A360345 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) for n >= 0.
1, 1, 5, 62, 1214, 31269, 973485, 34993597, 1412846469, 62926155294, 3053566438307, 160005640085764, 8992869671470675, 539298198547460797, 34364052537634696986, 2318526571023659653665, 165143229278977841236029, 12385688813185721332861730, 975844100444710104444582984
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 5*x^2 + 62*x^3 + 1214*x^4 + 31269*x^5 + 973485*x^6 + 34993597*x^7 + 1412846469*x^8 + 62926155294*x^9 + ... RELATED SERIES. G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins: B(x) = 1 + x + 6*x^2 + 78*x^3 + 1543*x^4 + 39810*x^5 + 1239252*x^6 + 44537587*x^7 + 1798314384*x^8 + ... + b(n)*x^n + ... such that b(n) = [x^n] (1 + x*A(x)^(2*n+1))^(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, 18/3, 312/4, 7715/5, 238860/6, 8674764/7, 356300696/8, ...]. ILLUSTRATION OF DEFINITION. The table of coefficients of x^k in A(x)^(n+1) begins: n=0: [1, 1, 5, 62, 1214, 31269, 973485, 34993597, ...]; n=1: [1, 2, 11, 134, 2577, 65586, 2025492, 72397390, ...]; n=2: [1, 3, 18, 217, 4104, 103212, 3161648, 112357788, ...]; n=3: [1, 4, 26, 312, 5811, 144428, 4387978, 155030276, ...]; n=4: [1, 5, 35, 420, 7715, 189536, 5710930, 200579975, ...]; n=5: [1, 6, 45, 542, 9834, 238860, 7137401, 249182232, ...]; n=6: [1, 7, 56, 679, 12187, 292747, 8674764, 301023241, ...]; n=7: [1, 8, 68, 832, 14794, 351568, 10330896, 356300696, ...]; ... Compare to the table of coefficients in (1 + x*A(x)^(2*n+1))^(n+1): n=0: [1, 1, 1, 5, 62, 1214, 31269, 973485, ...]; n=1: [1, 2, 7, 42, 479, 8750, 216258, 6562156, ...]; n=2: [1, 3, 18, 136, 1560, 26895, 633608, 18631701, ...]; n=3: [1, 4, 34, 312, 3767, 62888, 1412530, 40031684, ...]; n=4: [1, 5, 55, 595, 7715, 128041, 2763270, 75234930, ...]; n=5: [1, 6, 81, 1010, 14172, 238860, 5016947, 131313798, ...]; n=6: [1, 7, 112, 1582, 24059, 418166, 8674764, 219340759, ...]; n=7: [1, 8, 148, 2336, 38450, 696216, 14466592, 356300696, ...]; ... 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)^(2*m+1))^(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)^(2*n+1))^(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)^(2*n+1))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 1.635402029299..., c = 0.0308525091280143... - Vaclav Kotesovec, Feb 06 2023