A015128 -id:A015128 - cp's OEIS frontend

A195584 O.g.f.: exp( Sum_{n>=1} (sigma(2n^2)-sigma(n^2)) x^n/n ).

1, 2, 6, 18, 42, 102, 238, 522, 1130, 2394, 4926, 9978, 19890, 38942, 75254, 143598, 270506, 504126, 929926, 1698322, 3074010, 5516898, 9820550, 17349554, 30430610, 53007162, 91734262, 157771538, 269734714, 458542822, 775281982, 1303971722, 2182227546, 3634444634

Offset: 0

Paul D. Hanna, Sep 20 2011

nonn

Compare g.f. to the formula for Jacobi theta_4(x) given by:
 theta_4(x) = exp( Sum{n>=1} -(sigma(2*n)-sigma(n))*x^n/n )
where theta_4(x) = 1 + Sum_{n>=1} 2*(-x)^(n^2).
Here sigma(n) = A000203(n) is the sum of divisors of n.

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 18*x^3 + 42*x^4 + 102*x^5 + 238*x^6 +...
where
log(A(x)) = 2*x + 8*x^2/2 + 26*x^3/3 + 32*x^4/4 + 62*x^5/5 + 104*x^6/6 +...+ A195585(n)*x^n/n +...

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

Cf. A195585, A215603, A177399, A015128, A054785; variant: A225958.

nmax = 40; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2*n^2] - DivisorSigma[1, n^2])*(x^n/n), {n, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2015 *)

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

O.g.f.: exp( Sum_{n>=1} A054785(n^2)*x^n/n ), where exp( Sum_{n>=1} A054785(n)*x^n/n ) = 1/(1+2*Sum_{n>=1} (-x)^(n^2)), which is the g.f. of A015128.

A236001 Sum of positive ranks of all overpartitions of n.

Original entry on oeis.org

0, 2, 4, 10, 20, 36, 64, 110, 180, 288, 452, 696, 1052, 1568, 2304, 3346, 4808, 6838, 9636, 13464, 18664, 25684, 35104, 47672, 64348, 86368, 115304, 153152, 202452, 266404, 349032, 455406, 591856, 766284, 988544, 1270862, 1628380, 2079828, 2648296, 3362180

Offset: 1

Omar E. Pol, Jan 18 2014

nonn

Consider here that the rank of a overpartition is the largest part minus the number of parts (the same idea as the Dyson's rank of a partition).
It appears that the sum of all ranks of all overpartitions of n is equal to zero.
The equivalent sequence for partitions is A209616.

			For n = 4 we have:
---------------------------
Overpartitions
of 4               Rank
---------------------------
4               4 - 1 =  3
4               4 - 1 =  3
2+2             2 - 2 =  0
2+2             2 - 2 =  0
3+1             3 - 2 =  1
3+1             3 - 2 =  1
3+1             3 - 2 =  1
3+1             3 - 2 =  1
2+1+1           2 - 3 = -1
2+1+1           2 - 3 = -1
2+1+1           2 - 3 = -1
2+1+1           2 - 3 = -1
1+1+1+1         1 - 4 = -3
1+1+1+1         1 - 4 = -3
---------------------------
The sum of positive ranks of all overpartitions of 4 is 3+3+1+1+1+1 = 10 so a(4) = 10.

Cf. A015128, A209616, A235790, A235792, A235797, A235798, A236000.

a(n)={my(s=0); forpart(p=n, my(r=p[#p]-#p); if(r>0, s+=r*2^#Set(p))); s} \\ Andrew Howroyd, Feb 19 2020

Terms a(7) and beyond from Andrew Howroyd, Feb 19 2020

A261611 Expansion of Product_{k>=0} (1 + x^(4k+1))/(1 - x^(4k+1)).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 6, 6, 6, 8, 12, 14, 14, 16, 22, 28, 30, 32, 40, 50, 56, 60, 70, 86, 98, 106, 120, 144, 166, 180, 200, 234, 270, 296, 324, 372, 428, 472, 514, 580, 664, 736, 800, 890, 1010, 1124, 1222, 1346, 1514, 1684, 1834, 2008, 2240, 2488, 2712, 2956

Offset: 0

Vaclav Kotesovec, Aug 26 2015

nonn

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

Cf. A015128, A080054, A261610.

nmax = 60; CoefficientList[Series[Product[(1 + x^(4*k+1))/(1 - x^(4*k+1)), {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(n)/2) * Gamma(1/4) / (2^(9/4) * Pi^(3/4) * n^(5/8)).

A341366 Expansion of (1 / theta_4(x) - 1)^5 / 32.

Original entry on oeis.org

1, 10, 60, 275, 1060, 3612, 11210, 32310, 87665, 226130, 558684, 1329720, 3062905, 6853310, 14941330, 31820642, 66343150, 135659570, 272496680, 538427720, 1047788137, 2010303890, 3806292130, 7118038360, 13157217715, 24055170690, 43527162380, 77994164515, 138463246700

Offset: 5

Ilya Gutkovskiy, Feb 10 2021

nonn

Cf. A002448, A004406, A014968, A015128, A327383, A338223, A340481, A341223, A341364, A341365, A341367, A341368, A341369, A341370.

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..33);  # Alois P. Heinz, Feb 10 2021

nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^5/32, {x, 0, nmax}], x] // Drop[#, 5] &
nmax = 33; CoefficientList[Series[(1/32) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

G.f.: (1/32) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^5.

A341367 Expansion of (1 / theta_4(x) - 1)^6 / 64.

Original entry on oeis.org

1, 12, 84, 442, 1932, 7392, 25551, 81468, 243126, 686400, 1848156, 4775874, 11904215, 28737732, 67416756, 154122912, 344177823, 752310720, 1612395007, 3393652848, 7023685794, 14311193104, 28737793986, 56924936052, 111323290934, 215095157964, 410895944148, 776529566516

Offset: 6

Ilya Gutkovskiy, Feb 10 2021

nonn

Cf. A002448, A004407, A014968, A015128, A327384, A338223, A340905, A341225, A341364, A341365, A341366, A341368, A341369, A341370.

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..33);  # Alois P. Heinz, Feb 10 2021

nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^6/64, {x, 0, nmax}], x] // Drop[#, 6] &
nmax = 33; CoefficientList[Series[(1/64) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^6, {x, 0, nmax}], x] // Drop[#, 6] &

G.f.: (1/64) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^6.

A004402 Expansion of 1 / Sum_{n=-oo..oo} x^(n^2).

Original entry on oeis.org

1, -2, 4, -8, 14, -24, 40, -64, 100, -154, 232, -344, 504, -728, 1040, -1472, 2062, -2864, 3948, -5400, 7336, -9904, 13288, -17728, 23528, -31066, 40824, -53408, 69568, -90248, 116624, -150144, 192612, -246256, 313808

Offset: 0

N. J. A. Sloane

sign

Taylor series for 1/theta_3. Absolute values are coefficients in Taylor series for 1/theta_4.

Euler transform of period-4 sequence [-2,3,-2,1,...].

J. R. Newman, The World of Mathematics, Simon and Schuster, 1956, Vol. I p. 372.

Robert Israel, Table of n, a(n) for n = 0..10000
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 103.

See A015128 for a version without signs.

# JacobiTheta3 is defined in A000122.
A004402List(len) = JacobiTheta3(len, -1)
A004402List(35) |> println # Peter Luschny, Mar 12 2018

S:=series(1/JacobiTheta3(0,x),x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Dec 29 2015

terms = 35; 1/EllipticTheta[3, 0, x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Jul 05 2017 *)

a(n)=if(n<0,0,polcoeff(1/sum(k=1,sqrtint(n),2*x^k^2,1+x*O(x^n)),n))

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^4+A)^2/eta(x^2+A)^5, n))}

Ramanujan gave an asymptotic formula (see Almkvist).
G.f.: 1/Product_{m>0} ((1-q^(2m))(1+q^(2m-1))^2) = 1/theta_3(q).
a(n) = (-1)^n * A015128(n).

A004404 Expansion of 1 / (Sum_{n=-oo..oo} x^(n^2))^3.

Original entry on oeis.org

1, -6, 24, -80, 234, -624, 1552, -3648, 8184, -17654, 36816, -74544, 147056, -283440, 535008, -990912, 1803882, -3232224, 5707624, -9943536, 17106960, -29088352, 48922320, -81438528, 134261584, -219336630, 355242288

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..4000 from Robert Israel)
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. 8.

Cf. A001934, A004405-A004425, A015128.

S:= series(1/JacobiTheta3(0,x)^3,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Dec 29 2015

nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)

a(n) ~ (-1)^n * 3*exp(Pi*sqrt(3*n)) / (64*n^(3/2)) * (1 - sqrt(3)/(Pi*sqrt(n))). - Vaclav Kotesovec, Aug 18 2015, extended Jan 16 2017

A235797 Triangle read by rows in which T(n,k) is the sum of the k-th largest elements in all overpartitions of n.

Original entry on oeis.org

2, 6, 2, 16, 6, 2, 34, 14, 6, 2, 68, 30, 14, 6, 2, 128, 60, 30, 14, 6, 2

Offset: 1

Omar E. Pol, Jan 18 2014

It appears that T(n,k) is also the total number of parts >= k in all overpartitions of n.
It appears that the first differences of row n together with 2 give row n of triangle A235798.
The equivalent sequence for partitions is A181187.

			Triangle begins:
    2;
    6,  2;
   16,  6,  2;
   34, 14,  6,  2;
   68, 30, 14,  6,  2;
  128, 60, 30, 14,  6,  2;
  ...

Column 1 is A235792.

Cf. A015128, A181187, A235790, A235793, A235798, A236000, A236001.

A261389 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3*k).

Original entry on oeis.org

1, 6, 30, 128, 486, 1704, 5604, 17484, 52206, 150118, 417696, 1128984, 2973476, 7650720, 19272432, 47616568, 115570014, 275921460, 648771802, 1503889488, 3439990344, 7770915816, 17349229908, 38306180052, 83694778556, 181052778078, 387976101432, 823939048560

Offset: 0

Vaclav Kotesovec, Aug 17 2015

nonn

Convolution of A255610 and A027346.

In general, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(t*k) and t>=1, then a(n) ~ exp(t/12 + 3/2 * (7*t*Zeta(3)/2)^(1/3) * n^(2/3)) * t^(1/6 + t/36) * (7*Zeta(3))^(1/6 + t/36) / (A^t * 2^(2/3 + t/9) * sqrt(3*Pi) * n^(2/3 + t/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

G. C. Greubel, 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. 19.

Cf. A156616 (t=1), A261386 (t=2).

Cf. A015128, A027346, A027906, A193427, A255610.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(1/4 + 3/2 * (21*Zeta(3)/2)^(1/3) * n^(2/3)) * (7*Zeta(3)/3)^(1/4) / (2 * A^3 * sqrt(Pi) * n^(3/4)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A264686 Expansion of Product_{k>=1} (1 + 2*x^k)/(1 - x^k).

Original entry on oeis.org

1, 3, 6, 15, 27, 51, 93, 159, 264, 432, 696, 1086, 1683, 2553, 3837, 5700, 8367, 12147, 17505, 24972, 35361, 49728, 69402, 96243, 132657, 181782, 247692, 335838, 453042, 608289, 813102, 1082256, 1434519, 1894215, 2491644, 3265869, 4265973, 5553771, 7207167

Offset: 0

Vaclav Kotesovec, Nov 21 2015

nonn

Convolution of A000041 and A032302.

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

Cf. A000041, A006951, A015128, A032302, A261584, A264685, A266821.

Column k=3 of A321884.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 2*b(n-i, i-1))))
    end:
a:= n-> add(b(i$2)*combinat[numbpart](n-i), i=0..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 22 2017

nmax = 40; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

{ my(n=40); Vec(prod(k=1, n, 3/(1-x^k) - 2 + O(x*x^n))) } \\ Andrew Howroyd, Dec 22 2017

a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (4*Pi*sqrt(3)*n), where c = 2*Pi^2/3 + log(2)^2 + 2*polylog(2, -1/2) = 6.163360867463814765670634381079217086937812673723341... . - Vaclav Kotesovec, Jan 04 2016

cp's OEIS Frontend

A195584 O.g.f.: exp( Sum_{n>=1} (sigma(2*n^2)-sigma(n^2)) * x^n/n ).

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A236001 Sum of positive ranks of all overpartitions of n.

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A261611 Expansion of Product_{k>=0} (1 + x^(4*k+1))/(1 - x^(4*k+1)).

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A341366 Expansion of (1 / theta_4(x) - 1)^5 / 32.

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Maple

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A341367 Expansion of (1 / theta_4(x) - 1)^6 / 64.

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A004402 Expansion of 1 / Sum_{n=-oo..oo} x^(n^2).

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A004404 Expansion of 1 / (Sum_{n=-oo..oo} x^(n^2))^3.

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A235797 Triangle read by rows in which T(n,k) is the sum of the k-th largest elements in all overpartitions of n.

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A195584 O.g.f.: exp( Sum_{n>=1} (sigma(2n^2)-sigma(n^2)) x^n/n ).

A261611 Expansion of Product_{k>=0} (1 + x^(4k+1))/(1 - x^(4k+1)).