045873

Offset: 0

Vaclav Kotesovec, Mar 05 2015

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..3360 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 22.

Cf. A026007, A027998, A248882, A248884.

Column k=4 of A284992.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^4: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

b:= proc(n) option remember; add(
      (-1)^(n/d+1)*d^5, d=numtheory[divisors](n))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 16 2017

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^4),{k,1,nmax}],{x,0,nmax}],x]

x = 'x + O('x^50); Vec(prod(k=1, 50, (1 + x^k)^(k^4))) \\ Indranil Ghosh, Apr 06 2017

a(n) ~ 31^(1/12) * exp(1/5 * (31/7)^(1/6) * 6^(2/3) * Pi * n^(5/6)) / (2^(7/6) * 3^(2/3) * 7^(1/12) * n^(7/12)).
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A284926(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(k*(1 - x^k)^5)). - Ilya Gutkovskiy, May 30 2018

A284992 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, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 13, 8, 3, 1, 1, 16, 35, 31, 16, 4, 1, 1, 32, 97, 119, 83, 28, 5, 1, 1, 64, 275, 457, 433, 201, 49, 6, 1, 1, 128, 793, 1763, 2297, 1476, 487, 83, 8, 1, 1, 256, 2315, 6841, 12421, 11113, 4962, 1141, 142, 10, 1, 1

Offset: 0

Seiichi Manyama, Apr 07 2017

			Square array begins:
  1,  1,   1,    1,     1,      1,       1,        1, ...
  1,  1,   1,    1,     1,      1,       1,        1, ...
  1,  2,   4,    8,    16,     32,      64,      128, ...
  2,  5,  13,   35,    97,    275,     793,     2315, ...
  2,  8,  31,  119,   457,   1763,    6841,    26699, ...
  3, 16,  83,  433,  2297,  12421,   68393,   382573, ...
  4, 28, 201, 1476, 11113,  85808,  678101,  5466916, ...
  5, 49, 487, 4962, 52049, 561074, 6189117, 69540142, ...

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

Columns k=0-5 give A000009, A026007, A027998, A248882, A248883, A248884.
Rows (0+1),2-3 give: A000012, A000079, A007689.
Main diagonal gives A270917.
Cf. A283272, A284993.

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, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Oct 16 2017

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n - i*j, i - 1, k]*Binomial[i^k, j], {j, 0, n/i}]]];
A[n_, k_] := b[n, n, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 10 2021, after Alois P. Heinz *)

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

A284899 Expansion of Product_{k>=1} 1/(1+x^k)^(k^5) in powers of x.

Original entry on oeis.org

1, -1, -31, -212, -284, 4935, 43719, 160002, -96747, -4914512, -31358932, -94515285, 97642670, 2823746182, 16834776254, 51617810512, -11233909783, -1137004349695, -7267899354808, -25263858110877, -24537905293857, 319397811973578, 2523465326904492

Offset: 0

Seiichi Manyama, Apr 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..2647

Cf. A248884.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), A284896 (m=2), A284897 (m=3), A284898 (m=4), this sequence (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^5) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^5))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A284927(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A343325 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^(k^5).

Original entry on oeis.org

1, 32, 243, 1520, 3125, 15552, 16807, 70496, 88452, 200000, 161051, 867024, 371293, 1075648, 1518750, 3164792, 1419857, 6609600, 2476099, 11150000, 8168202, 10307264, 6436343, 44875296, 14646875, 23762752, 31059855, 59967376, 20511149, 121500000, 28629151, 138957472, 78270786, 90870848

Offset: 1

Ilya Gutkovskiy, Apr 11 2021

nonn

Cf. A000584, A045778, A050368, A248884, A343285, A343322, A343323, A343324.

cp's OEIS Frontend

A026007 Expansion of Product_{m>=1} (1 + q^m)^m; number of partitions of n into distinct parts, where n different parts of size n are available.

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A013665 Decimal expansion of zeta(7).

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A027998 Expansion of Product_{m>=1} (1+q^m)^(m^2).

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A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).

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A248883 Expansion of Product_{k>=1} (1+x^k)^(k^4).

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A284992 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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A284899 Expansion of Product_{k>=1} 1/(1+x^k)^(k^5) in powers of x.

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