A195584 -id:A195584 - cp's OEIS frontend

A215603 O.g.f.: exp( Sum_{n>=1} -(sigma(2n^2) - sigma(n^2)) (-x)^n/n ).

1, 2, -2, 2, 10, -10, 6, 10, -22, 58, -58, 10, 114, -210, 270, -242, 74, 382, -930, 1474, -1542, 1010, 446, -2798, 5682, -7718, 8030, -5182, -998, 11126, -23802, 35626, -42246, 39450, -20810, -15546, 69514, -133770, 194918, -234106, 227410, -147706, -19738, 282234

Offset: 0

Paul D. Hanna, Aug 17 2012

sign

Compare to the Jacobi theta_3 function:
1 + 2*Sum_{n>=1} x^(n^2)  =  exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n ).
Here sigma(n) = A000203(n) is the sum of divisors of n.

			O.g.f.: A(x) = 1 + 2*x - 2*x^2 + 2*x^3 + 10*x^4 - 10*x^5 + 6*x^6 + 10*x^7 +...
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 + 114*x^7/7 - 128*x^8/8 + 242*x^9/9 - 248*x^10/10 + 266*x^11/11 - 416*x^12/12 +...+ -A054785(n^2)*(-x)^n/n +...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A195584, A195585, A177399, A054785, A000203; variants: A225925, A225957.

{a(n)=polcoeff(exp(sum(m=1, n,-(sigma(2*m^2)-sigma(m^2))*(-x)^m/m)+x^2*O(x^n)),n)}
for(n=0,50,print1(a(n),", "))

O.g.f.: exp( Sum_{n>=1} -A054785(n^2)*(-x)^n/n ), where A054785(n^2) = A195585(n).

A225957 O.g.f.: exp( Sum_{n>=1} -(sigma(2n^3) - sigma(n^3)) (-x)^n/n ).

Original entry on oeis.org

1, 2, -6, 12, 38, -108, 148, 168, -922, 2294, -2656, -1732, 17908, -44516, 60896, -6936, -206474, 650848, -1181394, 1146324, 865832, -6609592, 16632596, -26643544, 22498916, 23275482, -144152248, 349896736, -563311472, 532552508, 233516176, -2378435472, 6264582710

Offset: 0

Paul D. Hanna, May 21 2013

sign

Compare to the Jacobi theta_3 function:
1 + 2*Sum_{n>=1} x^(n^2)  =  exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n ).
Here sigma(n) = A000203(n), the sum of the divisors of n.

			O.g.f.: A(x) = 1 + 2*x - 6*x^2 + 12*x^3 + 38*x^4 - 108*x^5 + 148*x^6 + 168*x^7 +...
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 + 114*x^7/7 - 128*x^8/8 + 242*x^9/9 - 248*x^10/10 + 266*x^11/11 - 416*x^12/12 +...+ -(-1)^n*A054785(n^3)*x^n/n +...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A225958, A225959, A054785, A000203; variants: A215603, A225925, A195584.

{a(n)=polcoeff(exp(sum(m=1, n, -(sigma(2*m^3)-sigma(m^3))*(-x)^m/m)+x^2*O(x^n)), n)}
for(n=0, 50, print1(a(n), ", "))

O.g.f.: exp( Sum_{n>=1} -A054785(n^3)*(-x)^n/n ).

A225958 O.g.f.: exp( Sum_{n>=1} (sigma(2n^3) - sigma(n^3)) x^n/n ).

Original entry on oeis.org

1, 2, 10, 44, 134, 468, 1524, 4584, 13862, 40566, 114880, 321052, 879092, 2360156, 6248864, 16297384, 41902454, 106437600, 267149022, 662979572, 1628437160, 3960377672, 9541519732, 22786066280, 53958062564, 126750346970, 295476011176, 683776368416, 1571299804688

Offset: 0

Paul D. Hanna, May 21 2013

nonn

Compare to the Jacobi theta_3 function:
1 + 2*Sum_{n>=1} x^(n^2)  =  exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n ).
Here sigma(n) = A000203(n), the sum of the divisors of n.

			O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 44*x^3 + 134*x^4 + 468*x^5 + 1524*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 + 114*x^7/7 + 128*x^8/8 + 242*x^9/9 + 248*x^10/10 + 266*x^11/11 +...+ A054785(n^3)*x^n/n +...

Cf. A225957, A225959, A054785, A000203; variants: A195584, A215603, A225925, A224902.

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m^3)-sigma(m^3))*x^m/m)+x^2*O(x^n)), n)}
for(n=0, 50, print1(a(n), ", "))

O.g.f.: exp( Sum_{n>=1} A054785(n^3)*x^n/n ).

Logarithmic derivative equals A225959.

A195585 sigma(2*n^2) - sigma(n^2).

Original entry on oeis.org

2, 8, 26, 32, 62, 104, 114, 128, 242, 248, 266, 416, 366, 456, 806, 512, 614, 968, 762, 992, 1482, 1064, 1106, 1664, 1562, 1464, 2186, 1824, 1742, 3224, 1986, 2048, 3458, 2456, 3534, 3872, 2814, 3048, 4758, 3968, 3446, 5928, 3786, 4256, 7502, 4424, 4514, 6656, 5602, 6248, 7982

Offset: 1

Paul D. Hanna, Sep 20 2011

nonn

			L.g.f.: L(x) = 2*x + 8*x^2/2 + 26*x^3/3 + 32*x^4/4 + 62*x^5/5 + 104*x^6/6 +...
where the g.f. of A195584 begins:
exp(L(x)) = 1 + 2*x + 6*x^2 + 18*x^3 + 42*x^4 + 102*x^5 + 238*x^6 +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A195584, A215603, A054785, A015128, A002117.

Table[DivisorSigma[1,2n^2]-DivisorSigma[1,n^2],{n,60}] (* Harvey P. Dale, May 05 2021 *)

PARI
```
{a(n)=sigma(2*n^2)-sigma(n^2)}
```

Equals the logarithmic derivative of A195584.
a(n) = A054785(n^2), where A054785 is the logarithmic derivative of A015128, which is the number of overpartitions of n.
Sum_{k=1..n} a(k) ~ c * n^3, where c = 7*zeta(3)/Pi^2 = 0.85255679763501158184... . - Amiram Eldar, Mar 17 2024

A219430 Number of overpartitions of n^2; a(n) = A015128(n^2).

Original entry on oeis.org

1, 2, 14, 154, 2062, 31066, 504886, 8652402, 154208270, 2832526306, 53287424374, 1022143389578, 19924535352374, 393685747760714, 7869272950148382, 158875743754158098, 3235672769357219854, 66405081412501161442, 1372115409786911859502, 28524372351269271839610

Offset: 0

Paul D. Hanna, Nov 19 2012

nonn

Limit a(n+1)/a(n) = exp(Pi) = 23.14069263...
a(n) ~ (cosh(Pi*n) - sinh(Pi*n)/(Pi*n)) / (4*n^2), a "remarkable approximation" due to "Ramanujan's false statement" (see formula 12 in "Jagged partitions" link).
By definition of A015128, an overpartition of n^2 is an ordered sequence of nonincreasing integers that sum to n^2, where the first occurrence of each integer may be overlined (see Hirschhorn and Sellers link).

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 154*x^3 + 2062*x^4 + 31066*x^5 + 504886*x^6 +...
It appears that the logarithmic derivative of the g.f. A(x),
A'(x)/A(x) = 2 + 24*x + 386*x^2 + 6832*x^3 + 128442*x^4 + 2505720*x^5 + 50153770*x^6 + 1022997344*x^7 + 21170657906*x^8 +...+ A219431(n+1)*x^n +...
is congruent to 2/(1-x^2) mod 4.

Vaclav Kotesovec, Table of n, a(n) for n = 0..730 (terms 0..180 from Paul D. Hanna)
J.-F. Fortin, P. Jacob and P. Mathieu, Jagged partitions
M. D. Hirschhorn and J. A. Sellers, AN INFINITE FAMILY OF OVERPARTITION CONGRUENCES MODULO 12

Cf. A219431, A015128, A195584.

Table[Sum[PartitionsP[n^2-k]*PartitionsQ[k], {k, 0, n^2}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 28 2015 *)

/* Formula: a(n) = [x^(n^2)] 1 / theta_4(x) */
{a(n)=polcoeff(1/(1+2*sum(k=1,n,(-x)^(k^2))+x*O(x^(n^2))),n^2)}
for(n=0,20,print1(a(n),", "))

/* Formula: a(n) = -2*Sum_{k=1..n} (-1)^k * A015128(n^2-k^2) */
{A015128(n)=polcoeff(1/(1+2*sum(k=1, sqrtint(n+1), (-x)^(k^2))+x*O(x^(n))), n)}
{a(n)=if(n==0,1,-2*sum(k=1, n, (-1)^k*A015128(n^2-k^2)))}
for(n=0, 25, print1(a(n), ", "))

a(n) = -2*Sum_{k=1..n} (-1)^k * A015128(n^2-k^2) for n>0 with a(0)=1.
a(n) = [x^(n^2)] 1 / ( Sum_{m=-inf..inf} (-x)^(m^2) ).
a(n) = [x^(n^2)] 1 / theta_4(x).
a(n) = [x^(n^2)] eta(x^2) / eta(x)^2.
a(n) = [x^(n^2)] Product_{m>=1} (1 + x^m) / (1 - x^m).
a(n) = [x^(n^2)] Product_{m>=1} 1 / ( (1 - x^(2*m)) * (1 - x^(2*m-1))^2 ).
a(n) = [x^(n^2)] exp( Sum_{m>=1} 2*x^(2*m-1)/(1 - x^(2*m-1))/(2*m-1) ).
a(n) = [x^(n^2)] exp( Sum_{m>=1} (sigma(2*m) - sigma(m)) * x^m/m ).

A224902 O.g.f.: exp( Sum_{n>=1} (sigma(2n^4) - sigma(n^4)) x^n/n ).

Original entry on oeis.org

1, 2, 18, 114, 450, 2298, 10466, 43314, 184402, 749490, 2942274, 11437026, 43364818, 161089130, 589901682, 2123791130, 7531395154, 26360805018, 91057065522, 310718196626, 1048405959266, 3499152601394, 11559430074418, 37818135048962, 122582070331106, 393830310786706, 1254654362883954

Offset: 0

Paul D. Hanna, Jul 24 2013

nonn

Compare to the Jacobi theta_3 function:
1 + 2*Sum_{n>=1} x^(n^2)  =  exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n ).
Here sigma(n) = A000203(n), the sum of the divisors of n.

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 114*x^3 + 450*x^4 + 2298*x^5 +...
where
log(A(x)) = 2*x + 32*x^2/2 + 242*x^3/3 + 512*x^4/4 + 1562*x^5/5 +...+ A224903(n)*x^n/n +...

Cf. A224903, A054785, A000203; variants: A195584, A215603, A225958.

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m^4)-sigma(m^4))*x^m/m)+x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

O.g.f.: exp( Sum_{n>=1} A054785(n^4)*x^n/n ).
Logarithmic derivative equals A224903.
a(n) == 2 (mod 4) for n>0 (conjecture).

cp's OEIS Frontend

A215603 O.g.f.: exp( Sum_{n>=1} -(sigma(2*n^2) - sigma(n^2)) * (-x)^n/n ).

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A225957 O.g.f.: exp( Sum_{n>=1} -(sigma(2*n^3) - sigma(n^3)) * (-x)^n/n ).

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A225958 O.g.f.: exp( Sum_{n>=1} (sigma(2*n^3) - sigma(n^3)) * x^n/n ).

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A195585 sigma(2*n^2) - sigma(n^2).

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A219430 Number of overpartitions of n^2; a(n) = A015128(n^2).

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A224902 O.g.f.: exp( Sum_{n>=1} (sigma(2*n^4) - sigma(n^4)) * x^n/n ).

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A215603 O.g.f.: exp( Sum_{n>=1} -(sigma(2n^2) - sigma(n^2)) (-x)^n/n ).

A225957 O.g.f.: exp( Sum_{n>=1} -(sigma(2n^3) - sigma(n^3)) (-x)^n/n ).

A225958 O.g.f.: exp( Sum_{n>=1} (sigma(2n^3) - sigma(n^3)) x^n/n ).

A224902 O.g.f.: exp( Sum_{n>=1} (sigma(2n^4) - sigma(n^4)) x^n/n ).