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 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Aug 17 2012

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) is the sum of divisors of n.

Examples

			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 +...

Links

Crossrefs

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

Programs

  • PARI
    {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),", "))

Formula

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

A225959 a(n) = sigma(2*n^3) - sigma(n^3).

Original entry on oeis.org

2, 16, 80, 128, 312, 640, 800, 1024, 2186, 2496, 2928, 5120, 4760, 6400, 12480, 8192, 10440, 17488, 14480, 19968, 32000, 23424, 25440, 40960, 39062, 38080, 59048, 51200, 50520, 99840, 61568, 65536, 117120, 83520, 124800, 139904, 104120, 115840, 190400, 159744, 141288, 256000
Offset: 1

Views

Author

Paul D. Hanna, May 22 2013

Keywords

Comments

Here sigma(n) = A000203(n), the sum of the divisors of n.

Examples

			L.g.f.: L(x) = 2*x + 16*x^2/2 + 80*x^3/3 + 128*x^4/4 + 312*x^5/5 + 640*x^6/6 +...
where
exp(L(x)) = 1 + 2*x + 10*x^2 + 44*x^3 + 134*x^4 + 468*x^5 + 1524*x^6 + 4584*x^7 + 13862*x^8 +...+ A225958(n)*x^n +...
exp(-L(-x)) = 1 + 2*x - 6*x^2 + 12*x^3 + 38*x^4 - 108*x^5 + 148*x^6 + 168*x^7 +...+ A225957(n)*x^n +...

Links

Crossrefs

Cf. A225958, A225957, A000203, A224903.
Cf. A013662, A330595.

Programs

  • Mathematica
    a[n_] := DivisorSigma[1, 2*n^3] - DivisorSigma[1, n^3]; Array[a, 50] (* Amiram Eldar, Mar 17 2024 *)
  • PARI
    {a(n)=sigma(2*n^3)-sigma(n^3)}
for(n=1, 50, print1(a(n), ", "))

Formula

a(n) = A054785(n^3).
Logarithmic derivative of A225958.
Sum_{k=1..n} a(k) ~ c * n^4, where c = (15/44) * zeta(4) * Product_{p prime} (1 + 1/p^2 + 1/p^3) = (15/44) * A013662 * A330595 = 0.64531050605789193162... . - Amiram Eldar, Mar 17 2024
Showing 1-3 of 3 results.

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