A052847 -id:A052847 - cp's OEIS frontend

A035528 Euler transform of A027656(n-1).

0, 1, 1, 3, 3, 6, 9, 13, 19, 28, 42, 57, 84, 115, 164, 227, 313, 429, 588, 799, 1079, 1461, 1952, 2617, 3480, 4627, 6111, 8072, 10604, 13905, 18181, 23701, 30828, 39990, 51763, 66822, 86124, 110687, 142039, 181841, 232409, 296401, 377419, 479635, 608558, 770818

Offset: 0

Christian G. Bower

nonn

Also the weigh transform of A003602. - John Keith, Nov 17 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Cf. A000219, A003602, A052847, A263150, A263352, A262876, A263136, A263141.

nmax = 50; CoefficientList[Series[-1 + Product[1/(1 - x^(2*k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)
nmax = 100; Flatten[{0, Rest[CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]]}] (* Vaclav Kotesovec, Oct 10 2015 *)

a(n) ~ A^(1/2) * Zeta(3)^(11/72) * exp(-1/24 - Pi^4/(1728*Zeta(3)) + Pi^2 * n^(1/3)/(3*2^(8/3)*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(4/3)) / (sqrt(3*Pi) * 2^(71/72) * n^(47/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Oct 02 2015

A069153 a(n) = Sum_{d|n} d*(d-1)/2.

Original entry on oeis.org

0, 1, 3, 7, 10, 19, 21, 35, 39, 56, 55, 91, 78, 113, 118, 155, 136, 208, 171, 252, 234, 287, 253, 395, 310, 404, 390, 497, 406, 614, 465, 651, 586, 698, 626, 910, 666, 875, 822, 1060, 820, 1202, 903, 1239, 1144, 1289, 1081, 1643, 1197, 1581, 1414, 1736

Offset: 1

Benoit Cloitre, Apr 08 2002

Inverse Mobius transform of A000217. - R. J. Mathar, Jan 19 2009

			x^2 + 3*x^3 + 7*x^4 + 10*x^5 + 19*x^6 + 21*x^7 + 35*x^8 + 39*x^9 + 56*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A000203, A001157, A002117, A007437, A086666, A134840.

Cf. A032741, A065608, A363604, A363605, A363606.

with(numtheory):
seq((1/2)*(sigma[2](n) - sigma[1](n)), n = 1..100); # Peter Bala, Jan 21 2021

A069153[n_]:=Plus@@Binomial[Divisors[n],2];Array[A069153,100] (* Enrique Pérez Herrero, Feb 21 2012 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 - d) / 2)}

a(n) = my(f = factor(n)); (sigma(f, 2) - sigma(f)) / 2; \\ Amiram Eldar, Jan 01 2025

G.f.: Sum_{k>0} x^(2*k)/(1-x^k)^3. - Vladeta Jovovic, Dec 17 2002
Row sums of triangle A134840. - Gary W. Adamson, Nov 12 2007
G.f. A(x) = (1/2) * x * d/dx log( B(x) ) where B() is g.f. for A052847. - Michael Somos, Feb 12 2008
G.f.: Sum_{k>0} ((k^2 - k) / 2) * x^k / (1 - x^k). - Michael Somos, Feb 12 2008
From Peter Bala, Jan 21 2021: (Start)
a(n) = (1/2)*(sigma_2(n) - sigma_1(n)) = (1/2)*(A001157(n)  A000203(n)) =  (1/2)*A086666.
G.f.: A(x) = (1/2)* Sum_{n >= 1} x^(n^2)*( n*(n-1)*x^(3*n) - (n^2 + n - 2)*x^(2*n) + n*(3 - n)*x^n + n*(n - 1) )/(1 - x^n)^3. - differentiate equation 5 in Arndt twice w.r.t x and set x = 1. (End)
From Amiram Eldar, Jan 01 2025: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-2) - zeta(s-1)) / 2.
Sum_{k=1..n} a(k) ~ (zeta(3)/6) * n^3. (End)

A052812 A simple grammar: power set of pairs of sequences.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 9, 16, 24, 42, 63, 102, 157, 244, 373, 570, 858, 1290, 1930, 2858, 4228, 6208, 9084, 13216, 19175, 27666, 39804, 57020, 81412, 115820, 164264, 232178, 327220, 459796, 644232, 900214, 1254554, 1743896, 2418071, 3344896, 4616026

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Number of partitions of n objects of two colors into distinct parts, where each part must contain at least one of each color. - Franklin T. Adams-Watters, Dec 28 2006

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 776

Cf. A026007, A052847, A219555, A255834, A255835.

spec := [S,{B=Sequence(Z,1 <= card),C=Prod(B,B),S= PowerSet(C)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k-1),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 07 2015 *)

G.f.: exp(Sum((-1)^(j[1]+1)*(x^j[1])^2/(x^j[1]-1)^2/j[1], j[1]=1 .. infinity))
G.f.: Product_{k>=1} (1+x^k)^(k-1). - Vladeta Jovovic, Sep 17 2002
Weigh transform of b(n) = n-1. - Franklin T. Adams-Watters, Dec 28 2006
a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4/(1296*Zeta(3)) - Pi^2 * n^(1/3) / (3^(4/3) * 2^(5/3) * Zeta(3)^(1/3)) + (3/2)^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(1/4) * 3^(1/3) * n^(2/3) * sqrt(Pi)), where Zeta(3) = A002117. - Vaclav Kotesovec, Mar 07 2015

More terms from Vladeta Jovovic, Sep 17 2002

A263150 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 1, 3, 2, 5, 6, 7, 11, 12, 21, 22, 34, 38, 59, 67, 95, 118, 155, 198, 252, 330, 409, 540, 662, 867, 1067, 1382, 1705, 2187, 2705, 3430, 4267, 5348, 6666, 8303, 10352, 12812, 15964, 19681, 24467, 30091, 37282, 45769, 56539, 69296, 85304

Offset: 0

Vaclav Kotesovec, Oct 10 2015

nonn

The side effect of this calculation is a formula: Integral_{x=0..infinity} exp(-3*x)/(x*(1-exp(-2*x))^2) - 1/(4*x^3) + 1/(4*x^2) - exp(-x)/(24*x) = log(2)/6 + log(A)/2 - 1/24, where A = A074962 is the Glaisher-Kinkelin constant.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
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

Cf. A035528, A052847, A263140, A263149, A263199, A263352, A263395, A263396, A263397.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d-1, 2)=0, (d-1)/2, 0),
       d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # after Alois P. Heinz, Oct 17 2015

nmax = 100; CoefficientList[Series[Product[1/(1-x^(2*k+1))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} 1/j*x^(3*j)/(1 - x^(2*j))^2).

a(n) ~ sqrt(A) * Zeta(3)^(11/72) * exp(-1/24 - Pi^4/(1728*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(8/3)* Zeta(3)^(1/3)) + 3 * (Zeta(3)/2)^(1/3) * n^(2/3)/2) / (2^(35/72) * sqrt(3*Pi) * n^(47/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A304966 Expansion of Product_{k>=1} 1/(1 - x^k)^(p(k)-1), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, 0, 1, 2, 5, 8, 18, 30, 61, 107, 203, 358, 663, 1162, 2093, 3666, 6481, 11258, 19652, 33874, 58464, 100046, 171032, 290563, 492745, 831393, 1399655, 2346707, 3924873, 6541472, 10875575, 18025629, 29804125, 49143254, 80841455, 132651457, 217179366, 354745107, 578215807

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Euler transform of A000065.

Convolution of the sequences A001970 and A010815.

N. J. A. Sloane, Transforms
Index entries for sequences related to partitions

Cf. A000041, A000065, A000219, A001383, A001970, A010815, A052847.

with(combinat): with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d)-d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

nmax = 38; CoefficientList[Series[Product[1/(1 - x^k)^(PartitionsP[k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (PartitionsP[d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 38}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A000065(k).

A054974 Number of nonnegative integer 2 X 2 matrices with no zero rows or columns and with sum of elements equal to n, up to row and column permutation.

Original entry on oeis.org

1, 2, 6, 9, 17, 23, 36, 46, 65, 80, 106, 127, 161, 189, 232, 268, 321, 366, 430, 485, 561, 627, 716, 794, 897, 988, 1106, 1211, 1345, 1465, 1616, 1752, 1921, 2074, 2262, 2433, 2641, 2831, 3060, 3270, 3521, 3752, 4026, 4279, 4577, 4853, 5176, 5476, 5825, 6150

Offset: 2

Vladeta Jovovic, May 28 2000

From Gus Wiseman, Jan 22 2019: (Start)
Also the number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices and exactly 2 (not necessarily distinct) edges. For example, non-isomorphic representatives of the a(2) = 1 through a(5) = 9 multiset partitions are:
  {{1}{2}}  {{1}{22}}  {{1}{122}}  {{11}{122}}
            {{2}{12}}  {{11}{22}}  {{1}{1222}}
                       {{12}{12}}  {{11}{222}}
                       {{1}{222}}  {{12}{122}}
                       {{12}{22}}  {{1}{2222}}
                       {{2}{122}}  {{12}{222}}
                                   {{2}{1122}}
                                   {{2}{1222}}
                                   {{22}{122}}
(End)

			There are 9 nonnegative integer 2 X 2 matrices with no zero rows or columns and with sum of elements equal to 5, up to row and column permutation:
[0 1] [0 1] [0 1] [0 1] [0 2] [0 2] [0 2] [0 3] [1 1]
[1 3] [2 2] [3 1] [4 0] [1 2] [2 1] [3 0] [1 1] [1 2].

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Column k=2 of A321615.

Cf. A007716, A052847, A053307, A323654, A323655, A323656.

gf := -x^2*(x^3-x^2-1)/((x^2-1)^2*(x-1)^2): s := series(gf, x, 101): for i from 2 to 100 do printf(`%d,`,coeff(s,x,i)) od:

Vec(-x^2*(x^3-x^2-1) / ((x^2-1)^2*(x-1)^2) + O(x^60)) \\ Colin Barker, Jan 16 2017

G.f.: -x^2*(x^3-x^2-1) / ((x^2-1)^2*(x-1)^2).
From Colin Barker, Jan 16 2017: (Start)
a(n) = (6 - 6*(-1)^n + (9*(-1)^n-17)*n + 12*n^2 + 2*n^3) / 48.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>7.
(End)

More terms from James Sellers, May 29 2000

A255803 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k+2).

Original entry on oeis.org

1, 5, 23, 86, 295, 926, 2748, 7732, 20891, 54401, 137355, 337249, 808043, 1893402, 4348634, 9805669, 21741925, 47463473, 102133056, 216841459, 454648373, 942113618, 1930779697, 3915946921, 7864385266, 15647363323, 30858285440, 60345383394, 117065924679

Offset: 0

Vaclav Kotesovec, Mar 07 2015

In general, if g.f. = Product_{k>=1} 1/(1-x^k)^(m*k+c), m > 0, then a(n) ~ (m*Zeta(3))^(m/36 + c/6 + 1/6) * exp(m/12 - c^2 * Pi^4 / (432*m*Zeta(3)) + c * Pi^2 * n^(1/3) / (3 * 2^(4/3) * (m*Zeta(3))^(1/3)) + 3 * (m*Zeta(3))^(1/3) * n^(2/3) / 2^(2/3)) / (A^m * 2^(c/3 + 1/3 - m/36) * 3^(1/2) * Pi^((c+1)/2) * n^(m/36 + c/6 + 2/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 08 2015

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

Cf. A000219 (k), A005380 (k+1), A052847 (k-1), A120844 (2k+1), A253289 (2k-1), A255802 (2k+3), A255271 (3k+1).

Cf. A255834 - A255837, A363601, A363602.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> 3*n+2): seq(a(n), n=0..50); # after Alois P. Heinz
with(numtheory):
series(exp(add((3*sigma[2](k) + 2*sigma[1](k))*x^k/k, k = 1..30)), x, 31):
seq(coeftayl(%, x = 0, n), n = 0..30); # Peter Bala, Jan 16 2025

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(3*k+2),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ Zeta(3)^(7/12) * 3^(1/12) * exp(1/4 - Pi^4/(324*Zeta(3)) + Pi^2 * n^(1/3) / (3^(4/3) * (2*Zeta(3))^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A^3 * 2^(11/12) * Pi^(3/2) * n^(13/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .

G.f.: exp(Sum_{k >= 1} (3*sigma_2(k) + 2*sigma_1(k))*x^k/k) = 1 + 5*x + 23*x^2 + 86*x^3 + 295*x^4 + .... - Peter Bala, Jan 16 2025

A263358 Expansion of Product_{k>=1} 1/(1-x^(k+2))^k.

Original entry on oeis.org

1, 0, 0, 1, 2, 3, 5, 7, 12, 18, 29, 43, 69, 101, 155, 231, 347, 509, 759, 1106, 1626, 2359, 3428, 4938, 7127, 10194, 14587, 20756, 29498, 41716, 58932, 82888, 116413, 162924, 227602, 316988, 440696, 610953, 845469, 1167118, 1608178, 2210888, 3034124, 4155111

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

In general, if v>=0 and g.f. = Product_{k>=1} 1/(1-x^(k+v))^k, then a(n) ~ d1(v) * n^(v^2/6 - 25/36) * exp(-Pi^4 * v^2 / (432*Zeta(3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(2/3) - v * Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * 2^(v^2/6 + 11/36) * Zeta(3)^(v^2/6 - 7/36)), where Zeta(3) = A002117.
d1(v) = exp(Integral_{x=0..infinity} (1/(x*exp((v-1)*x) * (exp(x)-1)^2) - (6*v^2-1) / (12*x*exp(x)) + v/x^2 - 1/x^3) dx).
d1(v) = (exp(Zeta'(-1) - v*Zeta'(0))) / Product_{j=0..v-1} j!, where Zeta'(0) = -A075700, Zeta'(-1) = A084448 and Product_{j=0..v-1} j! = A000178(v-1).
d1(v) = exp(1/12) * (2*Pi)^(v/2) / (A * G(v+1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
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
Vaclav Kotesovec, Graph - The asymptotic ratio (30000 terms)
Eric Weisstein's World of Mathematics, Barnes G-Function

Cf. A000219, A052847, A263359, A263360, A263361, A263362, A263363, A263364.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      max(0, d-2), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 16 2015

nmax = 50; CoefficientList[Series[Product[1/(1-x^(k+2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[E^Sum[x^(3*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(3*k)/(k*(1-x^k)^2)).

a(n) ~ exp(1/12 - Pi^4/(108*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(1/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(1/36) * sqrt(Pi) / (A * sqrt(3) * Zeta(3)^(17/36) * n^(1/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A263359 Expansion of Product_{k>=1} 1/(1-x^(k+3))^k.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 3, 4, 6, 8, 13, 18, 29, 40, 61, 86, 127, 178, 260, 364, 524, 734, 1042, 1454, 2051, 2848, 3981, 5510, 7652, 10542, 14558, 19970, 27428, 37480, 51222, 69720, 94870, 128634, 174306, 235506, 317899, 428018, 575688, 772540, 1035538, 1385264

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
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

Cf. A000219, A052847, A263358, A263360, A263361, A263362, A263363, A263364.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      max(0, d-3), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 16 2015

nmax = 50; CoefficientList[Series[Product[1/(1-x^(k+3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[E^Sum[x^(4*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(4*k)/(k*(1-x^k)^2)).

a(n) ~ exp(1/12 - Pi^4/(48*Zeta(3)) - Pi^2 * n^(1/3) / (2^(4/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(29/36) * Pi / (A * 2^(47/36) * sqrt(3) * Zeta(3)^(47/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A263360 Expansion of Product_{k>=1} 1/(1-x^(k+4))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 14, 19, 29, 40, 58, 79, 113, 153, 215, 294, 407, 555, 767, 1040, 1424, 1930, 2624, 3540, 4794, 6441, 8677, 11627, 15589, 20818, 27812, 37011, 49257, 65360, 86681, 114665, 151594, 199947, 263530, 346647, 455553, 597628

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
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

Cf. A000219, A052847, A263358, A263359, A263361, A263362, A263363, A263364.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      max(0, d-4), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 16 2015

nmax = 50; CoefficientList[Series[Product[1/(1-x^(k+4))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[E^Sum[x^(5*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(5*k)/(k*(1-x^k)^2)).

a(n) ~ exp(1/12 - Pi^4/(27*Zeta(3)) - 2^(2/3) * Pi^2 * n^(1/3) / (3 * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(71/36) * Pi^(3/2) / (12 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(89/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

cp's OEIS Frontend

A035528 Euler transform of A027656(n-1).

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A069153 a(n) = Sum_{d|n} d*(d-1)/2.

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A052812 A simple grammar: power set of pairs of sequences.

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A263150 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^k.

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A304966 Expansion of Product_{k>=1} 1/(1 - x^k)^(p(k)-1), where p(k) = number of partitions of k (A000041).

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Maple

Mathematica

Formula

A054974 Number of nonnegative integer 2 X 2 matrices with no zero rows or columns and with sum of elements equal to n, up to row and column permutation.

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Maple

PARI

Formula

Extensions

A255803 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k+2).

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