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.

A177399 O.g.f.: exp( Sum_{n>=1} (sigma(2n)-sigma(n))^n * x^n/n ).

Original entry on oeis.org

1, 2, 10, 188, 1414, 53596, 2923652, 44668152, 651967302, 605335444140, 7564881098284, 157357140966472, 96537385644719004, 695895399853879448, 86358988630956719304, 1103071610291574716763120
Offset: 0

Views

Author

Paul D. Hanna, May 30 2010

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 188*x^3 + 1414*x^4 + 53596*x^5 +...
log(A(x)) = 2*x + 4^2*x^2/2 + 8^3*x^3/3 + 8^4*x^4/4 + 12^5*x^5/5 +...+ A054785(n)^n*x^n/n +...
		

Crossrefs

Programs

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

A193538 O.g.f.: exp( Sum_{n>=1} (sigma(2*n)-sigma(n))^2/2 * x^n/n ).

Original entry on oeis.org

1, 2, 6, 20, 46, 116, 284, 632, 1414, 3102, 6536, 13636, 28020, 56300, 111888, 219608, 424694, 813104, 1540818, 2888060, 5366072, 9884616, 18050428, 32713048, 58851972, 105113942, 186505864, 328821408, 576153008, 1003687444, 1738735728, 2995837872
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2011

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 46*x^4 + 116*x^5 + 284*x^6 +...
log(A(x)) = 2^2*x/2 + 4^2*x^2/4 + 8^2*x^3/6 + 8^2*x^4/8 + 12^2*x^5/10 + 16^2*x^6/12 + 16^2*x^7/14 + 16^2*x^8/16 + 26^2*x^9/18 +...+ A054785(n)^2/2*x^n/n +...
		

Crossrefs

Programs

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

Formula

Self-convolution yields A177398.

A193539 O.g.f.: exp( Sum_{n>=1} (sigma(2*n)-sigma(n))^3 * x^n/n ).

Original entry on oeis.org

1, 8, 64, 512, 3200, 19392, 112128, 598016, 3088896, 15362408, 73331264, 340653056, 1538392064, 6762336448, 29072665600, 122299068416, 504128374784, 2040557142592, 8116582974656, 31760991869952, 122408808197120, 464983163273216, 1742277357389312
Offset: 0

Views

Author

Paul D. Hanna, Jul 30 2011

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + 8*x + 64*x^2 + 512*x^3 + 3200*x^4 + 19392*x^5 +...
log(A(x)) = 2^3*x + 4^3*x^2/2 + 8^3*x^3/3 + 8^3*x^4/4 + 12^3*x^5/5 + 16^3*x^6/6 + 16^3*x^7/7 + 16^3*x^8/8 + 26^3*x^9/9 +...+ A054785(n)^3*x^n/n +...
		

Crossrefs

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m)-sigma(m))^3*x^m/m)+x*O(x^n)), n)}
Showing 1-3 of 3 results.