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.

A177778 E.g.f.: A(x) = Sum_{n>=0} 2^n/n!*Product_{k=0..n-1} L(2^k*x), where L(x) is the e.g.f. of A177777.

Original entry on oeis.org

1, 2, 12, 160, 4272, 221648, 22347648, 4416360160, 1724182065408, 1336677590208512, 2064038664552586752, 6359502604300426739200, 39136760890428640414851072, 481344480930558145524346370048
Offset: 0

Views

Author

Paul D. Hanna, May 20 2010

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 160*x^3/3! + 4272*x^4/4! +...
Then e.g.f. A(x) is given by:
A(x) = 1 + 2*L(x) + 2^2*L(x)L(2x)/2! + 2^3*L(x)L(2x)L(4x)/3! + 2^4*L(x)L(2x)L(4x)L(8x)/4! +...
where L(x) is the e.g.f. of A177777:
. L(x) = x + 2*x^2/2! + 12*x^3/3! + 152*x^4/4! + 3640*x^5/5! +...
. L(x) = x*d/dx log( Sum_{n>=0} 2^(n(n-1)/2)*x^n/n! )
and satisfies:
. L(x)/x = 1 + L(x) + L(x)L(2x)/2! + L(x)L(2x)L(4x)/3! + L(x)L(2x)L(4x)L(8x)/4! +...

Crossrefs

Cf. A177777, A177780.

Programs

  • PARI
    {a(n,q=2)=local(Lq=x+x^2,A); for(i=1,n,Lq=x*sum(m=0,n,(q-1)^m/m!*prod(k=0,m-1,subst(Lq,x,q^k*x+x*O(x^n))))); A=sum(m=0,n,2^m/m!*prod(k=0,m-1,subst(Lq,x,q^k*x+x*O(x^n))));n!*polcoeff(A,n)}

A177780 E.g.f. satisfies: L(x) = x*Sum_{n>=0} (2^n/n!)*Product_{k=0..n-1} L(3^k*x).

Original entry on oeis.org

1, 4, 60, 2496, 276240, 83893248, 72508524480, 182341191057408, 1348995112077074688, 29528107099434111467520, 1918583757808453356238126080, 370812729559366641806998574727168
Offset: 1

Views

Author

Paul D. Hanna, May 20 2010

Keywords

Comments

More generally, we have the following conjecture.
Define the series E(,) and L(,) by:
E(x,q) = Sum_{n>=0} q^(n(n-1)/2)*x^n/n!,
L(x,q) = x*d/dx log(E(x,q)) = x*E'(x,q)/E(x,q),
then L(x,q) satisfies:
L(x,q) = x*Sum_{n>=0} (q-1)^n/n! * Product_{k=0..n-1} L(q^k*x,q),
1/E(x,q) = Sum_{n>=0} (-1)^n/n! * Product_{k=0..n-1} L(q^k*x,q).

Examples

			E.g.f.: L(x) = x + 4*x^2/2! + 60*x^3/3! + 2496*x^4/4! + 276240*x^5/5! + ... + n*A054941(n)*x^n/n! + ...
Given the related expansions:
  E(x) = 1 + x + 3*x^2/2! + 27*x^3/3! + 729*x^4/4! + 59049*x^5/5! + ...
  log(E(x)) = x + 2*x^2/2! + 20*x^3/3! + 624*x^4/4! + 55248*x^5/5! + ... + A054941(n)*x^n/n! + ...
then L(x) satisfies:
  L(x)/x = 1 + 2*L(x) + 2^2*L(x)L(3x)/2! + 2^3*L(x)L(3x)L(9x)/3! + 2^4*L(x)L(3x)L(9x)L(27x)/4! + ...
  1/E(x) = 1 - L(x) + L(x)L(3x)/2! - L(x)L(3x)L(9x)/3! + L(x)L(3x)L(9x)L(27x)/4! -+ ...

Crossrefs

Cf. A054941, A177779, A177777, A177781.

Programs

  • Mathematica
    m = 13; A[] = 0; Do[A[x] = x Sum[2^n/n! Product[A[3^k x], {k, 0, n-1}], {n, 0, m}] + O[x]^m // Normal, {m}]; CoefficientList[A[x]/x, x] * Range[1, m-1]! (* Jean-François Alcover, Nov 03 2019 *)
  • PARI
    {a(n,q=3)=local(Lq=x+x^2);for(i=1,n,Lq=x*sum(m=0,n,(q-1)^m/m!*prod(k=0,m-1,subst(Lq,x,q^k*x+x*O(x^n)))));n!*polcoeff(Lq,n)}

Formula

a(n) = n*A054941(n), where A054941(n) is the number of connected oriented graphs on n nodes.
Define the series E(x) and L(x) by:
E(x) = Sum_{n>=0} 3^(n(n-1)/2)*x^n/n!,
L(x) = x*d/dx log(E(x)) = x*E'(x)/E(x),
then L(x) satisfies:
L(x) = x*Sum_{n>=0} 2^n/n! * Product_{k=0..n-1} L(3^k*x),
1/E(x) = Sum_{n>=0} (-1)^n/n! * Product_{k=0..n-1} L(3^k*x).

A177781 E.g.f. satisfies: L(x) = x*Sum_{n>=0} 3^n/n!*Product_{k=0..n-1} L(4^k*x).

Original entry on oeis.org

1, 6, 162, 15336, 5135400, 6403850928, 30733361357328, 576178771105452672, 42495458789243292762240, 12378928091101498820594407680, 14278666564505879853034906179788544
Offset: 1

Views

Author

Paul D. Hanna, May 20 2010

Keywords

Comments

More generally, we have the following conjecture.
Define the series E(,) and L(,) by:
. E(x,q) = Sum_{n>=0} q^(n(n-1)/2)*x^n/n!,
. L(x,q) = x*d/dx log(E(x,q)) = x*E'(x,q)/E(x,q),
then L(x,q) satisfies:
. L(x,q) = x*Sum_{n>=0} (q-1)^n/n! * Product_{k=0..n-1} L(q^k*x,q),
. 1/E(x,q) = Sum_{n>=0} (-1)^n/n! * Product_{k=0..n-1} L(q^k*x,q).
...
Explicitly, L(x,q) = [Sum_{n>=1} q^(n(n-1)/2)*x^n/(n-1)! ]/[Sum_{n>=0} q^(n(n-1)/2)*x^n/n! ]. - Paul D. Hanna, Aug 31 2010

Examples

			E.g.f.: L(x) = x + 6*x^2/2! + 162*x^3/3! + 15336*x^4/4! + 5135400*x^5/5! + ... + n*A003027(n)*x^n/n! + ...
Given the related expansions:
. E(x) = 1 + x + 4*x^2/2! +64*x^3/3! +4096*x^4/4! +1048576*x^5/5! + ...
. log(E(x)) = x + 3*x^2/2! +54*x^3/3! +3834*x^4/4! +1027080*x^5/5! + ... + A003027(n)*x^n/n! + ...
then L(x) satisfies:
. L(x)/x = 1 + 3*L(x) + 3^2*L(x)L(4x)/2! + 3^3*L(x)L(4x)L(16x)/3! + 3^4*L(x)L(4x)L(16x)L(64x)/4! + ...
. 1/E(x) = 1 - L(x) + L(x)L(4x)/2! - L(x)L(4x)L(16x)/3! + L(x)L(4x)L(16x)L(64x)/4! -+ ...

Crossrefs

Cf. A003027, A177777, A177780.

Programs

  • PARI
    {a(n,q=4)=local(Lq=x+x^2);for(i=1,n,Lq=x*sum(m=0,n,(q-1)^m/m!*prod(k=0,m-1,subst(Lq,x,q^k*x+x*O(x^n)))));n!*polcoeff(Lq,n)}
  • PARI
    {a(n,q=4)=n!*polcoeff(sum(m=1,n,q^(m*(m-1)/2)*x^m/(m-1)!)/sum(m=0,n,q^(m*(m-1)/2)*x^m/m!+x*O(x^n)),n)} \\ Paul D. Hanna, Aug 31 2010

Formula

a(n) = n*A003027(n), where A003027(n) is the number of weakly connected digraphs with n nodes.
Define the series E(x) and L(x) by:
. E(x) = Sum_{n>=0} 4^(n(n-1)/2)*x^n/n!,
. L(x) = x*d/dx log(E(x)) = x*E'(x)/E(x),
then L(x) satisfies:
. L(x) = x*Sum_{n>=0} 3^n/n! * Product_{k=0..n-1} L(4^k*x),
. 1/E(x) = Sum_{n>=0} (-1)^n/n! * Product_{k=0..n-1} L(4^k*x).
...
E.g.f.: L(x) = [Sum_{n>=1} 4^(n(n-1)/2)*x^n/(n-1)! ]/[Sum_{n>=0} 4^(n(n-1)/2)*x^n/n! ]. - Paul D. Hanna, Aug 31 2010
Showing 1-3 of 3 results.

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