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.

A216604 G.f. satisfies: A(x) = (1 + x*(1-x)*A(x)) * (1 + x^2*A(x)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 12, 19, 33, 59, 102, 181, 329, 593, 1076, 1979, 3643, 6723, 12494, 23289, 43498, 81557, 153356, 288925, 545687, 1032997, 1958978, 3721819, 7083716, 13503311, 25778612, 49283755, 94345179, 180830195, 347006694, 666636809, 1282024484, 2467964693
Offset: 0

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Author

Paul D. Hanna, Sep 10 2012

Keywords

Comments

The radius of convergence of the g.f. A(x) equals 1/2, with A(1/2) = 4.
More generally, if A(x) = (1 + x*(t-x)*A(x)) * (1 + x^2*A(x)), |t|>0, then
A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k) )
where the radius of convergence r of the g.f. A(x) satisfies
r = (1-r)^2/(t-r) = (1-t*r)/(2*(1-r)) with A(r) = 1/(r*(1-r)) = 2/(1-t*r).
Number of Motzkin excursions of length n that avoid the patterns UU, UD and DH. A Motzkin excursion is a lattice path with steps from the set {D=-1, H=0, U=1} that starts at (0,0), never goes below the x-axis, and terminates at the altitude 0. - Andrei Asinowski, Dec 20 2019

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 12*x^7 + 19*x^8 + ...
The logarithm of the g.f. begins:
log(A(x)) = ((1-x) + x)*x + ((1-x)^2 + 2^2*x*(1-x) + x^2)*x^2/2 +
((1-x)^3 + 3^2*x*(1-x)^2 + 3^2*x^2*(1-x) + x^3)*x^3/3 +
((1-x)^4 + 4^2*x*(1-x)^3 + 6^2*x^2*(1-x)^2 + 4^2*x^3*(1-x) + x^4)*x^4/4 +
((1-x)^5 + 5^2*x*(1-x)^4 + 10^2*x^2*(1-x)^3 + 10^2*x^3*(1-x)^2 + 5^2*x^4*(1-x) + x^5)*x^5/5 + ...
Explicitly,
log(A(x)) = x + x^2/2 + 4*x^3/3 + 5*x^4/4 + 6*x^5/5 + 16*x^6/6 + 29*x^7/7 + 45*x^8/8 + 94*x^9/9 + 186*x^10/10 + ... + A217464(n)*x^n/n + ...

Links

Crossrefs

Cf. A000108, A217464, A105633, A216434, A216454, A216447, A192415, A216616, A216617.

Programs

  • Mathematica
    CoefficientList[Series[((1 - x) - Sqrt[(1 - x)^2 - 4*x^3*(1 - x)])/(2*x^3 *(1 - x)), {x,0,50}], x] (* G. C. Greubel, Jan 24 2017 *)
  • Maxima
    a(n):=sum(sum(binomial(n-2*q-2,n-r-q)*binomial(q+1,r-1)*binomial(q+1,r) ,r,0,q+1)/(q+1), q,0,n); /* Vladimir Kruchinin, Jan 22 2019 */
a(n):=sum((binomial(2*m,m)*binomial(n-2*m+1,n-3*m))/(n-2*m+1),m,0,n/3);
/*Vladimir Kruchinin, Jan 27 2022 */
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(1-x)^(m-k) + x*O(x^n)))),n)}
  • PARI
    {a(n)=polcoeff(2/(1-x+sqrt((1-x)^2-4*x^3*(1-x) +x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))
  • PARI
    a(n) = sum(k=0, n\3, binomial(n-2*k, k)*binomial(2*k, k)/(k+1)); \\ Seiichi Manyama, Jan 22 2023

Formula

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k) ).
G.f.: ((1-x) - sqrt( (1-x)^2 - 4*x^3*(1-x) )) / (2*x^3*(1-x)).
a(n) ~ 2^(n+2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 16 2013
a(n) = Sum_{q=0..n} 1/(q+1)*Sum_{r=0..q+1} C(n-2*q-2,n-r-q)*C(q+1,r-1)*C(q+1,r). - Vladimir Kruchinin, Jan 22 2019
a(n) = 1 + Sum_{k=0..n-3} a(k) * a(n-k-3). - Ilya Gutkovskiy, Jan 28 2021
a(n) = Sum_{m=0..n/3} C(2*m,m)*C(n-2*m+1,n-3*m)/(n-2*m+1). - Vladimir Kruchinin, Jan 27 2022

A216447 G.f. satisfies: A(x) = (1 + x*(1-x)*A(x)^2) * (1 + x^2*A(x)^2).

Original entry on oeis.org

1, 1, 2, 6, 19, 64, 228, 841, 3181, 12277, 48156, 191400, 769168, 3120044, 12758080, 52533265, 217637308, 906511243, 3793989118, 15947205096, 67290484581, 284934164506, 1210374907352, 5156562941596, 22027291990432, 94325712634264, 404842107811880
Offset: 0

Views

Author

Paul D. Hanna, Sep 10 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 228*x^6 + 841*x^7 +...
The logarithm of the g.f. begins:
log(A(x)) = ((1-x) + x)*x*A(x) + ((1-x)^2 + 2^2*x*(1-x) + x^2)*x^2*A(x)^2/2 +
((1-x)^3 + 3^2*x*(1-x)^2 + 3^2*x^2*(1-x) + x^3)*x^3*A(x)^3/3 +
((1-x)^4 + 4^2*x*(1-x)^3 + 6^2*x^2*(1-x)^2 + 4^2*x^3*(1-x) + x^4)*x^4*A(x)^4/4 +
((1-x)^5 + 5^2*x*(1-x)^4 + 10^2*x^2*(1-x)^3 + 10^2*x^3*(1-x)^2 + 5^2*x^4*(1-x) + x^5)*x^5*A(x)^5/5 +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 51*x^4/4 + 206*x^5/5 + 861*x^6/6 + 3641*x^7/7 + 15523*x^8/8 + 66676*x^9/9 +...+ L(n)*x^n/n +...
where L(n) = [x^n] ((1-x^4)/(1-x-x^3))^n.

Crossrefs

Cf. A216604, A216616, A216617.

Programs

  • Mathematica
    nmax=20;aa=ConstantArray[0,nmax]; aa[[1]]=1;Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1+x*(1-x)*AGF^2)*(1+x^2*AGF^2)-AGF,x,j]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}];Flatten[{1,aa}] (* Vaclav Kotesovec, Aug 18 2013 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=(1 + x*(1-x)*A^2)*(1+x^2*A^2) +x*O(x^n));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n+1,x^m*A^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(1-x)^(m-k) +x*O(x^n)))));polcoeff(A,n)}
  • PARI
    {a(n)=polcoeff(((1-x^4)/(1-x-x^3 +x*O(x^n)))^(n+1)/(n+1),n)}
  • PARI
    {a(n)=polcoeff((1/x)*serreverse(x*(1-x-x^3)/(1-x^4 +x^2*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k) ).
(2) A(x) = exp( Sum_{n>=1} L(n)*x^n/n ) where L(n) = [x^n] ((1-x^4)/(1-x-x^3))^n.
(3) A(x) = (1/x) * Series_Reversion( x*(1-x-x^3)/(1-x^4) ).
(4) A(x) = (1 - x^4*A(x)^4) / (1 - x*A(x) - x^3*A(x)^3).
(5) A( x*(1-x-x^3)/(1-x^4) ) = (1-x^4)/(1-x-x^3).
a(n) = [x^n] ((1-x^4)/(1-x-x^3))^(n+1) / (n+1).
Recurrence: 31*(n-1)*n*(n+1)*(877*n^4 - 9157*n^3 + 32799*n^2 - 48009*n + 24592)*a(n) = 2*(n-1)*n*(95593*n^5 - 1032316*n^4 + 3953542*n^3 - 6623680*n^2 + 4768887*n - 1162262)*a(n-1) - (n-1)*(402543*n^6 - 4844150*n^5 + 22130482*n^4 - 48931238*n^3 + 54934501*n^2 - 29672070*n + 6219456)*a(n-2) + 4*(143828*n^7 - 2089338*n^6 + 12282478*n^5 - 37816765*n^4 + 65573867*n^3 - 63612965*n^2 + 31782331*n - 6379188)*a(n-3) - 16*(n-3)*(49112*n^6 - 639080*n^5 + 3201244*n^4 - 7806610*n^3 + 9609821*n^2 - 5589051*n + 1241722)*a(n-4) + 64*(n-4)*(n-3)*(10524*n^5 - 108130*n^4 + 387266*n^3 - 581563*n^2 + 354497*n - 75516)*a(n-5) - 256*(n-5)*(n-4)*(n-3)*(877*n^4 - 5649*n^3 + 10590*n^2 - 6374*n + 1102)*a(n-6). - Vaclav Kotesovec, Aug 18 2013
a(n) ~ c*d^n/n^(3/2), where d=4.54477579... is the root of the equation -256 + 512*d - 384*d^2 + 272*d^3 - 187*d^4 + 31*d^5 = 0 and c = 0.448853665050529472948816... - Vaclav Kotesovec, Aug 18 2013

A216616 G.f. satisfies: A(x) = (1 + x*(1-x)*A(x)) * (1 + x^2*A(x)^2).

Original entry on oeis.org

1, 1, 1, 3, 7, 15, 39, 103, 267, 719, 1975, 5447, 15199, 42863, 121647, 347455, 998559, 2884143, 8367599, 24377503, 71282351, 209132511, 615447711, 1816255583, 5373748287, 15937008575, 47368376255, 141075930495, 420957812863, 1258317356799, 3767538459391
Offset: 0

Views

Author

Paul D. Hanna, Sep 10 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 15*x^5 + 39*x^6 + 103*x^7 +...
The logarithm of the g.f. begins:
log(A(x)) = ((1-x) + x*A(x))*x + ((1-x)^2 + 2^2*x*(1-x)*A(x) + x^2*A(x)^2)*x^2/2 +
((1-x)^3 + 3^2*x*(1-x)^2*A(x) + 3^2*x^2*(1-x)*A(x)^2 + x^3*A(x)^3)*x^3/3 +
((1-x)^4 + 4^2*x*(1-x)^3*A(x) + 6^2*x^2*(1-x)^2*A(x)^2 + 4^2*x^3*(1-x)*A(x)^3 + x^4*A(x)^4)*x^4/4 +
((1-x)^5 + 5^2*x*(1-x)^4*A(x) + 10^2*x^2*(1-x)^3*A(x)^2 + 10^2*x^3*(1-x)^2*A(x)^3 + 5^2*x^4*(1-x)*A(x)^4 + x^5*A(x)^5)*x^5/5 +...
Explicitly,
log(A(x)) = x + x^2/2 + 7*x^3/3 + 17*x^4/4 + 41*x^5/5 + 133*x^6/6 + 393*x^7/7 + 1121*x^8/8 + 3373*x^9/9 + 10161*x^10/10 +...

Crossrefs

Cf. A216617, A216604, A216447.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=(1 + x*(1-x)*A)*(1+x^2*A^2) +x*O(x^n));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(1-x)^(m-k)*A^k +x*O(x^n)))));polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k) * A(x)^k ).
Recurrence: 2*(n+1)*(2*n+3)*(1367*n^5 - 22152*n^4 + 133091*n^3 - 342822*n^2 + 320852*n - 46056)*a(n) = (25973*n^7 - 386713*n^6 + 1995969*n^5 - 3489635*n^4 - 1128062*n^3 + 4916268*n^2 + 111144*n - 1105344)*a(n-1) - (42377*n^7 - 677143*n^6 + 3973061*n^5 - 9668137*n^4 + 6180338*n^3 + 7421072*n^2 - 9286128*n + 2230560)*a(n-2) + 4*(20505*n^7 - 365088*n^6 + 2506402*n^5 - 7967959*n^4 + 10412513*n^3 + 122671*n^2 - 10870392*n + 6017148)*a(n-3) - 2*(76552*n^7 - 1463333*n^6 + 10992560*n^5 - 39638307*n^4 + 64155864*n^3 - 18295816*n^2 - 55893000*n + 38191680)*a(n-4) + 2*(76552*n^7 - 1553555*n^6 + 12502721*n^5 - 49103379*n^4 + 89371647*n^3 - 37863250*n^2 - 72758304*n + 53344368)*a(n-5) - 4*(20505*n^7 - 443007*n^6 + 3803465*n^5 - 16027031*n^4 + 31715194*n^3 - 16244518*n^2 - 25294728*n + 19078920)*a(n-6) + 4*(n-6)*(6835*n^6 - 116228*n^5 + 733337*n^4 - 1976556*n^3 + 1537404*n^2 + 1530848*n - 1351680)*a(n-7) - 4*(n-7)*(n-6)*(1367*n^5 - 15317*n^4 + 58153*n^3 - 62791*n^2 - 47292*n + 44280)*a(n-8). - Vaclav Kotesovec, Dec 21 2013
a(n) ~ c*d^n/n^(3/2), where d = 3.14415377058430689... is the root of the equation -4 + 16*d - 44*d^2 + 68*d^3 - 44*d^4 + 16*d^5 - 15*d^6 + 4*d^7 = 0 and c = 0.77951549908443860621183... - Vaclav Kotesovec, Dec 21 2013
Constant c = 1/(8*sqrt(Pi*r)), where r = 0.008185036943737927662526644... is the root of the equation -5468 + 436073*r + 8837888*r^2 + 2268581888*r^3 - 4115660800*r^4 + 2178940928000*r^5 + 3543348019200*r^6 + 1717986918400*r^7 = 0. - Vaclav Kotesovec, Jan 04 2014
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