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 1-5 of 5 results.

A098616 Product of Pell and Catalan numbers: a(n) = A000129(n+1)*A000108(n).

Original entry on oeis.org

1, 2, 10, 60, 406, 2940, 22308, 175032, 1408550, 11561836, 96425836, 814773960, 6960289532, 60012947800, 521582661000, 4564643261040, 40190674554630, 355772529165900, 3164408450118300, 28266363849505320, 253466716153665300, 2280803103062033160, 20588945107316958840
Offset: 0

Views

Author

Paul D. Hanna, Oct 09 2004

Keywords

Comments

Radius of convergence: r = (sqrt(2)-1)/4, where A(r) = sqrt(2+sqrt(2)).
More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2), |b|>0, |c|>0, then Sum_{n>=0} S(n)*Catalan(n)*x^n = sqrt( (1-2*b*x - sqrt(1-4*b*x-16*c*x^2))/(2*b^2+8*c) )/x.

Examples

			Sequence begins: [1*1, 2*1, 5*2, 12*5, 29*14, 70*42, 169*132, 408*429,...].

Links

Crossrefs

Cf. A000129, A000108, A098614, A098617, A098618, A000984, A214377.

Programs

  • Mathematica
    With[{nn=30},Times@@@Thread[{LinearRecurrence[{2,1},{1,2},nn], CatalanNumber[ Range[0,nn-1]]}]] (* Harvey P. Dale, Jan 04 2012 *)
a[n_] := Fibonacci[n + 1, 2] * CatalanNumber[n]; Array[a, 25, 0] (* Amiram Eldar, May 05 2023 *)
  • PARI
    a(n) = binomial(2*n,n)/(n+1)*round(((1+sqrt(2))^(n+1)-(1-sqrt(2))^(n+1))/(2*sqrt(2)))

Formula

G.f.: A(x) = sqrt( (1-4*x - sqrt(1-8*x-16*x^2))/16 )/x.
Run lengths of zeros (mod 10) equal (5^k - (-1)^k)/2 - 1 starting at index (5^k + (-1)^k)/2:
a(n) == 0 (mod 10) for n = (5^k + (-1)^k)/2 through n = 5^k - 1 when k>=1.
a(n) ~ 2^(2*n-3/2) * (1+sqrt(2))^(n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, May 09 2014
A(-x) = 1/x * series reversion( x*(2*x + sqrt(1 - 4*x^2)) ). Compare with the o.g.f. B(x) of the central binomial numbers A000984, which satisfies B(-x) = 1/x * series reversion( x*(2*x + sqrt(1 + 4*x^2)) ). See also A214377. - Peter Bala, Oct 19 2015
n*(n+1)*a(n) -4*n*(2*n-1)*a(n-1) -4*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 17 2018
Sum_{n>=0} a(n)/16^n = 2*sqrt(3-sqrt(7)). - Amiram Eldar, May 05 2023
G.f. A(x) satisfies A(x) = sqrt( 1 + 4*x*A(x)^2 + 8*x^2*A(x)^4 ). - Paul D. Hanna, Dec 14 2024

A098619 G.f. A(x) satisfies: A(x*G098618(x)) = G098618(x), where G098618 is the g.f. for A098618(n) = A007482(n)*Catalan(n).

Original entry on oeis.org

1, 3, 13, 51, 213, 867, 3589, 14739, 60853, 250563, 1033605, 4259571, 17565909, 72412707, 298586661, 1231016019, 5075753589, 20927272323, 86286346693, 355763629491, 1466857936405, 6047981701347, 24936516122469, 102815688922899, 423920292507061, 1747866711689283, 7206641564551429
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2004

Keywords

Comments

G.f. satisfies: A(x) = x/(series reversion of x*G098618(x)), where G098618 is the g.f. for A098618 = {1*1,3*1,11*2,39*5,139*14,495*42,1763*132,...}.

Links

Crossrefs

Cf. A098618, A098615, A098617, A007482, A000108.

Programs

  • Mathematica
    Flatten[{1,3,13,51,Table[17^(n/2)*(1/2+1/2*(-1)^n + 3/34*Sqrt[17]*(1-(-1)^n) + Sum[(-1)^j*(4/17 + Sum[Binomial[2*k-1,k-1]*2^(k+3)/ ((k+1)*17^(k+1)), {k,1,Floor[(j-1)/2]}]),{j,3,n-1}]),{n,4,20}]}] (* Vaclav Kotesovec, Oct 29 2012 *)
  • PARI
    a(n)=polcoeff((sqrt(1-8*x^2+x^2*O(x^n))+3*x)/(1-17*x^2),n);
  • PARI
    x='x+O('x^66); Vec((sqrt(1-8*x^2) + 3*x)/(1-17*x^2)) \\ Joerg Arndt, May 12 2013

Formula

G.f.: (sqrt(1-8*x^2) + 3*x)/(1-17*x^2).
a(2*n+1) = 3*17^n.
Recurrence: n*a(n) = (25*n-24)*a(n-2) - 136*(n-3)*a(n-4). - Vaclav Kotesovec, Oct 29 2012

A200376 G.f.: 1/sqrt(1-10*x^2 + x^4/(1-8*x^2)) + x/(1-9*x^2).

Original entry on oeis.org

1, 1, 5, 9, 37, 81, 301, 729, 2549, 6561, 22045, 59049, 193029, 531441, 1703469, 4782969, 15111573, 43046721, 134539837, 387420489, 1200901157, 3486784401, 10739313997, 31381059609, 96172251061, 282429536481, 862142190941, 2541865828329, 7734936371269, 22876792454961, 69439155241581
Offset: 0

Views

Author

Paul D. Hanna, Nov 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 9*x^3 + 37*x^4 + 81*x^5 + 301*x^6 + 729*x^7 +...
The g.f. of A200375(n) = A000108(n)*A001045(n) begins:
G(x) = 1 + x + 2*3*x^2 + 5*5*x^3 + 14*11*x^4 + 42*21*x^5 + 132*43*x^6 +...
where A(x) = G(x/A(x)) and G(x) = A(x*G(x)).

Crossrefs

Cf. A200375, A098615, A098617.

Programs

  • Mathematica
    CoefficientList[Series[1/Sqrt[1-10x^2+x^4/(1-8x^2)]+x/(1-9x^2),{x,0,30}], x] (* Harvey P. Dale, Nov 19 2011 *)
  • PARI
    {a(n)=polcoeff(1/sqrt(1-10*x^2 + x^4/(1-8*x^2 +x*O(x^n))) + x/(1-9*x^2 +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=local(G=sum(m=0,n,binomial(2*m, m)/(m+1)*polcoeff(1/(1-x-2*x^2+x*O(x^m)), m)*x^m)+x*O(x^n)); polcoeff(x/serreverse(x*G),n)}
for(n=0,30,print1(a(n),", "))

Formula

D-finite with recurrence: n*a(n) +(n-1)*a(n-1) +(24-17*n)*a(n-2) +(41-17*n)*a(n-3) +72*(n-3)*a(n-4) +72*(n-4)*a(n-5)=0. - R. J. Mathar, Nov 17 2011
G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x) + 9*x^2*A(x)^2). - Paul D. Hanna, Nov 18 2014
Let G(x) = g.f. of A200375, then g.f. A(x) satisfies:
(1) A(x) = x/Series_Reversion(x*G(x)),
(2) A(x) = G(x/A(x)) and G(x) = A(x*G(x)),
where A200375(n) = A000108(n)*A001045(n), the product of Catalan and Jacobsthal numbers.
a(n) ~ 3^(n-1). - Vaclav Kotesovec, Jun 29 2013

A194349 E.g.f.: -log( sqrt(1-x^2) - x ).

Original entry on oeis.org

1, 2, 5, 24, 129, 960, 7965, 80640, 903105, 11612160, 163451925, 2554675200, 43259364225, 797058662400, 15764670046125, 334764638208000, 7571150452490625, 182111963185152000, 4634731528895593125, 124564582818643968000
Offset: 1

Views

Author

Paul D. Hanna, Aug 21 2011

Keywords

Comments

Compare e.g.f. to arccosh(x) = log(sqrt(x^2-1) + x).

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 5*x^3/3! + 24*x^4/4! + 129*x^5/5! + ...
where
exp(A(x)) = 1 + 2*(x/2) + 6*(x/2)^2 + 16*(x/2)^3 + 46*(x/2)^4 + 128*(x/2)^5 + ... + A098617(n)*(x/2)^n + ...

Crossrefs

Cf. A098617, A100097.

Programs

  • Mathematica
    With[{nn=30},Rest[CoefficientList[Series[-Log[Sqrt[1-x^2]-x],{x,0,nn}], x] Range[0,nn]!]] (* Harvey P. Dale, Dec 01 2011 *)
  • PARI
    {a(n)=n!*polcoeff(-log(sqrt(1-x^2+x*O(x^n))-x),n)}
  • PARI
    {A000129(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=if(n<1,0,sum(k=0,floor((n+1)/2),binomial(n+1, k)*A000129(n+1-2*k))*(n-1)!/2^n)}

Formula

a(2*n) = 2^n*(2*n-1)! for n>=1.
a(n) = A100097(n+1)*(n-1)!/2^n for n>=1.
a(n) = (n-1)!/2^n * Sum_{k=0..floor((n+1)/2)} C(n+1,k)*A000129(n+1-2*k) for n >= 1. [From a formula of Paul Barry in A100097]
E.g.f.: log( (sqrt(1-x^2) + x)/(1-2*x^2) ).

A201093 Triangle T(n,k), read by rows, given by (0,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,...) DELTA (1,0,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 3, 1, 0, 0, 2, 3, 4, 1, 0, 2, 2, 4, 6, 5, 1, 0, 0, 4, 6, 8, 10, 6, 1, 0, 5, 5, 9, 13, 15, 15, 7, 1, 0, 0, 10, 15, 20, 25, 26, 21, 8, 1, 0, 14, 14, 24, 34, 41, 45, 42, 28, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Nov 26 2011

Keywords

Comments

Riordan array (1,xf(x)) where f(x) is g.f. of A097331.
Row sums are in A001405.

Examples

			Triangle begins :
1
0, 1
0, 1, 1
0, 0, 2, 1
0, 1, 1, 3, 1
0, 0, 2, 3, 4, 1
0, 2, 2, 4, 6, 5, 1
0, 0, 4, 6, 8, 10, 6, 1
0, 5, 5, 9, 13, 15, 15, 7, 1
0, 0, 10, 15, 20, 25, 26, 21, 8, 1
0, 14, 14, 24, 34, 41, 45, 42, 28, 9, 1
0, 0, 28, 42, 56, 70, 78, 77, 64, 36, 10, 1
0, 42, 42, 70, 98, 120, 136, 140, 126, 93, 45, 11, 1

Crossrefs

Cf. Diagonals : A000012, A001477, A000217, A000215,
Columns :A000007, A097331, A089408

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A001405(n), A098617(n) for x = 0,1,2 respectively.
Showing 1-5 of 5 results.

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