cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Sep 20 2011

Keywords

Comments

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.

Examples

			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 +...

Links

Crossrefs

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

Programs

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

Formula

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.

A225925 G.f.: exp( Sum_{n>=1} A002129(n^2)*x^n/n ), where A002129(n) is the excess of sum of odd divisors of n over sum of even divisors of n.

Original entry on oeis.org

1, 1, -2, 2, -1, -7, 8, -14, 1, 11, -23, 43, -54, 38, 17, -55, 162, -198, 257, -175, 69, 141, -518, 764, -1049, 1215, -1241, 549, 161, -1625, 3192, -5176, 6782, -7568, 7267, -4263, -788, 8394, -17866, 29782, -39041, 46101, -45857, 36551, -14591, -20937, 70638, -129520, 190994, -245846, 280560
Offset: 0

Views

Author

Paul D. Hanna, May 20 2013

Keywords

Comments

Compare to: Sum_{n>=0} x^(n*(n+1)/2) = exp( Sum_{n>=1} A002129(n)*x^n/n ).

Examples

			G.f.: A(x) = 1 + x - 2*x^2 + 2*x^3 - x^4 - 7*x^5 + 8*x^6 - 14*x^7 + x^8 +...
where
log(A(x)) = x - 5*x^2/2 + 13*x^3/3 - 29*x^4/4 + 31*x^5/5 - 65*x^6/6 + 57*x^7/7 - 125*x^8/8 + 121*x^9/9 - 155*x^10/10 +...+ A002129(n^2)*x^n/n +...

Links

Crossrefs

Cf. A224340, A224339, A002129; variant: A215603.

Programs

  • PARI
    {A002129(n)=if(n<1, 0, -sumdiv(n, d, (-1)^d*d))}
{a(n)=polcoeff(exp(sum(k=1,n,A002129(k^2)*x^k/k)+x*O(x^n)),n)}
for(n=0,50,print1(a(n),", "))

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

Views

Author

Paul D. Hanna, May 21 2013

Keywords

Comments

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.

Examples

			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 +...

Links

Crossrefs

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

Programs

  • PARI
    {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), ", "))

Formula

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

A225958 O.g.f.: exp( Sum_{n>=1} (sigma(2*n^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

Views

Author

Paul D. Hanna, May 21 2013

Keywords

Comments

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.

Examples

			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 +...

Crossrefs

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

Programs

  • PARI
    {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), ", "))

Formula

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

Views

Author

Paul D. Hanna, Sep 20 2011

Keywords

Examples

			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 +...

Links

Crossrefs

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

Programs

  • Mathematica
    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)}

Formula

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

A224902 O.g.f.: exp( Sum_{n>=1} (sigma(2*n^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

Views

Author

Paul D. Hanna, Jul 24 2013

Keywords

Comments

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.

Examples

			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 +...

Crossrefs

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

Programs

  • PARI
    {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), ", "))

Formula

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).
Showing 1-6 of 6 results.

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