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A185072
G.f.: A(x) satisfies: [x^n] A(x)^(n^2-2*n+2) = 0 for n>=2.
7
1, 1, -2, 6, -28, 70, -1446, -22302, -855032, -33435486, -1541257250, -80299386706, -4675923739764, -300809006015466, -21184444811987030, -1620869900459370150, -133878027649528854000, -11872222666784936265342, -1125045987661214982721602, -113458738692543731877937418
OFFSET
0,3
COMMENTS
It appears that (n+1) divides [x^n] A(x)^2 for n>=0 (A229128).
LINKS
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 ).
EXAMPLE
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.
PROG
(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), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jan 22 2012
EXTENSIONS
Name changed and entry revised by Paul D. Hanna, Oct 11 2013
STATUS
approved