A185072 G.f.: A(x) satisfies: [x^n] A(x)^(n^2-2*n+2) = 0 for n>=2.

1, 1, -2, 6, -28, 70, -1446, -22302, -855032, -33435486, -1541257250, -80299386706, -4675923739764, -300809006015466, -21184444811987030, -1620869900459370150, -133878027649528854000, -11872222666784936265342, -1125045987661214982721602, -113458738692543731877937418

Offset: 0

Author

Paul D. Hanna, Jan 22 2012

Keywords

• sign

Comments

It appears that (n+1) divides [x^n] A(x)^2 for n>=0 (A229128).

Examples

			G.f.: A(x) = 1 + x - 2*x^2 + 6*x^3 - 28*x^4 + 70*x^5 - 1446*x^6 -...
Coefficients of x^k in the powers A(x)^(n^2-2*n+2) of g.f. A(x) begin:
n=1: [1,  1,   -2,     6,    -28,     70,   -1446,   -22302, ...];
n=2: [1,  2,   -3,     8,    -40,     60,   -2604,   -48112, ...];
n=3: [1,  5,    0,     0,    -35,   -189,   -5760,  -140700, ...];
n=4: [1, 10,   25,     0,    -70,   -728,  -13410,  -339000, ...];
n=5: [1, 17,  102,   238,      0,  -2142,  -32198,  -743886, ...];
n=6: [1, 26,  273,  1456,   3822,      0,  -80366, -1638312, ...];
n=7: [1, 37,  592,  5328,  29045,  89947,       0, -3630588, ...];
n=8: [1, 50, 1125, 15000, 130900, 769860, 2823600,        0, ...]; ...
where the coefficients of x^n in A(x)^(n^2-2*n+2) all equal zero for n>1.
RELATED FUNCTIONS.
The coefficients in G_n(x) that satisfy G_n(x) = A(x*G_n(x)^n) begin:
G_1: [1, 1,-1,  1,  -7,   -49, -1191, -31569,-1051695, -41520593, ...];
G_2: [1, 1, 0, -1,  -6,   -78, -1544, -40605,-1328178, -51857806, ...];
G_3: [1, 1, 1,  0,  -9,  -117, -2118, -53232,-1699905, -65386779, ...];
G_4: [1, 1, 2,  4,   0,  -141, -2958, -71900,-2216860, -83454920, ...];
G_5: [1, 1, 3, 11,  37,     0, -3245, -95286,-2941059,-108180433, ...];
G_6: [1, 1, 4, 21, 118,   581,     0, -99086,-3760182,-141280086, ...];
G_7: [1, 1, 5, 34, 259,  2002, 13212,      0,-3775221,-176047295, ...];
G_8: [1, 1, 6, 50, 476,  4788, 47578, 397090,       0,-172383145, ...];
G_9: [1, 1, 7, 69, 785,  9589,120333,1468749,14889577,         0, ...];
G_10:[1, 1, 8, 91,1202, 17180,256056,3859425,56018694, 669865615, 0, ...]; ...
Note how that the coefficients of x^n in G_n(x) are zero for n>1.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A185082, A229041, A229044, A229128, A230218.

Programs

PARI

{a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0);A[#A]=-Vec((1/x*serreverse(x/Ser(A)^(#A-1)))^(1/(#A-1)))[#A]);A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = F(x/A(x)) where F(x) = A(x*F(x)) is the g.f. of A229041.

(2) A(x) = G(x/A(x)^2) where G(x) = A(x*G(x)^2) is the g.f. of A229044.

(3) A(x) = H(x*A(x)) where H(x) = A(x/H(x)) is the g.f. of A230218.

(4) [x^n] G_n(x) = 0 for n>1 where G_n(x) = A( x*G_n(x)^n ) and A(x) = G_n( x/A(x)^n ).

Extensions

Name changed and entry revised by Paul D. Hanna, Oct 11 2013