OFFSET
0,4
COMMENTS
Compare e.g.f. to: [x^(n-1)] exp(x)^n = [x^n] exp(x)^n for n>=1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
The logarithm of the e.g.f. A(x) is an integer series:
_ log(A(x)) = Sum_{n>=1} A296171(n) * x^n.
E.g.f. A(x) satisfies:
_ 1/n! * d^n/dx^n A(x)^(n^2) = 1/(n-1)! * d^(n-1)/dx^(n-1) A(x)^(n^2) for n>=1, when evaluated at x = 0.
EXAMPLE
E.g.f.: A(x) = 1 + x - x^2/2! - 11*x^3/3! - 239*x^4/4! - 17059*x^5/5! - 2145689*x^6/6! - 412595231*x^7/7! - 111962826751*x^8/8! - 40590007936199*x^9/9! - 18900753214178609*x^10/10! - 10974885891916507219*x^11/11! - 7765167486697279401071*x^12/12! - 6571694718107813687003051*x^13/13! - 6551841491106355785902247049*x^14/14! - 7597507878436131044487467850599*x^15/15! +...
To illustrate [x^(n-1)] A(x)^(n^2) = [x^n] A(x)^(n^2), form a table of coefficients of x^k in A(x)^(n^2) that begins as
n=1: [(1), (1), -1/2, -11/6, -239/24, -17059/120, -2145689/720, ...];
n=2: [1, (4), (4), -28/3, -196/3, -10472/15, -614264/45, ...];
n=3: [1, 9, (63/2), (63/2), -1701/8, -98217/40, -3168081/80, ...];
n=4: [1, 16, 112, (1232/3), (1232/3), -95648/15, -4835264/45, ...];
n=5: [1, 25, 575/2, 11725/6, (190225/24), (190225/24), ...];
n=6: [1, 36, 612, 6444, 45684, (1043784/5), (1043784/5), ...];
n=7: [1, 49, 2303/2, 102949/6, 4313617/24, 164086349/120, (5086480231/720), (5086480231/720), ...];
...
in which the diagonals indicated by parenthesis are equal.
Dividing the coefficients of x^(n-1)/(n-1)! in A(x)^(n^2) by n^2, we obtain the following sequence:
[1, 1, 7, 154, 7609, 695856, 103805719, 23134327168, 7227250033329, 3017857024161280, 1623903877812828871, ..., A296232(n), ...].
LOGARITHMIC PROPERTY.
Amazingly, the logarithm of the e.g.f. A(x) is an integer series:
log(A(x)) = x - x^2 - x^3 - 9*x^4 - 134*x^5 - 2852*x^6 - 79096*x^7 - 2699480*x^8 - 109201844*x^9 - 5100872244*x^10 - 269903909820*x^11 - 15944040740604*x^12 - 1039553309158964*x^13 - 74123498185170292*x^14 - 5736368141560365292*x^15 - 478780244956262592748*x^16 - 42865943103053965559668*x^17 - 4097785410628237071311764*x^18 - 416572537937169684523985420*x^19 - 44873737158384968851319470220*x^20 +...+ A296171(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)^2)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^2 ); n!*A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Dec 07 2017
STATUS
editing