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%I #17 Oct 24 2024 09:26:26
%S 1,1,1,-1,-5,12,81,-293,-2361,11365,104562,-630172,-6493832,47143346,
%T 538611615,-4581889465,-57623005154,562546009728,7739224455922,
%U -85309456282000,-1276419913050610,15682410921426105,253801993058469530,-3439337745753797445,-59903911856917937325,887628418264985947932
%N G.f. A(x) satisfies A(x) = 1/A(-x*A(x)) such that [x^(2*n-1)] A(x)^n = 0 for n >= 2, with A(0) = A'(0) = 1.
%H Paul D. Hanna, <a href="/A377250/b377250.txt">Table of n, a(n) for n = 0..500</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F (1.a) A(x) = 1/A(-x*A(x)),
%F (1.b) A(x) = (-1/x) * Series_Reversion(-x*A(x)),
%F (1.c) A(x) = Sum_{n>=0} (-x)^n * [x^n] 1/A(x)^(n+1) / (n+1),
%F (1.d) A(x)^m = Sum_{n>=0} (-x)^n * [x^n] 1/A(x)^(n+m) * m/(n+m) for |m| > 0,
%F (1.e) A(x) = exp( Sum_{n>=1} (-x)^n * [x^n] 1/A(x)^n / n ).
%F (2.a) [x^(2*n-1)] A(x)^n = 0 for n >= 2,
%F (2.b) [x^(2*n)] 1/A(x)^n = 0 for n >= 1,
%F (2.c) [x^(2*n-1)] 1/A(x)^(3*n-1) = 0 for n >= 2.
%F A related power series B(x) may be defined by:
%F (3.a) A(x) = B( x/A(x) )
%F (3.b) A(x) = 1 / B( -x*A(x)^2 ),
%F (3.c) B(x) = A( x*B(x) ),
%F (3.d) B(x) = 1 / A( -x*B(x)^2 ),
%F (3.e) B(x) = (1/x) * Series_Reversion(x/A(x)),
%F (3.f) B(x) = ( (-1/x) * Series_Reversion(-x*A(x)^2) )^(1/2),
%F (3.g) B(x) = Sum_{n>=0} x^n * [x^n] A(x)^(n+1) / (n+1),
%F (3.h) B(x) = Sum_{n>=0} (-x)^n * [x^n] 1/A(x)^(2*n+1) / (2*n+1),
%F (3.i) B(x) = exp( Sum_{n>=1} x^n * [x^n] A(x)^n / n ),
%F (3.j) B(x) = exp( Sum_{n>=1} (-x)^n * [x^n] 1/A(x)^(2*n) / (2*n) ).
%F A related power series C(x) may be defined by:
%F (4.a) A(x) = C( x/A(x)^2 ),
%F (4.b) A(x) = 1 / C( -x*A(x)^3 ),
%F (4.c) C(x) = A( x*C(x)^2 ),
%F (4.d) C(x) = 1 / A( -x*C(x)^3 ),
%F (4.e) C(x) = ( (1/x) * Series_Reversion(x/A(x)^2) )^(1/2),
%F (4.f) C(x) = ( (-1/x) * Series_Reversion(-x*A(x)^3) )^(1/3),
%F (4.g) C(x) = Sum_{n>=0} x^n * [x^n] A(x)^(2*n+1) / (2*n+1),
%F (4.h) C(x) = Sum_{n>=0} (-x)^n * [x^n] 1/A(x)^(3*n+1) / (3*n+1),
%F (4.i) C(x) = exp( Sum_{n>=1} x^n * [x^n] A(x)^(2*n) / (2*n) ),
%F (4.j) C(x) = exp( Sum_{n>=1} (-x)^n * [x^n] 1/A(x)^(3*n) / (3*n) ).
%F For fixed integer k, there exists a power series F(x,k) that satisfies:
%F (5.a) A(x) = F( x/A(x)^k, k),
%F (5.b) A(x) = 1 / F( -x*A(x)^(k+1), k),
%F (5.c) F(x,k) = A( x*F(x,k)^k ),
%F (5.d) F(x,k) = 1 / A( -x*F(x,k)^(k+1) ),
%F (5.e) F(x,k) = ( (1/x) * Series_Reversion(x/A(x)^k) )^(1/k),
%F (5.f) F(x,k) = ( (-1/x) * Series_Reversion(-x*A(x)^(k+1)) )^(1/(k+1)),
%F (5.g) F(x,k) = Sum_{n>=0} x^n * [x^n] A(x)^(k*n+1) / (k*n+1),
%F (5.h) F(x,k) = Sum_{n>=0} (-x)^n * [x^n] 1/A(x)^((k+1)*n+1) / ((k+1)*n+1),
%F (5.i) F(x,k)^m = Sum_{n>=0} x^n * [x^n] A(x)^(k*n+m) * m/(k*n+m) for |m| > 0,
%F (5.j) F(x,k)^m = Sum_{n>=0} (-x)^n * [x^n] 1/A(x)^((k+1)*n+m) * m/((k+1)*n+m) for |m| > 0,
%F (5.k) F(x,k) = exp( Sum_{n>=1} x^n * [x^n] A(x)^(k*n) / (k*n) ),
%F (5.l) F(x,k) = exp( Sum_{n>=1} (-x)^n * [x^n] 1/A(x)^((k+1)*n) / ((k+1)*n) ).
%e G.f.: A(x) = 1 + x + x^2 - x^3 - 5*x^4 + 12*x^5 + 81*x^6 - 293*x^7 - 2361*x^8 + 11365*x^9 + 104562*x^10 - 630172*x^11 - 6493832*x^12 + ...
%e RELATED SERIES.
%e A related power series B(x) = A(x*B(x)) begins
%e B(x) = 1 + x + 2*x^2 + 3*x^3 - 7*x^5 + 52*x^6 + 247*x^7 - 1560*x^8 - 9715*x^9 + 73924*x^10 + 554683*x^11 + ...
%e where A(x) = B(x/A(x)) and A(x) = 1/B(-x*A(x)^2).
%e B(x)^2 = (-1/x)*Series_Reversion(-x*A(x)^2) = 1 + 2*x + 5*x^2 + 10*x^3 + 10*x^4 - 2*x^5 + 99*x^6 + 570*x^7 - 2460*x^8 + ...
%e where B(x) = 1/A(-x*B(x)^2).
%e A related power series C(x) = A(x*C(x)^2) begins
%e C(x) = 1 + x + 3*x^2 + 10*x^3 + 33*x^4 + 114*x^5 + 468*x^6 + 2145*x^7 + 8445*x^8 + 24618*x^9 + 111930*x^10 + ...
%e where A(x) = C(x/A(x)^2) and A(x) = 1/C(-x*A(x)^3).
%e C(x)^2 = (1/x)*Series_Reversion(x/A(x)^2) = 1 + 2*x + 7*x^2 + 26*x^3 + 95*x^4 + 354*x^5 + 1462*x^6 + 6570*x^7 + 27357*x^8 + ...
%e C(x)^3 = (-1/x)*Series_Reversion(-x*A(x)^3) = 1 + 3*x + 12*x^2 + 49*x^3 + 195*x^4 + 777*x^5 + 3288*x^6 + 14781*x^7 + 63963*x^8 + ...
%e where C(x) = 1/A(-x*C(x)^3).
%e RELATED TABLES.
%e The table of coefficients of x^k in A(x)^n begins
%e A^1: [1, 1, 1, -1, -5, 12, 81, -293, -2361, 11365, ...];
%e A^2: [1, 2, 3, 0, -11, 12, 177, -390, -5145, 17140, ...];
%e A^3: [1, 3, 6, 4, -15, 0, 268, -285, -8019, 17000, ...];
%e A^4: [1, 4, 10, 12, -13, -20, 336, 0, -10667, 11096, ...];
%e A^5: [1, 5, 15, 25, 0, -39, 370, 420, -12825, 0, ...];
%e A^6: [1, 6, 21, 44, 30, -42, 372, 918, -14307,-15390, 711480, 0, ...];
%e A^7: [1, 7, 28, 70, 84, -7, 364, 1443, -15015,-33971, 791210, 830060, -53403077, 0, ...]; ...
%e in which zeros are found at [x^(2*n-1)] A(x)^n for n >= 2.
%e The table of coefficients of x^k in 1/A(x)^n begins
%e 1/A^1: [1, -1, 0, 2, 2, -21, -48, 455, 1626, -16146, ...];
%e 1/A^2: [1, -2, 1, 4, 0, -46, -50, 1014, 2262, -35820, ...];
%e 1/A^3: [1, -3, 3, 5, -6, -69, 0, 1602, 1740, -57409, ...];
%e 1/A^4: [1, -4, 6, 4, -15, -84, 100, 2136, 0, -79060, ...];
%e 1/A^5: [1, -5, 10, 0, -25, -86, 240, 2535, -2900, -98825, 0, ...];
%e 1/A^6: [1, -6, 15, -8, -33, -72, 403, 2730, -6786,-114818, 126585, 6327630, 0, ...]; ...
%e in which zeros are found at [x^(2*n)] 1/A(x)^n for n >= 1.
%e Notice that the main diagonal of this table equals (-1)^n*(n+1)*a(n):
%e [1, -2, 3, 4, -25, -72, ...] = [1, -2*(1), 3*(1), -4*(-1), 5*(-5), -6*(12), ...];
%e that is, a(n) = (-1)^n * [x^n] 1/A(x)^(n+1)/(n+1) for n >= 0.
%o (PARI) \\ Using [x^(2*n-1)] A(x)^n = 0 and [x^(2*n)] 1/A(x)^n = 0
%o {a(n) = my(A=[1,1]); for(m=1,n, A=concat(A,0);
%o A[#A] = (1/(-(-1)^m*(m\2+1)))*polcoeff( Ser(A)^((-1)^m*(m\2+1)),m+1); ); A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A377251.
%K sign
%O 0,5
%A _Paul D. Hanna_, Oct 21 2024