A179270
G.f. satisfies: inverse of function A(x) + i*A(x)^2 equals the conjugate, A(x) - i*A(x)^2, where i=sqrt(-1).
Original entry on oeis.org
1, 0, -1, 0, 4, 0, -21, 0, 122, 0, -758, 0, 4958, 0, -33509, 0, 233810, 0, -1641150, 0, 12364368, 0, -71807506, 0, 1354944972, 0, 33794258600, 0, 2524565441138, 0, 186642439700891, 0, 16196862324254354, 0, 1602823227559245434, 0, 179707702260054046760, 0
Offset: 1
G.f.: A(x) = x - x^3 + 4*x^5 - 21*x^7 + 122*x^9 - 758*x^11 +...
A(x)^2 = x^2 - 2*x^4 + 9*x^6 - 50*x^8 + 302*x^10 - 1928*x^12 +...
A(x) + i*A(x)^2 = x - i*x^2 - x^3 + 2*i*x^4 + 4*x^5 - 9*i*x^6 - 21*x^7 - 50*i*x^8 + 122*x^9 +...
where Series_Reversion[A(x) + i*A(x)^2] = A(x) - i*A(x)^2.
The i-th iteration of A(x) + i*A(x)^2 is a real-valued series in x, and begins:
x - x^2 + x^3 - 2*x^5 + 3*x^6 + x^7 - 38*x^8/3 + 70*x^9/3 - 2*x^10 - 266*x^11/3 + 214*x^12 - 436*x^13/3 - 469*x^14 + 12649*x^15/9 +...
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T(n, m):=if n=m then 1 else 1/2*(m/n*binomial(2*n-m-1, n-1)*(%i)^(n+m)*(-1)^n-(sum(T(k, m)*sum(T(n, i)*binomial(k, i-k)*(-%i)^(i-k), i, k, n), k, m+1, n-1)+sum(T(n, i)*binomial(m, i-m)*(-%i)^(i-m), i, m+1, n)));
makelist(T(n, 1), n, 1, 10); /* Vladimir Kruchinin, Apr 30 2012 */
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{a(n)=local(A=x+sum(k=3,n-1,a(k)*x^k)+x*O(x^n));if(n==1,1,if(n%2==0,0,-polcoeff((subst(A,x,A-I*A^2)+I*subst(A,x,A-I*A^2+x*O(x^n))^2),n)/2))}
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/* Faster vectorized version: */
{ooo=100;A=[1];B=x;C=(1-sqrt(1-4*(x+x^2+x*O(x^ooo))))/2;A182399=[1];
for(n=1,ooo,A182399=concat(A182399,0);B=x*Ser(A182399);
A182399[n]=Vec((B+subst(C+x*O(x^n),x,serreverse(B)))/2)[n];
A=Vec(-I*subst(x*Ser(A182399),x,I*serreverse(x+I*x^2+x^2*O(x^n))));
print1(A[n],", "))}
A318008
G.f. A(x) satisfies: A( A( x - x^2 ) ) = x + x^2.
Original entry on oeis.org
1, 1, 1, 2, 4, 9, 21, 50, 122, 302, 758, 1928, 4958, 12849, 33509, 88122, 233810, 621022, 1641150, 4411180, 12364368, 33073210, 71807506, 206985492, 1354944972, 3153779248, -33794258600, -62697691948, 2524565441138, 5004344042337, -186642439700891, -368380986364150, 16196862324254354, 32039943659306982, -1602823227559245434
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 21*x^7 + 50*x^8 + 122*x^9 + 302*x^10 + 758*x^11 + 1928*x^12 + 4958*x^13 + 12849*x^14 + 33509*x^15 + ...
such that A(A(x - x^2)) = x + x^2.
RELATED SERIES.
(a) A(A(x)) = x + 2*x^2 + 4*x^3 + 10*x^4 + 28*x^5 + 84*x^6 + 264*x^7 + 858*x^8 + 2860*x^9 + 9724*x^10 + ... + 2*A000108(n-2)*x^n + ...
(b) The odd bisection B(x) = (A(x) - A(-x))/2 begins
B(x) = x + x^3 + 4*x^5 + 21*x^7 + 122*x^9 + 758*x^11 + 4958*x^13 + 33509*x^15 + 233810*x^17 + 1641150*x^19 + 12364368*x^21 + ... + A277292(n)*x^(2*n-1) + ...
such that B(x)^2 yields the even bisection (A(x) + A(-x))/2
B(x)^2 = x^2 + 2*x^4 + 9*x^6 + 50*x^8 + 302*x^10 + 1928*x^12 + 12849*x^14 + 88122*x^16 + 621022*x^18 + 4411180*x^20 + ... + a(2*n)*x^(2*n) + ...
thus A(x) = B(x) + B(x)^2.
(c) Also, the Catalan series equals
B( B(x) + B(x)^2 ) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + ... + A000108(n-2)*x^n + ...
(d) Note that A(x - x^2) equals the series reversion of B(x):
A(x - x^2) = x - x^3 - x^5 - x^7 + 3*x^9 + 11*x^11 - 9*x^13 + 71*x^15 - 1685*x^17 + 31683*x^19 - 845729*x^21 + 28968319*x^23 + ...
where B( A(x -x^2) ) = x and A( B(x) - B(x)^2 ) = x.
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/* Using A(A( x - x^2 )) = x + x^2. */
{a(n) = my(A=x+x*O(x^n)); for(i=1,n, A = A + (x+x^2 - subst(A,x,subst(A,x,x-x^2)) )/2 ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))
A277293
G.f. A(x) satisfies: Series_Reversion( A(x) + 2*A(x)^3 ) = A(x) - 2*A(x)^3.
Original entry on oeis.org
1, 6, 230, 11676, 663174, 41413812, 2680851420, 188202575736, 11594488853190, 1403241101718852, -168694608332762892, 147980852726994346248, -105389748185249598629220, 93603955226272690108355208, -97621599698823169573607867400, 118585871916859034002263802974192, -165925215611130932978517186291637434, 264930310299987157013624532678164003556, -478773962141712793295304310624498264046460
Offset: 1
G.f.: A(x) = x + 6*x^5 + 230*x^9 + 11676*x^13 + 663174*x^17 + 41413812*x^21 + 2680851420*x^25 + 188202575736*x^29 + 11594488853190*x^33 +...
such that Series_Reversion( A(x) + 2*A(x)^3 ) = A(x) - 2*A(x)^3, where
A(x)^3 = x^3 + 18*x^7 + 798*x^11 + 43524*x^15 + 2593398*x^19 + 166441788*x^23 + 11038085612*x^27 + 778999431816*x^31 + 50589526452390*x^35 +...
A(x) + 2*A(x)^3 = x + 2*x^3 + 6*x^5 + 36*x^7 + 230*x^9 + 1596*x^11 + 11676*x^13 + 87048*x^15 + 663174*x^17 + 5186796*x^19 + 41413812*x^21 + 332883576*x^23 + 2680851420*x^25 + 22076171224*x^27 + 188202575736*x^29 + 1557998863632*x^31 + 11594488853190*x^33 + 101179052904780*x^35 +...
Also,
A( A(x) + 2*A(x)^3 ) = x + 2*x^3 + 12*x^5 + 96*x^7 + 880*x^9 + 8736*x^11 + 91392*x^13 +...+ 2^(n-1)*binomial(3*n-2,n-1)/(3*n-2)*x^(2*n-1) +...
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{a(n) = my(Oxn=x*O(x^(4*n)), A = x +Oxn); for(i=1, 4*n, A = A + (x - subst(A + 2*A^3, x, A - 2*A^3 ))/2); polcoeff(A, 4*n-3)}
for(n=1,25,print1(a(n),", "))
A277294
G.f. A(x) satisfies: Series_Reversion( A(x) + A(x)^4 ) = A(x) - A(x)^4.
Original entry on oeis.org
1, 2, 35, 812, 21359, 623244, 18568947, 638475040, 13249877870, 2024051330358, -355660668390645, 130426094235366208, -54120354853298252400, 27045033537893084984896, -15918675371944450999486319, 10905983125914263654567255488, -8603776324190250513027830925715, 7743542274281960968631431340349870, -7886327135586560316787947739703112447, 9023297352140462809043434127286176617288, -11524615288427474577090651960651636283169590
Offset: 1
G.f.: A(x) = x + 2*x^7 + 35*x^13 + 812*x^19 + 21359*x^25 + 623244*x^31 + 18568947*x^37 + 638475040*x^43 + 13249877870*x^49 + 2024051330358*x^55 +...
such that Series_Reversion( A(x) + A(x)^4 ) = A(x) - A(x)^4, where
A(x)^4 = x^4 + 8*x^10 + 164*x^16 + 4120*x^22 + 113970*x^28 + 3416128*x^34 + 104776764*x^40 + 3565389600*x^46 + 88390775151*x^52 +...
A(x) + A(x)^4 = x + x^4 + 2*x^7 + 8*x^10 + 35*x^13 + 164*x^16 + 812*x^19 + 4120*x^22 + 21359*x^25 + 113970*x^28 + 623244*x^31 + 3416128*x^34 + 18568947*x^37 + 104776764*x^40 + 638475040*x^43 + 3565389600*x^46 + 13249877870*x^49 + 88390775151*x^52 + 2024051330358*x^55 +...
Also,
A( A(x) + A(x)^4 ) = x + x^4 + 4*x^7 + 22*x^10 + 140*x^13 + 969*x^16 + 7084*x^19 + 53820*x^22 +...+ binomial(4*n-3,n-1)/(4*n-3)*x^(3*n-2) +...
which is a g.f. of A002293.
Note that the following is an integer series:
sqrt(A(x)/x) = 1 + x^6 + 17*x^12 + 389*x^18 + 10146*x^24 + 294863*x^30 + 8741468*x^36 + 301536587*x^42 + 6008625027*x^48 + 994498807123*x^54 - 179176440388960*x^60 + 65367342797524884*x^66 - 27123073712119583646*x^72 +...
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{a(n) = my(Oxn=x*O(x^(6*n)), A = x +Oxn); for(i=1, 6*n, A = A + (x - subst(A + A^4, x, A - A^4 ))/2); polcoeff(A, 6*n-5)}
for(n=1,25,print1(a(n),", "))
Showing 1-4 of 4 results.
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