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 1-4 of 4 results.

A209440 G.f.: 1 = Sum_{n>=0} a(n)*x^n * (1-x)^((n+1)^2).

Original entry on oeis.org

1, 1, 4, 30, 340, 5235, 102756, 2464898, 70120020, 2313120225, 86962820000, 3674969314090, 172615622432040, 8928295918586815, 504561763088722500, 30946605756915149850, 2048137516834986743700, 145535818715694311408181, 11054204297079333714850260
Offset: 0

Views

Author

Paul D. Hanna, Apr 07 2012

Keywords

Comments

Compare to a g.f. of the Catalan numbers: 1 = Sum_{n>=0} A000108(n)*x^n*(1-x)^(n+1).

Examples

			G.f.: 1 = 1*(1-x) + 1*x*(1-x)^4 + 4*x^2*(1-x)^9 + 30*x^3*(1-x)^16 + 340*x^4*(1-x)^25 +...

Links

Crossrefs

Cf. A000108, A107878, A177450, A180716.
Column k=2 of A355614.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
      *(-1)^(n-j)*binomial((j+1)^2, n-j), j=0..n-1))
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Jul 08 2022
  • Mathematica
    a[0] := 1; a[n_] := a[n] = Sum[(-1)^(n + 1 - k)*a[k]*Binomial[(k + 1)^2, n - k], {k, 0, n - 1}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jan 02 2018 *)
  • PARI
    {a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*(1-x+x*O(x^n))^((m+1)^2)), n))}
  • PARI
    {a(n)=if(n==0,1,sum(k=0,n-1,(-1)^(n+1-k)*a(k)*binomial((k+1)^2,n-k)))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n-1} (-1)^(n+1-k) * a(k) * binomial((k+1)^2,n-k) for n>=1, with a(0)=1.

A181167 G.f.: 1 = Sum_{n>=0} a(n)*x^n* Sum_{k>=0} C(2n+k,k)^2*(-x)^k.

Original entry on oeis.org

1, 1, 8, 165, 6384, 397320, 36273600, 4566166605, 757975618400, 160424015864112, 42164387189608320, 13473505313334666600, 5144136790654611953280, 2312696796696904699224000, 1209297981696245764641077760, 727688337054213932985609546525
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2010

Keywords

Comments

Compare g.f. to a g.f of the Catalan numbers: 1 = Sum_{n>=0} A000108(n)*x^n * Sum_{k>=0} C(2n+k,k)*(-x)^k.

Examples

			E.g.f.: E(x) = 1 + x^2/2! + 8*x^4/4! + 165*x^6/6! + 6384*x^8/8! +...
where the e.g.f. equals the continued fraction:
E(x) = 1/(1 - x^2/(2 - x^2/(3 - x^2/(4 - x^2/(5 - x^2/(6 - x^2/(7 - x^2/(8 - x^2/(9 - x^2/(10 - ...)))))))))). [Due to Matthieu Josuat-Vergès]
Illustrate the g.f. by the series:
1 = 1*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*x*(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 + 15^2*x^4 - 21^2*x^5 +...)
+ 8*x^2*(1 - 5^2*x + 15^2*x^2 - 35^2*x^3 + 70^2*x^4 - 126^2*x^5 +...)
+ 165*x^3*(1 - 7^2*x + 28^2*x^2 - 84^2*x^3 + 210^2*x^4 - 462^2*x^5+...)
+ 6384*x^4*(1 - 9^2*x + 45^2*x^2 - 165^2*x^3 + 495^2*x^4 +...)
+ 397320*x^5*(1 - 11^2*x + 66^2*x^2 - 286^2*x^3 + 1001^2*x^4 +...)
+ 36273600*x^6*(1 - 13^2*x + 91^2*x^2 - 455^2*x^3 + 1820^2*x^4 +...)
+ 4566166605*x^7*(1 - 15^2*x + 120^2*x^2 - 680^2*x^3 + 3060^2*x^4 +...)
+...
Compare to a g.f. of the Catalan numbers (A000108):
1 = 1*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*x*(1 - 3*x + 6*x^2 - 10*x^3 + 15*x^4 - 21*x^5 +...)
+ 2*x^2*(1 - 5*x + 15*x^2 - 35*x^3 + 70*x^4 - 126*x^5 +...)
+ 5*x^3*(1 - 7*x + 28*x^2 - 84*x^3 + 210*x^4 - 462*x^5 +...)
+ 14*x^4*(1 - 9*x + 45*x^2 - 165*x^3 + 495*x^4 - 1287*x^5 +...)
+ 42*x^5*(1 - 11*x + 66*x^2 - 286*x^3 + 1001*x^4 - 3003*x^5 +...)
+ 132*x^6*(1 - 13*x + 91*x^2 - 455*x^3 + 1820*x^4 - 6188*x^5 +...)
+...
Surprisingly, terms a(n) are divisible by n*A000108(n) for n>0:
a(2)=2*2*2, a(3)=3*5*11, a(4)=4*14*114, a(5)=5*42*1892, a(6)=6*132*45800, a(7)=7*429*1520535, ..., a(n)=n*A000108(n)*A181168(n).

Links

Crossrefs

Cf. A181168, A180716.
Cf. A002190. [From Paul D. Hanna, Oct 09 2010]
Cf. A115368.

Programs

  • Maple
    b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, x/(y+1)*(b(x-1, y-1)+b(x-1, y+1))))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 08 2018
  • Mathematica
    nmax=20; Table[(CoefficientList[Series[BesselJ[1,2*x]/x/BesselJ[0,2*x],{x,0,2*nmax}],x] * Range[0,2*nmax]!)[[2*n-1]],{n,1,nmax}] (* Vaclav Kotesovec, Jul 31 2014 *)
  • PARI
    {a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*sum(k=0, n-m, binomial(2*m+k, k)^2*(-x)^k)+x*O(x^n)), n))}
  • PARI
    /* Formula: a(n) = A000108(n)*A002190(n+1) implies: */
{a(n)=binomial(2*n,n)/(n+1)*(n+1)!^2*polcoeff(-log(sum(m=0,n+1,(-x)^m/m!^2)+O(x^(n+2))),n+1)} \\ Paul D. Hanna, Oct 09 2010
  • PARI
    /* Continued Fraction expansion of the E.G.F.: */
{a(n)=local(CF=1+O(x));for(i=0,n,CF=1/((n-i+1)-x^2*CF));(2*n)!*polcoeff(CF,2*n)}

Formula

a(n) = n*A000108(n)*A181168(n) = C(2n,n-1)*A181168(n) for n>0, with a(0)=1.
a(n) = A000108(n)*A002190(n+1), where A002190 describes the coefficients in -log(BesselJ(0,2*sqrt(x))) and A000108 is the Catalan numbers. - Paul D. Hanna, Oct 09 2010
Differentiating -log(BesselJ(0,2*sqrt(x))) and substituting z=z^2 gives the e.g.f. Sum_{n>=0} a(n) * z^(2*n)/(2n)! = BesselJ(1,2*z)/z/BesselJ(0,2*z). Consequently, using Gauss' continued fraction, this e.g.f. is also: 1/(1-z^2/(2-z^2/(3-z^2/(4-z^2/(5-z^2/...))))). - Matthieu Josuat-Vergès, Apr 17 2011
E.g.f.: U(0) where U(k) = 1 - x^2/(x^2 - (k+1)*(k+2)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Nov 15 2012
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 16/BesselJZero[0,1]^2 = 2.76664110449031883070186935..., c = 4/(sqrt(Pi)*BesselJZero[0,1]^2) = 0.390227523142124366836071453... . - Vaclav Kotesovec, Jul 31 2014

A217042 G.f.: 1 = Sum_{n>=0} a(n) * x^n * Sum_{k=0..2*n+1} binomial(2*n+1,k)^2 * (-x)^k.

Original entry on oeis.org

1, 1, 9, 216, 9685, 690129, 71218224, 10016312400, 1839013713405, 426795483514725, 122096137679279577, 42196285096882327872, 17327812666870134181584, 8338575020551966129589776, 4647348123388957546230426120, 2969504710005383652330487580832
Offset: 0

Views

Author

Paul D. Hanna, Sep 25 2012

Keywords

Comments

Compare g.f. to: 1 = Sum_{n>=0} A001764(n)*x^n * Sum_{k=0..2*n+1} C(2*n+1,k)*(-x)^k where A001764(n) = C(3*n+1,n)/(3*n+1).

Examples

			G.f.: A(x) = 1 + x + 9*x^2 + 216*x^3 + 9685*x^4 + 690129*x^5 +...
The coefficients satisfy:
1 = 1*(1 - x) + 1*x*(1 - 3^2*x^1 + 3^2*x^2 - x^3) +
9*x^2*(1 - 5^2*x^1 + 10^2*x^2 - 10^2*x^3 + 5^2*x^4 - x^5) +
216*x^3*(1 - 7^2*x^1 + 21^2*x^2 - 35^2*x^3 + 35^2*x^4 - 21^2*x^5 + 7^2*x^6 - x^7) +
9685*x^4*(1 - 9^2*x^1 + 36^2*x^2 - 84^2*x^3 + 126^2*x^4 - 126^2*x^5 + 84^2*x^6 - 36^2*x^7 + 9^2*x^8 - x^9) +
690129*x^5*(1 - 11^2*x^1 + 55^2*x^2 - 165^2*x^3 + 330^2*x^4 - 462^2*x^5 + 462^2*x^6 - 330^2*x^7 + 165^2*x^8 - 55^2*x^9 + 11^2*x^10 - x^11) +...

Crossrefs

Cf. A180716, A001764.

Programs

  • PARI
    {a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*sum(k=0, 2*m+1, binomial(2*m+1, k)^2*(-x)^k)+x*O(x^n)), n))}
for(n=0,31,print1(a(n),", "))

A247032 G.f.: 1 = Sum_{n>=0} a(n) * x^n * Sum_{k=0..n} C(n,k)^3 * (-x)^k.

Original entry on oeis.org

1, 1, 8, 215, 13544, 1646568, 342128448, 111806434449, 54089613731960, 36991616761628936, 34487632073741256512, 42564197996724997147672, 67876867685905911079322176, 137043021921732373141812704320, 344286933629331983612822165758464, 1060279482920092978432461141783224583
Offset: 0

Views

Author

Paul D. Hanna, Sep 09 2014

Keywords

Comments

Compare g.f. to a g.f. of the Catalan numbers (A000108):
1 = Sum_{n>=0} A000108(n)*x^n * Sum_{k=0..n+1} C(n+1,k)*(-x)^k.

Crossrefs

Cf. A180716.

Programs

  • PARI
    {a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*sum(k=0, m+1, binomial(m+1, k)^3*(-x)^k+x*O(x^n))^1 ), n))}
for(n=0,10,print1(a(n),", "))

Formula

G.f.: 1 = 1*(1-x) + 1*x*(1-2^3*x+x^2) + 8*x^2*(1-3^3*x+3^3*x^2-x^3) + 215*x^3*(1-4^3*x+6^3*x^2-4^3*x^3+x^4) + 13544*x^4*(1-5^3*x+10^3*x^2-10^3*x^3+5^3*x^4-x^5) +...
Showing 1-4 of 4 results.

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