A107699
Bisection (odd-indexed terms) of A107700.
Original entry on oeis.org
1, 1, -2, 13, -145, 2328, -49784, 1358965, -46076544, 1902202515, -94104681660, 5503867176832, -376096374571125, 29714871818774044, -2689473418781240320, 276562260699626541509, -32073434441440654231749, 4167256779254698145624064, -603002974281204853457239860
Offset: 1
-
a[n_] := Module[{A, B, F}, A = Series[x+x^2+x^(2*n), {x, 0, 2*n}]; If[n<1, 0, For[i=0, i <= 2*n, i++, F = x+2*A^2; B = InverseSeries[A]; A = (A+(B /. x -> F))/2]; SeriesCoefficient[A, {x, 0, 2*n-1}]]]; Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Feb 04 2014, after PARI *)
-
{a(n)=local(A,B,F);A=x+x^2+x*O(x^(2*n));if(n<1,0, for(i=0,2*n,F=x+2*A^2;B=serreverse(A);A=(A+subst(B,x,F))/2); polcoeff(A,2*n-1,x))}
A107701
Self-COMPOSE of A107700; thus g.f. A(x) = G(G(x)) = x + 2*G(x)^2, where G(x) is the g.f. of A107700.
Original entry on oeis.org
1, 2, 4, 6, 4, -6, -8, 44, 52, -520, -580, 8628, 9312, -188326, -199136, 5210560, 5435860, -178308938, -184306176, 7409693156, 7608810060, -368330924160, -376418726640, 21620184591972, 22015468707328, -1481473941622200, -1504385498284500, 117304374434860032
Offset: 1
-
{a(n)=local(A,B,F);A=x+x^2+x*O(x^n);if(n<1,0, for(i=0,n,F=x+2*A^2;B=serreverse(A);A=(A+subst(B,x,F))/2); polcoeff(x+2*A^2,n,x))}
A213422
G.f. satisfies: A( A(x) - 4*A(x)^2 ) = x.
Original entry on oeis.org
0, 1, 2, 12, 96, 880, 8720, 90752, 975936, 10737152, 120093056, 1360051456, 15556087296, 179424700416, 2084953411584, 24393551634432, 287204585508864, 3400978267127808, 40480500900446208, 484006813958356992, 5810240353159839744, 70001749695581061120
Offset: 0
G.f.: A(x) = x + 2*x^2 + 12*x^3 + 96*x^4 + 880*x^5 + 8720*x^6 + 90752*x^7 +...
where
A(A(x)) = x + 4*x^2 + 32*x^3 + 320*x^4 + 3584*x^5 + 43008*x^6 + 540672*x^7 +...+ A000108(n-1)*4^(n-1)*x^n +...
The series reversion of the g.f. A(x) begins:
A(x) - 4*A(x)^2 = x - 2*x^2 - 4*x^3 - 16*x^4 - 80*x^5 - 432*x^6 - 2304*x^7 -...
-
max = 21; a[0] = 0; a[1] = 1; f[x_] := Sum[a[n]*x^n, {n, 0, max}]; se = Series[f[f[x]] - (1 - Sqrt[1 - 16*x])/8 , {x, 0, max}]; coes = CoefficientList[se, x]; sol = Solve[Thread[coes == 0]]; Table[a[n], {n, 1, max}] /. sol // First (* Jean-François Alcover, Feb 19 2013, from 1st formula *)
T[0, 1]=0; T[n_, n_]=1; T[n_, m_]:= T[n, m]= 1/2*(4^(n-m)* m/n * Binomial[2*n-m-1, n-1] - Sum[T[n, i]*T[i, m], {i, m+1, n-1}]);
a[n_] := T[n, 1];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jan 11 2018, after Dmitry Kruchinin *)
-
T(n,m):=if n=m then 1 else 1/2*(4^(n-m)*m/n*binomial(2*n-m-1,n-1) -sum(T(n,i) *T(i,m), i,m+1,n-1));
makelist(T(n,1),n,1,10); /* Dmitry Kruchinin, Dec 02 2012 */
-
{a(n)=local(A, B, F); A=x+2*x^2; if(n<1, 0, for(i=0, n, B=subst(A,x,A+x*O(x^n)); B=x+4*B^2; F=serreverse(A+x*O(x^n)); A=(A+subst(B, x, F))/2); polcoeff(A, n, x))}
for(n=1,31,print1(a(n),", "))
-
@CachedFunction
def T(n,k):
if (k<0 or k>n): return 0
elif (n==0): return 0
elif (k==n): return 1
else: return 2^(2*n-2*k-1)*(k/(2*n-k))*binomial(2*n-k, n) - (1/2)*sum( T(n, n-j-1)*T(n-j-1, k) for j in range(n-k-1) )
def A213422(n): return T(n,1)
[A213422(n) for n in range(31)] # G. C. Greubel, Mar 08 2023
A191557
G.f. satisfies: A(A(x))^2 = x^2 + 4*A(x)^3.
Original entry on oeis.org
1, 1, 1, -1, -5, 6, 57, -68, -996, 1151, 23487, -26316, -703858, 769268, 25912425, -27791388, -1146924362, 1212941187, 60112150656, -62911402588, -3686975047595, 3828485422340, 262043300715095, -270475215554448, -21394371719691000
Offset: 1
G.f.: A(x) = x + x^2 + x^3 - x^4 - 5*x^5 + 6*x^6 + 57*x^7 - 68*x^8 +...
Note that x^3 is the only odd power of x in A(x)^2:
A(x)^2 = x^2 + 2*x^3 + 3*x^4 - 11*x^6 + 117*x^8 - 2001*x^10 +...
Illustrate A(A(x))^2 = x^2 + 4*A(x)^3 by the expansions:
A(A(x))^2 = x^2 + 4*x^3 + 12*x^4 + 24*x^5 + 16*x^6 - 60*x^7 - 72*x^8 + 640*x^9 + 768*x^10 - 11160*x^11 - 12916*x^12 +...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 4*x^6 - 15*x^7 - 18*x^8 + 160*x^9 + 192*x^10 - 2790*x^11 - 3229*x^12 +...
G.f. of odd bisection B(x) = (A(x) - A(-x))/2 begins:
B(x) = x + x^3 - 5*x^5 + 57*x^7 - 996*x^9 + 23487*x^11 +...
where A(x) = B(x) + x^3/B(x).
-
{a(n)=local(A=x+x^2+x*O(x^n));for(i=1,n,A=A-(subst(A,x,A)-x*sqrt(1+4*A^3/x^2))/2);polcoeff(A,n)}
Showing 1-4 of 4 results.
Comments