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A300873 E.g.f. A(x) satisfies: [x^n] A(x)^(n*(n+1)) = 2*n * [x^(n-1)] A(x)^(n*(n+1)) for n>=1. 3
1, 1, 3, 43, 2041, 197721, 31094251, 7086479443, 2187876597873, 874871971357681, 438740658523346131, 269314248304239932091, 198529013874402868930153, 173067121551267519897494473, 176154202119865662835343738811, 207099741506845262022248534098531, 278645958801870115911315221474653921, 425605862347493892454320041743878801633 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare to: [x^n] exp(x)^(n*(n+1)) = (n+1) * [x^(n-1)] exp(x)^(n*(n+1)) for n>=1.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

EXAMPLE

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 43*x^3/3! + 2041*x^4/4! + 197721*x^5/5! + 31094251*x^6/6! + 7086479443*x^7/7! + 2187876597873*x^8/8! + 874871971357681*x^9/9! + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k in A(x)^(n*(n+1)) begins:

n=1: [(1), (2), 4, 52/3, 560/3, 52304/15, 4048864/45, 914958416/315, ...];

n=2: [1, (6), (24), 108, 864, 67104/5, 1601424/5, 348254352/35, ...];

n=3: [1, 12, (84), (504), 3600, 211968/5, 4273776/5, 860107104/35, ...];

n=4: [1, 20, 220, (5560/3), (44480/3), 438400/3, 20480720/9, 3534944800/63, ...];

n=5: [1, 30, 480, 5580, (55440), (554400), 6991920, 947466000/7, ...];

n=6: [1, 42, 924, 14364, 181440, (10403568/5), (124842816/5), 1922103792/5, ...];

n=7: [1, 56, 1624, 98224/3, 1566992/3, 107909312/15, (4208547616/45), (58919666624/45), ...]; ...

in which the coefficients in parenthesis are related by

2 = 2*1*(1); 24 = 2*2*(6); 504 = 2*3*(84); 44480/3 = 2*4*(5560/3); 554400 = 2*5*(55440); 124842816/5 = 2*6*(10403568/5); ...

illustrating that: [x^n] A(x)^(n*(n+1)) = 2*n * [x^(n-1)] A(x)^(n*(n+1)).

LOGARITHMIC PROPERTY.

The logarithm of the e.g.f. is the integer series:

log(A(x)) = x + x^2 + 6*x^3 + 78*x^4 + 1560*x^5 + 41484*x^6 + 1361640*x^7 + 52824144*x^8 + 2355612192*x^9 + 118455668960*x^10 + ... + A300874(n)*x^n + ...

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)*(#A))); A[#A] = (2*(#A-1)*V[#A-1] - V[#A])/(#A-1)/(#A) ); EGF=Ser(A); n!*A[n+1]}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A300874, A300870, A295811, A300590, A296170, A182962.

Sequence in context: A317343 A307248 A009720 * A333329 A201173 A290777

Adjacent sequences:  A300870 A300871 A300872 * A300874 A300875 A300876

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 14 2018

STATUS

approved

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Last modified April 10 14:07 EDT 2021. Contains 342845 sequences. (Running on oeis4.)