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.

A209199 G.f. satisfies: A(x) = 1 + x*A(x)*A(-x) + x^2*A(x)/A(-x).

Original entry on oeis.org

1, 1, 1, 3, 2, 5, 10, 21, 30, 76, 114, 257, 448, 1052, 1706, 4093, 6928, 16284, 28266, 67580, 116288, 278582, 488152, 1168105, 2060388, 4959066, 8772450, 21133812, 37675236, 90901086, 162659624, 393382077, 706479172, 1710430178, 3084264618, 7477512244, 13522121028
Offset: 0

Views

Author

Paul D. Hanna, Mar 08 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 10*x^6 + 21*x^7 +...
Related series:
A(x)*A(-x) = 1 + x^2 - x^4 + 5*x^6 + 12*x^8 + 25*x^10 + 164*x^12 +...
A(x)/A(-x) = 1 + 2*x + 2*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 30*x^6 +...

Links

Crossrefs

Cf. A208889, A208887.

Programs

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

Formula

G.f. satisfies: 1 - x^4 - y + x^2*y + x^3*y^2 - (x*y^2)/(x^2 - y - x*y^2) = 0, where y = A(x). - Vaclav Kotesovec, Mar 13 2014
a(n) ~ (C(r,s1) - (-1)^n*C(-r,s2)) / (sqrt(Pi) * n^(3/2) * r^n), where {r1 = r = 0.45889975689289..., s1 = 3.7914195980097...} and {r2 = -r, s2 = 0.3725313335801...} are roots of the system of equations r^2*(1 + 2*r*s) = 1 + (2*r*s)/(r^2 - s - r*s^2) + (r*s^2*(1 + 2*r*s))/(-r^2 + s + r*s^2)^2, 1 + r^2*s + r^3*s^2 = r^4 + s + (r*s^2)/(r^2 - s - r*s^2), and C(r,s) = sqrt((r*s^2 - r^2 + s)*(4*r^7 - 11*r^6*s^2 - s^3 - 2*r*s^3 - 3*r^4*s^3*(s^3-6) + 10*r^5*s*(s^3-1) - 8*r^3*s^2*(s^3-1) - r^2*s^2*(7*s^2+1)) / (4*r*(r^7 - 3*r^6*s^2 + s^3 - r*s^3 - r^4*s^3*(s^3-6) + 3*r^5*s*(s^3-1) - 3*r^2*s^2*(s^2+1) + r^3*(3*s^2 - 3*s^5 - 1)))), C(r,s1) = 4.083478805997458527..., C(-r,s2) = 0.26836221180354127... - Vaclav Kotesovec, Mar 13 2014

A233895 G.f. satisfies: A(x) = 1 + x*A(x)*A(-x) + x^2*(A(x)^2 + A(-x)^2).

Original entry on oeis.org

1, 1, 2, 3, 10, 18, 60, 115, 410, 822, 2996, 6174, 22980, 48324, 182328, 389187, 1484410, 3205710, 12329988, 26876586, 104080812, 228606012, 890262984, 1967830254, 7699472676, 17110322908, 67215426440, 150058534620, 591517612616, 1325828841480, 5241992235888
Offset: 0

Views

Author

Paul D. Hanna, Dec 17 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 10*x^4 + 18*x^5 + 60*x^6 + 115*x^7 +...
Related series:
A(x)^2 = 1 + 2*x + 5*x^2 + 10*x^3 + 30*x^4 + 68*x^5 + 205*x^6 + 482*x^7 +...
A(x)*A(-x) = 1 + 3*x^2 + 18*x^4 + 115*x^6 + 822*x^8 + 6174*x^10 +...
A(x)^2+A(-x)^2 = 2 + 10*x^2 + 60*x^4 + 410*x^6 + 2996*x^8 + 22980*x^10 +...

Links

Crossrefs

Cf. A208887, A143926, A233896.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x*A*subst(A, x, -x)+x^2*(A^2+subst(A^2, x, -x+x*O(x^n))) ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

Formula

Recurrence: n*(n+1)*(n+2)*(81*n^6 - 756*n^5 + 2565*n^4 - 3630*n^3 + 1150*n^2 + 1998*n - 1648)*a(n) = - 6*n*(n+1)*(81*n^5 - 1026*n^4 + 4851*n^3 - 10458*n^2 + 10280*n - 3872)*a(n-1) + 12*n*(486*n^8 - 4536*n^7 + 14634*n^6 - 14085*n^5 - 23214*n^4 + 69355*n^3 - 66518*n^2 + 29510*n - 6544)*a(n-2) - 72*(81*n^7 - 297*n^6 - 2124*n^5 + 13899*n^4 - 28141*n^3 + 20794*n^2 + 540*n - 4800)*a(n-3) - 144*(n-3)*(243*n^8 - 2268*n^7 + 7182*n^6 - 5976*n^5 - 13488*n^4 + 33301*n^3 - 30472*n^2 + 20528*n - 10400)*a(n-4) - 864*(n-4)*(n-3)*(n-2)*(108*n^3 - 441*n^2 + 75*n + 200)*a(n-5) - 1728*(n-5)*(n-4)*(n-3)*(81*n^6 - 270*n^5 + 690*n^3 - 695*n^2 + 374*n - 240)*a(n-6). - Vaclav Kotesovec, Dec 21 2013
a(n) ~ c*d^n/n^(3/2), where d = sqrt(24 - 3*I*2^(2/3)*3^(5/6)*(3 + I*sqrt(3))^(1/3) + 6*I*2^(1/3)*3^(1/6)*(3 + I*sqrt(3))^(2/3) - 3*2^(2/3)*(9 + 3*I*sqrt(3))^(1/3)) = 3.12769717670219... is the root of the equation 1728 + 432*d^2 - 72*d^4 + d^6 = 0 and c = sqrt((34 - 4*sqrt(247) * sin(arccsc(494 * sqrt(247)/7687)/3)) / Pi) = 1.281119572461999772722... if n is even, and c = 2*sqrt(6 - sqrt(129) * sin(arcsin(323*sqrt(3/43)/86)/3)) / sqrt(Pi) = 0.970593260725094233562... if n is odd. - Vaclav Kotesovec, Dec 21 2013, updated Mar 18 2024

A233896 G.f. satisfies: A(x) = 1 + x*A(x)^2*A(-x)^2 + x^2*(A(x)^2 + A(-x)^2).

Original entry on oeis.org

1, 1, 2, 6, 10, 33, 72, 236, 572, 1964, 4800, 16910, 42354, 150670, 386992, 1390176, 3622696, 13128940, 34580568, 126131776, 335409928, 1229708169, 3295834080, 12137093684, 32738710652, 121016095812, 328220839472, 1217132137132, 3316783066620, 12333770952588
Offset: 0

Views

Author

Paul D. Hanna, Dec 17 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 10*x^4 + 33*x^5 + 72*x^6 + 236*x^7 +...
Related series:
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 36*x^4 + 110*x^5 + 286*x^6 + 868*x^7 +...
A(x)*A(-x) = 1 + 3*x^2 + 12*x^4 + 82*x^6 + 664*x^8 + 5479*x^10 + 47568*x^12 +...
A(x)^2*A(-x)^2 = 1 + 6*x^2 + 33*x^4 + 236*x^6 + 1964*x^8 + 16910*x^10 +...
A(x)^2+A(-x)^2 = 2 + 10*x^2 + 72*x^4 + 572*x^6 + 4800*x^8 + 42354*x^10 +...

Links

Crossrefs

Cf. A208887, A143926, A233895.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^2*subst(A^2, x, -x)+x^2*(A^2+subst(A^2, x, -x+x*O(x^n))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A208888 G.f. satisfies: A(x) = 1 + x*(A(x) + A(-x)) + x^2*A(x)*A(-x).

Original entry on oeis.org

1, 2, 1, 2, -2, -4, -11, -22, -14, -28, 58, 116, 316, 632, 397, 794, -2198, -4396, -11954, -23908, -14684, -29368, 95170, 190340, 517492, 1034984, 623764, 1247528, -4462472, -8924944, -24270275, -48540550, -28820966, -57641932, 220608454, 441216908, 1200216340
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + x^2 + 2*x^3 - 2*x^4 - 4*x^5 - 11*x^6 - 22*x^7 +...
Related series:
A(x)+A(-x) = 2 + 2*x^2 - 4*x^4 - 22*x^6 - 28*x^8 + 116*x^10 + 632*x^12 +...
A(x)*A(-x) = 1 - 2*x^2 - 11*x^4 - 14*x^6 + 58*x^8 + 316*x^10 + 397*x^12 +...

Links

Crossrefs

Cf. A208887.

Programs

  • Maple
    A208888_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1] - add((-1)^j*a[j]*a[w-j-1], j=1..w-1) od;
convert(a, list); subsop(1=NULL,%); end: A208888_list(37); # Peter Luschny, Feb 29 2016
  • Mathematica
    CoefficientList[Series[(1-Sqrt[1-4*x^2+16*x^4])/(2*x^2*(1-2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 19 2013 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*(A+subst(A,x,-x+x*O(x^n)))+x^2*A*subst(A,x,-x+x*O(x^n)));polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: A(x) = (1 - sqrt(1 - 4*x^2 + 16*x^4)) / (2*x^2*(1-2*x)).
Recurrence: (n+2)*a(n) = 2*(n+2)*a(n-1) + 4*(n-1)*a(n-2) - 8*(n-1)*a(n-3) - 16*(n-4)*a(n-4) + 32*(n-4)*a(n-5). - Vaclav Kotesovec, Aug 19 2013
|a(n)| ~ c * 3^(1/4)*2^(n+2)/(sqrt(Pi)*n^(3/2)), where c=(sqrt(3)+1)/2 if n=6k+0 or n=6k+1, c=(sqrt(3)-1)/2 if n=6k+2 or n=6k+3 and c=1 if n=6k+4 or n=6k+5. - Vaclav Kotesovec, Aug 19 2013
Showing 1-4 of 4 results.

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