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A303559 G.f. A(x) satisfies: 0 = [x^(n-1)] (x*A(x))' / (1 + x*A(x)^(n+1))^n for n>1. 2
1, 1, 4, 27, 236, 2405, 27322, 341359, 4714996, 72864319, 1265303986, 24429044332, 514629734620, 11658131151120, 282702829140564, 7356326626391035, 205875441726381570, 6178380749014085048, 197325821666554796930, 6653680184371365887508, 235653641615055767872038, 8749040544777960685341423 (list; graph; refs; listen; history; text; internal format)
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 with n.
Related identity: 0 = [x^(n-1)] (x*F(x))' / F(x)^n for n>1 holds when F(0) = 1.
LINKS
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 27*x^3 + 236*x^4 + 2405*x^5 + 27322*x^6 + 341359*x^7 + 4714996*x^8 + 72864319*x^9 + 1265303986*x^10 + ...
such that 0 = [x^(n-1)] (A(x) + x*A'(x)) / (1 + x*A(x)^(n+1))^n for n>1.
RELATED SERIES.
(x*A(x))' = 1 + 2*x + 12*x^2 + 108*x^3 + 1180*x^4 + 14430*x^5 + 191254*x^6 + 2730872*x^7 + 42434964*x^8 + ...
A'(x)/A(x) = 1 + 7*x + 70*x^2 + 819*x^3 + 10501*x^4 + 144208*x^5 + 2120511*x^6 + 33844851*x^7 + 596093380*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (x*A(x))'/(1 + x*A(x)^(n+1))^n begins:
n=1: [1, 1, 9, 88, 1003, 12566, 169292, 2449518, 38542987, ...];
n=2: [1, 0, 5, 62, 770, 10114, 140612, 2085848, 33572317, ...];
n=3: [1, -1, 0, 32, 503, 7272, 107140, 1660276, 27755613, ...];
n=4: [1, -2, -6, 0, 233, 4344, 71912, 1205724, 21476038, ...];
n=5: [1, -3, -13, -32, 0, 1744, 38977, 765450, 15238101, ...];
n=6: [1, -4, -21, -62, -147, 0, 13205, 388412, 9601263, ...];
n=7: [1, -5, -30, -88, -150, -242, 0, 121754, 5068535, ...];
n=8: [1, -6, -40, -108, 58, 1786, 4918, 0, 1922345, ...];
n=9: [1, -7, -51, -120, 553, 6978, 33190, 30542, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating: 0 = [x^(n-1)] (x*A(x))' / (1 + x*A(x)^(n+1))^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, 7, 70, 819, 10501, 144208, 2120511, 33844851, ...];
n=2: [1, 0, 3, 40, 531, 7324, 105650, 1613214, 26592811, ...];
n=3: [1, -1, 0, 17, 305, 4782, 74272, 1193530, 20486577, ...];
n=4: [1, -2, -2, 0, 131, 2772, 48936, 848064, 15359375, ...];
n=5: [1, -3, -3, -12, 0, 1204, 28657, 565245, 11066901, ...];
n=6: [1, -4, -3, -20, -96, 0, 12587, 335114, 7484601, ...];
n=7: [1, -5, -2, -25, -164, -907, 0, 149131, 4505231, ...];
n=8: [1, -6, 0, -28, -210, -1574, -9722, 0, 2036675, ...];
n=9: [1, -7, 3, -30, -239, -2049, -17102, -118489, 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+1))^m/m )[m]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A303558.
Sequence in context: A328978 A185655 A181146 * A161120 A183430 A212559
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 27 2018
STATUS
approved

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Last modified May 23 06:14 EDT 2024. Contains 372760 sequences. (Running on oeis4.)