OFFSET
0,3
COMMENTS
Compare to: 0 = [x^(n-1)] (x*F(x))' / (1 + x*F(x)^2)^n for n>1 holds when F(x) = 1 + x*F(x)^2 is a g.f. of the Catalan numbers (A000108).
Compare to: 0 = [x^(n-1)] (x*G(x))' / (1 + x*G(x)^k)^n for n>1 holds when G(x) = 1 + x*G(x)^k and k is fixed; this sequence explores the case where k varies directly with n.
Related identity: 0 = [x^(n-1)] (x*F(x))' / F(x)^n for n>1 holds when F(0) = 1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 93*x^4 + 620*x^5 + 4046*x^6 + 24077*x^7 + 138709*x^8 + 1448110*x^9 + 32196907*x^10 + ...
such that 0 = [x^(n-1)] (A(x) + x*A'(x)) / (1 + x*A(x)^n)^n for n>1.
RELATED SERIES.
(x*A(x))' = 1 + 2*x + 9*x^2 + 60*x^3 + 465*x^4 + 3720*x^5 + 28322*x^6 + 192616*x^7 + 1248381*x^8 + ...
A'(x)/A(x) = 1 + 5*x + 37*x^2 + 305*x^3 + 2516*x^4 + 19205*x^5 + 126624*x^6 + 792081*x^7 + 10741078*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (x*A(x))'/(1 + x*A(x)^n)^n begins:
n=1: [1, 1, 7, 49, 391, 3151, 23815, 158425, 1013803, ...];
n=2: [1, 0, 4, 34, 287, 2354, 17532, 110690, 681487, ...];
n=3: [1, -1, 0, 17, 169, 1431, 10215, 55203, 297810, ...];
n=4: [1, -2, -5, 0, 62, 554, 3111, 1872, -56195, 5924106, ...];
n=5: [1, -3, -11, -15, 0, -31, -2145, -36993, -281119, 4826695, ...];
n=6: [1, -4, -18, -26, 26, 0, -3742, -50288, -295155, 5468804, ...];
n=7: [1, -5, -26, -31, 192, 1053, 0, -34031, -94648, 7795307, ...];
n=8: [1, -6, -35, -28, 559, 3620, 10228, 0, 155958, 10425666, ...];
n=9: [1, -7, -45, -15, 1197, 8283, 27059, 12493, 0, 9907761, ...];
n=10: [1, -8, -56, 10, 2185, 15718, 48988, -78722, -1474424, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that: 0 = [x^(n-1)] (x*A(x))' / (1 + x*A(x)^n)^n for n>1.
RELATED TABLE.
The table of coefficients of x^k in (x*A(x))' / A(x)^n begins:
n=1: [1, 1, 5, 37, 305, 2516, 19205, 126624, 792081, 10741078, ...];
n=2: [1, 0, 2, 20, 186, 1620, 12495, 79302, 474495, 7998088, ...];
n=3: [1, -1, 0, 8, 100, 969, 7680, 46440, 261588, 5988094, ...];
n=4: [1, -2, -1, 0, 40, 510, 4337, 24608, 126090, 4504280, ...];
n=5: [1, -3, -1, -5, 0, 199, 2120, 11037, 46376, 3386810, ...];
n=6: [1, -4, 0, -8, -25, 0, 749, 3514, 5499, 2514128, ...];
n=7: [1, -5, 2, -10, -39, -116, 0, 290, -9640, 1795607, ...];
n=8: [1, -6, 5, -12, -45, -172, -304, 0, -9000, 1165374, ...];
n=9: [1, -7, 9, -15, -45, -186, -301, 1594, 0, 577155, ...];
n=10: [1, -8, 14, -20, -40, -172, -97, 4278, 11976, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating the identity: 0 = [x^(n-1)] (x*A(x))' / A(x)^n for n>1 holds since A(0) = 1.
PROG
(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A = concat(A, 0); m=#A; A[m] = -Vec( (Ser(A) + x*Ser(A)')/(1 + x*Ser(A)^m)^m/m )[m]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 26 2018
STATUS
approved