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The series reversion of which begins

Series_Reversion(log(A(x))) = x + x^2 + 3*x^3 + 19*x^4 + 226*x^5 + 4259*x^6 + 110514*x^7 + 3626207*x^8 + 143043592*x^9 + 6567931068*x^10 + 343278693103*x^11 + 20092744961109*x^12 + 1300754163383700*x^13 + 92223505422990050*x^14 + 7104166647498916816*x^15 +...

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in which the pairs of terms enclosed in diagonals indicated by parenthesis are equal.

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Cf. A296171, A296172, A296174, A296176.

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Paul D. Hanna, <a href="/A296170/b296170.txt">Table of n, a(n) for n = 0..200</a>

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E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = [x^n] A(x)^(n^2) for n>=1.

1, 1, -1, -11, -239, -17059, -2145689, -412595231, -111962826751, -40590007936199, -18900753214178609, -10974885891916507219, -7765167486697279401071, -6571694718107813687003051, -6551841491106355785902247049, -7597507878436131044487467850599, -10136619271768255373949409579309439, -15416099624633773180711565727641136271, -26508391106594400233543066679525341764961

0,4

Compare e.g.f. to: [x^(n-1)] exp(x)^n = [x^n] exp(x)^n for n>=1.

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) = 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 pairs of terms enclosed in 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, ...].

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 +...

The series reversion of which begins

Series_Reversion(log(A(x))) = x + x^2 + 3*x^3 + 19*x^4 + 226*x^5 + 4259*x^6 + 110514*x^7 + 3626207*x^8 + 143043592*x^9 + 6567931068*x^10 + 343278693103*x^11 + 20092744961109*x^12 + 1300754163383700*x^13 + 92223505422990050*x^14 + 7104166647498916816*x^15 +...

(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), ", "))

Cf. A296171, A296174, A296176.

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Paul D. Hanna, Dec 07 2017

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