A360236 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+4))^(n+1) for n >= 0.
1, 1, 6, 66, 1028, 20138, 464863, 12162876, 351915528, 11075859686, 374858234365, 13530279602015, 517628371405448, 20890826296067329, 886175281852068632, 39393952245422498344, 1830781283537184304756, 88768944166701791039297, 4482797026386165709436753, 235417696462456105986818505
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 6*x^2 + 66*x^3 + 1028*x^4 + 20138*x^5 + 464863*x^6 + 12162876*x^7 + 351915528*x^8 + 11075859686*x^9 + ... RELATED SERIES. G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins: B(x) = 1 + x + 7*x^2 + 85*x^3 + 1401*x^4 + 28339*x^5 + 666638*x^6 + 17651052*x^7 + 514911165*x^8 + ... + b(n)*x^n + ... such that b(n) = [x^n] (1 + x*A(x)^(n+4))^(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, 21/3, 340/4, 7005/5, 170034/6, 4666466/7, 141208416/8, ...]. ILLUSTRATION OF DEFINITION. The table of coefficients of x^k in A(x)^(n+1) begins: n=0: [1, 1, 6, 66, 1028, 20138, 464863, 12162876, ...]; n=1: [1, 2, 13, 144, 2224, 43124, 986694, 25632830, ...]; n=2: [1, 3, 21, 235, 3606, 69264, 1571169, 40527480, ...]; n=3: [1, 4, 30, 340, 5193, 98888, 2224444, 56974172, ...]; n=4: [1, 5, 40, 460, 7005, 132351, 2953185, 75110670, ...]; n=5: [1, 6, 51, 596, 9063, 170034, 3764599, 95085882, ...]; n=6: [1, 7, 63, 749, 11389, 212345, 4666466, 117060623, ...]; n=7: [1, 8, 76, 920, 14006, 259720, 5667172, 141208416, ...]; ... Compare to the table of coefficients in (1 + x*A(x)^(n+4))^(n+1): n=0: [1, 1, 4, 30, 340, 5193, 98888, 2224444, ...]; n=1: [1, 2, 11, 90, 1025, 15330, 284912, 6277922, ...]; n=2: [1, 3, 21, 190, 2220, 32862, 597579, 12884601, ...]; n=3: [1, 4, 34, 340, 4131, 61208, 1094268, 23093756, ...]; n=4: [1, 5, 50, 550, 7005, 104951, 1856360, 38416740, ...]; n=5: [1, 6, 69, 830, 11130, 170034, 2996425, 61005672, ...]; n=6: [1, 7, 91, 1190, 16835, 263956, 4666466, 93880165, ...]; n=7: [1, 8, 116, 1640, 24490, 395968, 7067220, 141208416, ...]; ... 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+4))^(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+4))^(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+4))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 5.036315811445755451051893091437..., c = 0.000317937301879544729612100255927... - Vaclav Kotesovec, Jan 31 2023