A248883 -id:A248883 - 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

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

A206624 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^4).

Original entry on oeis.org

1, 2, 34, 228, 1414, 8872, 52876, 301136, 1662614, 8929406, 46738920, 239036116, 1197187780, 5882369976, 28397283056, 134864166352, 630819797174, 2908948327780, 13236421303742, 59477002686404, 264104800719672, 1159649708139680, 5037895127964316

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

Convolution of A023873 and A248883. - Vaclav Kotesovec, Aug 19 2015
In general, for m >= 0, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(k^m), then a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + Zeta'(-m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015
If m is even and m >= 2, then can be simplified as: a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^(1/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + (-1)^(m/2) * Gamma(m+1) * Zeta(m+1) / (2^(m+1) * Pi^m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 88*x^3 + 398*x^4 + 1768*x^5 + 7508*x^6 +...
where A(x) = (1+x)/(1-x) * (1+x^2)^16/(1-x^2)^16 * (1+x^3)^81/(1-x^3)^81 *...
Also, A(x) = Euler transform of [2,31,162,496,1250,2511,4802,7936,...]:
A(x) = 1/((1-x)^2*(1-x^2)^31*(1-x^3)^162*(1-x^4)^496*(1-x^5)^1250*(1-x^6)^2511*...).

Seiichi Manyama, Table of n, a(n) for n = 0..2916 (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. 23.

Cf. A015128 (m=0), A156616 (m=1), A206622 (m=2), A206623 (m=3), A001160 (sigma_5).

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k^4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)

{a(n)=polcoeff(prod(m=1,n+1,((1+x^m)/(1-x^m+x*O(x^n)))^(m^4)),n)}

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m, 5)-sigma(m, 5))/16*x^m/m)+x*O(x^n)), n)}

{a(n)=local(InvEulerGF=x*(2+31*x+152*x^2+341*x^3+460*x^4+341*x^5+152*x^6+31*x^7+2*x^8)/(1-x^2+x*O(x^n))^5); polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^polcoeff(InvEulerGF,k)),n)}
for(n=0,30,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} (sigma_5(2*n) - sigma_5(n))/16 * x^n/n ), where sigma_5(n) is the sum of 5th powers of divisors of n (A001160).
Inverse Euler transform has g.f.: x*(2 + 31*x + 152*x^2 + 341*x^3 + 460*x^4 + 341*x^5 + 152*x^6 + 31*x^7 + 2*x^8)/(1-x^2)^5.
a(n) ~ exp(3*2^(2/3)*Pi*n^(5/6)/5 + 3*Zeta(5)/(4*Pi^4)) / (2^(7/6) * 3^(1/2) * n^(7/12)), where Zeta(5) = A013663. - Vaclav Kotesovec, Aug 19 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A096960(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017

A248884 Expansion of Product_{k>=1} (1+x^k)^(k^5).

Original entry on oeis.org

1, 1, 32, 275, 1763, 12421, 85808, 561074, 3535678, 21815897, 131733641, 778099521, 4505634324, 25635135074, 143507764032, 791243636305, 4300983535471, 23070300486656, 122213931799869, 639848848696540, 3312824859756453, 16972058378914997, 86082216143323410

Offset: 0

Vaclav Kotesovec, Mar 05 2015

nonn

In general, for m > 0, if g.f. = Product_{k>=1} (1+x^k)^(k^m), then a(n) ~ 2^(zeta(-m)) * ((1-2^(-m-1)) * Gamma(m+2) * zeta(m+2))^(1/(2*m+4)) * exp((m+2)/(m+1) * ((1-2^(-m-1)) * Gamma(m+2) * zeta(m+2))^(1/(m+2)) * n^((m+1)/(m+2))) / (sqrt(2*Pi*(m+2)) * n^((m+3)/(2*m+4))).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
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 (m=1), A027998 (m=2), A248882 (m=3), A248883 (m=4).

Column k=5 of A284992.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^5: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

b:= proc(n) option remember; add(
      (-1)^(n/d+1)*d^6, 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^5),{k,1,nmax}],{x,0,nmax}],x]

m=50; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^k^5)) \\ G. C. Greubel, Oct 31 2018

a(n) ~ (5*zeta(7))^(1/14) * 3^(2/7) * exp(zeta(7)^(1/7) * 2^(-9/7) * 3^(-3/7) * 5^(1/7) * 7^(8/7) * n^(6/7)) / (2^(163/252) * 7^(3/7) * sqrt(Pi) * n^(4/7)), where zeta(7) = A013665.

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).

A284898 Expansion of Product_{k>=1} 1/(1+x^k)^(k^4) in powers of x.

Original entry on oeis.org

1, -1, -15, -66, -54, 725, 4580, 12739, 3346, -149076, -791226, -2182124, -1656973, 16553206, 100646954, 318795473, 506196578, -818806580, -9148048880, -36415709566, -87180585636, -70923559814, 484810027389, 2992082912770, 9866919438716, 19936695359140

Offset: 0

Seiichi Manyama, Apr 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..3995

Cf. A248883.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), A284896 (m=2), A284897 (m=3), this sequence (m=4), A284899 (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^4) , {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^4))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A284926(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A343324 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^(k^4).

Original entry on oeis.org

1, 16, 81, 376, 625, 2592, 2401, 8752, 9801, 20000, 14641, 71928, 28561, 76832, 101250, 196252, 83521, 366768, 130321, 555000, 388962, 468512, 279841, 1859760, 585625, 913952, 1148202, 2132088, 707281, 4050000, 923521, 4307216, 2371842, 2672672, 3001250, 11242800, 1874161, 4170272

Offset: 1

Ilya Gutkovskiy, Apr 11 2021

nonn

Cf. A000583, A045778, A050368, A248883, A343284, A343322, A343323, A343325.

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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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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A206624 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^4).

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A248884 Expansion of Product_{k>=1} (1+x^k)^(k^5).

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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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A284898 Expansion of Product_{k>=1} 1/(1+x^k)^(k^4) in powers of x.

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