A360234 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+2))^(n+1) for n >= 0.
1, 1, 4, 33, 414, 6750, 131963, 2957899, 73968136, 2027178710, 60143834893, 1914750144642, 64984397381766, 2339387034919340, 88976089246855623, 3563952072597604091, 149941204887915187568, 6610797722288579969347, 304837386103152855175255, 14675559490665539299350303
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 4*x^2 + 33*x^3 + 414*x^4 + 6750*x^5 + 131963*x^6 + 2957899*x^7 + 73968136*x^8 + 2027178710*x^9 + 60143834893*x^10 + ... RELATED SERIES. G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins: B(x) = 1 + x + 5*x^2 + 46*x^3 + 603*x^4 + 10011*x^5 + 197357*x^6 + 4444483*x^7 + 111520277*x^8 + ... + b(n)*x^n + ... such that b(n) = [x^n] (1 + x*A(x)^(n+2))^(n+1) / (n+1), as well as b(n) = [x^n] A(x)^(n+1) / (n+1), so that b(n) begin: [1/1, 2/2, 15/3, 184/4, 3015/5, 60066/6, 1381499/7, 35555864/8, ...]. ILLUSTRATION OF DEFINITION. The table of coefficients of x^k in A(x)^(n+1) begins: n=0: [1, 1, 4, 33, 414, 6750, 131963, 2957899, ...]; n=1: [1, 2, 9, 74, 910, 14592, 281827, 6261048, ...]; n=2: [1, 3, 15, 124, 1500, 23673, 451690, 9944484, ...]; n=3: [1, 4, 22, 184, 2197, 34156, 643878, 14046740, ...]; n=4: [1, 5, 30, 255, 3015, 46221, 860965, 18610170, ...]; n=5: [1, 6, 39, 338, 3969, 60066, 1105794, 23681298, ...]; n=6: [1, 7, 49, 434, 5075, 75908, 1381499, 29311192, ...]; n=7: [1, 8, 60, 544, 6350, 93984, 1691528, 35555864, ...]; ... Compare to the table of coefficients in (1 + x*A(x)^(n+2))^(n+1): n=0: [1, 1, 2, 9, 74, 910, 14592, 281827, ...]; n=1: [1, 2, 7, 36, 287, 3338, 51315, 963446, ...]; n=2: [1, 3, 15, 91, 744, 8337, 122662, 2227101, ...]; n=3: [1, 4, 26, 184, 1591, 17600, 249194, 4361112, ...]; n=4: [1, 5, 40, 325, 3015, 33656, 463710, 7824385, ...]; n=5: [1, 6, 57, 524, 5244, 60066, 816474, 13339956, ...]; n=6: [1, 7, 77, 791, 8547, 101619, 1381499, 22023891, ...]; n=7: [1, 8, 100, 1136, 13234, 164528, 2263888, 35555864, ...]; ... to see that the main diagonals of the tables are the same.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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PARI
{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+2))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]} for(n=0, 20, print1(a(n), ", "))
Formula
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+2))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(n+2))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 3.146325060582260657459991059461810..., c = 0.007037477865521004701131626931596125... - Vaclav Kotesovec, Jan 31 2023