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.

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A231291 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + k*x) / (1 - x - k*x^2).

Original entry on oeis.org

1, 1, 3, 9, 29, 99, 355, 1333, 5213, 21163, 88899, 385413, 1720637, 7894827, 37166563, 179254501, 884548253, 4460597131, 22962705027, 120557527941, 644952640253, 3512995320939, 19468234666531, 109694091843109, 628027149163613, 3651429293510731, 21547912967252163
Offset: 0

Views

Author

Paul D. Hanna, Nov 06 2013

Keywords

Comments

Compare to the identity:
Sum_{n>=0} x^n * Product_{k=1..n} (1 + k*x)/(1 + x + k*x^2) = 1/(1-x).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 29*x^4 + 99*x^5 + 355*x^6 + 1333*x^7 +...
where
A(x) = 1 + x*(1+x)/(1-x-x^2) + x^2*(1+x)*(1+2*x)/((1-x-x^2)*(1-x-2*x^2)) + x^3*(1+x)*(1+2*x)*(1+3*x)/((1-x-x^2)*(1-x-2*x^2)*(1-x-3*x^2)) + x^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)/((1-x-x^2)*(1-x-2*x^2)*(1-x-3*x^2)*(1-x-4*x^2)) +...

Links

Crossrefs

Cf. A204064, A231274.

Programs

  • PARI
    {a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (1+k*x)/(1-x-k*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

A231352 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k - x) / (1 - k*x + x^2).

Original entry on oeis.org

1, 1, 2, 8, 50, 382, 3434, 35694, 421682, 5582158, 81860978, 1317457646, 23087951666, 437673142510, 8924179990322, 194763818998638, 4530072136715954, 111870258525352174, 2923319958390174770, 80590596894930389102, 2337567736223817582002, 71162943130933082039278
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2013

Keywords

Comments

Compare to the identity: Sum_{n>=0} x^n*Product_{k=1..n} -(k + x)/(1 - k*x - x^2) = 1 - x.
Compare also to the identity: Sum_{n>=0} x^n*Product_{k=1..n} (k + x)/(1 + k*x + x^2) = (1+x^2)/(1-x).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 50*x^4 + 382*x^5 + 3434*x^6 +...
where
A(x) = 1 + x*(1-x)/(1-x+x^2) + x^2*(1-x)*(2-x)/((1-x+x^2)*(1-2*x+x^2)) + x^3*(1-x)*(2-x)*(3-x)/((1-x+x^2)*(1-2*x+x^2)*(1-3*x+x^2)) + x^4*(1-x)*(2-x)*(3-x)*(4-x)/((1-x+x^2)*(1-2*x+x^2)*(1-3*x+x^2)*(1-4*x+x^2)) +...

Links

Crossrefs

Cf. A231274.

Programs

  • PARI
    {a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (k-x)/(1-k*x+x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Oct 30 2014
G.f. (conjecture): 1/2 + (1/2)*Sum_{n >= 0} (2*x)^n * Product_{k = 1..n} (k - x)/(1 + k*x). - Peter Bala, Jul 12 2025
Showing 1-2 of 2 results.

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