A078306 -id:A078306 - 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

A255528 G.f.: Product_{k>=1} 1/(1+x^k)^k.

Original entry on oeis.org

1, -1, -1, -2, 1, 0, 4, 2, 8, -2, 4, -11, -1, -25, -5, -35, 13, -26, 49, -6, 110, 6, 159, -23, 182, -141, 129, -358, 62, -640, 39, -897, 237, -1013, 771, -914, 1793, -664, 3143, -565, 4635, -1157, 5727, -3119, 6121, -7041, 5642, -13088, 5097, -20758, 5879

Offset: 0

Vaclav Kotesovec, Feb 24 2015

sign

In general, if m >= 1 and g.f. = Product_{k>=1} 1/(1 + x^k)^(m*k), then a(n, m) ~ (-1)^n * exp(-m/12 + 3 * 2^(-5/3) * m^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(m/18 - 5/6) * A^m * m^(1/6 - m/36) * Zeta(3)^(1/6 - m/36) * n^(m/36 - 2/3) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Vaclav Kotesovec, Graph - the asymptotic ratio (250000 terms)
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. 20.

Cf. A000219, A026007, A026011, A073592, A262736.

Cf. A278710 (m=2), A279031 (m=3), A279411 (m=4), A279932 (m=5).

with(numtheory): A000219:=proc(n) option remember; if n = 0 then 1 else add(sigma[2](k)*A000219(n-k), k = 1..n)/n fi: end: A073592:=proc(n) option remember; if n = 0 then 1 else -add(sigma[2](k)*A073592(n-k), k = 1..n)/n fi: end: a:=proc(n); add(A073592(n-2*m)*A000219(m), m = 0..floor(n/2)): end: seq(a(n), n = 0..50); # Vaclav Kotesovec, Mar 09 2015

nmax=100; CoefficientList[Series[Product[1/(1+x^k)^k,{k,1,nmax}],{x,0,nmax}],x]

{a(n) = if(n<0, 0, polcoeff(exp(sum(k=1, n, (-1)^k * x^k / (1-x^k)^2 / k, x*O(x^n))), n))}
for(n=0, 100, print1(a(n), ", "))

a(n) ~ (-1)^n * A * Zeta(3)^(5/36) * exp(3*Zeta(3)^(1/3)*n^(2/3)/2^(5/3) - 1/12) / (2^(7/9) * sqrt(3*Pi) * n^(23/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 29 2015

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A078306(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A078307 a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^3.

Original entry on oeis.org

1, 7, 28, 55, 126, 196, 344, 439, 757, 882, 1332, 1540, 2198, 2408, 3528, 3511, 4914, 5299, 6860, 6930, 9632, 9324, 12168, 12292, 15751, 15386, 20440, 18920, 24390, 24696, 29792, 28087, 37296, 34398, 43344, 41635, 50654, 48020, 61544, 55314, 68922, 67424

Offset: 1

Vladeta Jovovic, Nov 22 2002

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
Index entries for sequences mentioned by Glaisher.

Cf. A000593, A078306, A027998.

with(numtheory):
a:= n-> add((-1)^(n/d+1)*d^3, d=divisors(n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Aug 03 2013

a[n_] := Sum[(-1)^(n/d+1)*d^3, {d, Divisors[n]}]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Jan 17 2014 *)
f[p_, e_] := (p^(3*e + 3) - 1)/(p^3 - 1); f[2, e_] := (6*8^e + 1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 42] (* Amiram Eldar, Oct 27 2022 *)

a(n) = sumdiv(n, d, (-1)^(n/d + 1)*d^3); \\ Indranil Ghosh, Apr 05 2017

from sympy import divisors
print([sum((-1)**(n//d + 1)*d**3 for d in divisors(n)) for n in range(1, 51)]) # Indranil Ghosh, Apr 05 2017

G.f.: Sum_{n >= 1} n^3*x^n/(1+x^n).
Multiplicative with a(2^e) = (6*8^e+1)/7, a(p^e) = (p^(3*e+3)-1)/(p^3-1), p > 2.
L.g.f.: log(Product_{k>=1} (1 + x^k)^(k^2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 12 2018
Sum_{k=1..n} a(k) ~ c * n^4, where c = 7*Pi^4/2880 = 0.236758... . - Amiram Eldar, Oct 27 2022

A284900 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^4.

Original entry on oeis.org

1, 15, 82, 239, 626, 1230, 2402, 3823, 6643, 9390, 14642, 19598, 28562, 36030, 51332, 61167, 83522, 99645, 130322, 149614, 196964, 219630, 279842, 313486, 391251, 428430, 538084, 574078, 707282, 769980, 923522, 978671, 1200644, 1252830, 1503652, 1587677

Offset: 1

Seiichi Manyama, Apr 05 2017

Multiplicative because this sequence is the Dirichlet convolution of A000583 and A062157 which are both multiplicative. - Andrew Howroyd, Jul 20 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Sum_{d|n} (-1)^(n/d+1)*d^k: A000593 (k=1), A078306 (k=2), A078307 (k=3), this sequence (k=4), A284926 (k=5), A284927 (k=6), A321552 (k=7), A321553 (k=8), A321554 (k=9), A321555 (k=10), A321556 (k=11), A321557 (k=12).

Cf. A000583, A013663, A062157, A386729.

Table[Sum[(-1)^(n/d + 1)*d^4, {d, Divisors[n]}], {n, 50}] (* Indranil Ghosh, Apr 05 2017 *)
f[p_, e_] := (p^(4*e + 4) - 1)/(p^4 - 1); f[2, e_] := (7*2^(4*e + 1) + 1)/15; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

a(n) = sumdiv(n, d, (-1)^(n/d + 1)*d^4); \\ Indranil Ghosh, Apr 05 2017

from sympy import divisors
print([sum([(-1)**(n//d + 1)*d**4 for d in divisors(n)]) for n in range(1, 51)]) # Indranil Ghosh, Apr 05 2017

G.f.: Sum_{k>=1} k^4*x^k/(1 + x^k). - Ilya Gutkovskiy, Apr 07 2017
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (7*2^(4*e+1)+1)/15, and a(p^e) = (p^(4*e+4) - 1)/(p^4 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^5, where c = 3*zeta(5)/16 = 0.194423... . (End)

A284926 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^5.

Original entry on oeis.org

1, 31, 244, 991, 3126, 7564, 16808, 31711, 59293, 96906, 161052, 241804, 371294, 521048, 762744, 1014751, 1419858, 1838083, 2476100, 3097866, 4101152, 4992612, 6436344, 7737484, 9768751, 11510114, 14408200, 16656728, 20511150, 23645064, 28629152, 32472031, 39296688

Offset: 1

Seiichi Manyama, Apr 06 2017

Multiplicative because this sequence is the Dirichlet convolution of A000584 and A062157 which are both multiplicative. - Andrew Howroyd, Jul 20 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Sum_{d|n} (-1)^(n/d+1)*d^k: A000593 (k=1), A078306 (k=2), A078307 (k=3), A284900 (k=4), this sequence (k=5), A284927 (k=6), A321552 (k=7), A321553 (k=8), A321554 (k=9), A321555 (k=10), A321556 (k=11), A321557 (k=12).

Cf. A000584, A013664, A062157.

Table[Sum[(-1)^(n/d + 1)*d^5, {d, Divisors[n]}], {n, 50}] (* Indranil Ghosh, Apr 06 2017 *)
f[p_, e_] := (p^(5*e + 5) - 1)/(p^5 - 1); f[2, e_] := (15*2^(5*e + 1) + 1)/31; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

a(n) = sumdiv(n, d, (-1)^(n/d + 1)*d^5); \\ Indranil Ghosh, Apr 06 2017

from sympy import divisors
print([sum((-1)**(n//d + 1)*d**5 for d in divisors(n)) for n in range(1, 51)]) # Indranil Ghosh, Apr 06 2017

G.f.: Sum_{k>=1} k^5*x^k/(1 + x^k). - Ilya Gutkovskiy, Apr 07 2017
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (15*2^(5*e+1)+1)/31, and a(p^e) = (p^(5*e+5) - 1)/(p^5 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^6, where c = 31*zeta(6)/192 = 0.164258... . (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A284927 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^6.

Original entry on oeis.org

1, 63, 730, 4031, 15626, 45990, 117650, 257983, 532171, 984438, 1771562, 2942630, 4826810, 7411950, 11406980, 16510911, 24137570, 33526773, 47045882, 62988406, 85884500, 111608406, 148035890, 188327590, 244156251, 304089030, 387952660, 474247150, 594823322

Offset: 1

Seiichi Manyama, Apr 06 2017

Multiplicative because this sequence is the Dirichlet convolution of A001014 and A062157 which are both multiplicative. - Andrew Howroyd, Jul 20 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Sum_{d|n} (-1)^(n/d+1)*d^k: A000593 (k=1), A078306 (k=2), A078307 (k=3), A284900 (k=4), A284926 (k=5), this sequence (k=6), A321552 (k=7), A321553 (k=8), A321554 (k=9), A321555 (k=10), A321556 (k=11), A321557 (k=12).

Cf. A001014, A013665, A062157.

Table[Sum[(-1)^(n/d + 1)*d^6, {d, Divisors[n]}], {n, 50}] (* Indranil Ghosh, Apr 06 2017 *)
f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); f[2, e_] := (31*2^(6*e + 1) + 1)/63; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

a(n) = sumdiv(n, d, (-1)^(n/d + 1)*d^6); \\ Indranil Ghosh, Apr 06 2017

from sympy import divisors
print([sum([(-1)**(n//d + 1)*d**6 for d in divisors(n)]) for n in range(1, 51)]) # Indranil Ghosh, Apr 06 2017

G.f.: Sum_{k>=1} k^6*x^k/(1 + x^k). - Ilya Gutkovskiy, Apr 07 2017
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (31*2^(6*e+1)+1)/63, and a(p^e) = (p^(6*e+6) - 1)/(p^6 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^7, where c = 9*zeta(7)/64 = 0.141799... . (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A321552 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^7.

Original entry on oeis.org

1, 127, 2188, 16255, 78126, 277876, 823544, 2080639, 4785157, 9922002, 19487172, 35565940, 62748518, 104590088, 170939688, 266321791, 410338674, 607714939, 893871740, 1269938130, 1801914272, 2474870844, 3404825448, 4552438132, 6103593751, 7969061786, 10465138360, 13386707720, 17249876310

Offset: 1

N. J. A. Sloane, Nov 23 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Sum_{k>=1} k^b*x^k/(1 + x^k): A000593 (b=1), A078306 (b=2), A078307 (b=3), A284900 (b=4), A284926 (b=5), A284927 (b=6), this sequence (b=7), A321553 (b=8), A321554 (b=9), A321555 (b=10), A321556 (b=11), A321557 (b=12).
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Cf. A013666.

f[p_, e_] := (p^(7*e + 7) - 1)/(p^7 - 1); f[2, e_] := (63*2^(7*e + 1) + 1)/127; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

apply( A321552(n)=sumdiv(n, d, (-1)^(n\d-1)*d^7), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^7*x^k/(1 + x^k). - Seiichi Manyama, Nov 23 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (63*2^(7*e+1)+1)/127, and a(p^e) = (p^(7*e+7) - 1)/(p^7 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^8, where c = 127*zeta(8)/1024 = 0.124529... . (End)

A321558 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^2.

Original entry on oeis.org

1, -5, 10, -13, 26, -50, 50, -45, 91, -130, 122, -130, 170, -250, 260, -173, 290, -455, 362, -338, 500, -610, 530, -450, 651, -850, 820, -650, 842, -1300, 962, -685, 1220, -1450, 1300, -1183, 1370, -1810, 1700, -1170, 1682, -2500, 1850, -1586, 2366

Offset: 1

N. J. A. Sloane, Nov 23 2018

			G.f. = x - 5*x^2 + 10*x^3 - 13*x^4 + 26*x^5 - 50*x^6 + 50*x^7 + ... - _Michael Somos_, Oct 24 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)
Peter Bala, A signed Dirichlet product of arithmetical functions
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher

Column k=2 of A322083.
Glaisher's xi_i (i=0..12): A228441, A109506, A321558, A321559, A321560, A321561, A321562, A321563, A321564, A321565, A321807, A321808, A321809
Cf. A321543 - A321557, A321810 - A321836 for similar sequences.
Cf. A078306, A328667.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^2*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^2 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)

apply( A321558(n)=sumdiv(n, d, (-1)^(n\d-d)*d^2), [1..30]) \\ M. F. Hasler, Nov 26 2018

s=(sum((-1)^(k+1)*k^2*x^k/(1 + x^k)  for k in (1..50))).series(x, 30); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

G.f.: Sum_{k>=1} (-1)^(k+1)*k^2*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018
G.f.: Sum_{k>=1} (-1)^(k+1)*(x^k - x^(2*k))/(1 + x^k)^3. - Michael Somos, Oct 24 2019
a(n) = -(-1)^n A328667(n). a(2*n + 1) = A078306(2*n + 1). a(2*n) = A078306(2*n) - 8*A078306(n). - Michael Somos, Oct 24 2019
From Peter Bala, Jan 29 2022: (Start)
Multiplicative with a(2^k) = - (2^(2*k+1) + 7)/3 for k >= 1 and a(p^k) = (p^(2*k+2) - 1)/(p^2 - 1) for odd prime p.
n^2 = (-1)^(n+1)*Sum_{d divides n} A067856(n/d)*a(d). (End)

A322081 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+1)*d^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 3, 4, -1, 1, 7, 10, 1, 2, 1, 15, 28, 11, 6, 0, 1, 31, 82, 55, 26, 4, 2, 1, 63, 244, 239, 126, 30, 8, -2, 1, 127, 730, 991, 626, 196, 50, 1, 3, 1, 255, 2188, 4031, 3126, 1230, 344, 43, 13, 0, 1, 511, 6562, 16255, 15626, 7564, 2402, 439, 91, 6, 2, 1, 1023, 19684, 65279, 78126, 45990, 16808, 3823, 757, 78, 12, -2

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
   1,  1,   1,    1,     1,     1,  ...
   0,  1,   3,    7,    15,    31,  ...
   2,  4,  10,   28,    82,   244,  ...
  -1,  1,  11,   55,   239,   991,  ...
   2,  6,  26,  126,   626,  3126,  ...
   0,  4,  30,  196,  1230,  7564,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A048272, A000593, A078306, A078307, A284900, A284926, A284927, A321552, A321553, A321554, A321555, A321556, A321557.

Cf. A109974, A279394, A279396, A285425, A321438 (diagonal), A322082, A322083, A322084.

Table[Function[k, Sum[(-1)^(n/d + 1) d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j^k x^j/(1 + x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

T(n,k)={sumdiv(n, d, (-1)^(n/d+1)*d^k)}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

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

A356391 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) * d^2 ) /k.

Original entry on oeis.org

1, 5, 35, 206, 1654, 13524, 130668, 1262064, 15027696, 178581600, 2407111200, 33276182400, 514020643200, 8130342124800, 144621487584000, 2537556118272000, 49206063078144000, 982811803276800000, 20991083543732736000, 454612169591580672000, 10763306565511514112000

Offset: 1

Seiichi Manyama, Aug 05 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..448

Cf. A356389, A356390.

Cf. A078306, A356298, A356394.

Table[n! * Sum[Sum[(-1)^(k/d + 1)*d^2, {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)

a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/k);

my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (-x)^k/(k*(1-x^k)^2))/(1-x)))

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k*log(1+x^k))/(1-x)))

a(n) = n! * Sum_{k=1..n} A078306(k)/k.
E.g.f.: -(1/(1-x)) * Sum_{k>0} (-x)^k/(k * (1 - x^k)^2).
E.g.f.: (1/(1-x)) * Sum_{k>0} k * log(1 + x^k).
a(n) ~ n! * n^2 * 3 * zeta(3) / 8. - Vaclav Kotesovec, Aug 07 2022

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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A255528 G.f.: Product_{k>=1} 1/(1+x^k)^k.

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A078307 a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^3.

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A284900 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^4.

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A284926 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^5.

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A284927 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^6.

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A321552 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^7.

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