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A300871 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1) * A(x) ) = n*(n+1) * [x^(n-1)] exp( n*(n+1) * A(x) ) for n>=1. 5
1, 3, 48, 1510, 71280, 4511808, 361640832, 35516910960, 4184770003200, 582762638275840, 94800017774905344, 17836975939663156224, 3847898790157443653632, 944223655310222217584640, 261663903298936561335828480, 81353978185283974468642093056, 28208743160867030634605718994944, 10849126423364041648181194666082304, 4605289001051501407092469612444385280 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

O.g.f. equals the logarithm of the e.g.f. of A300870.

The e.g.f. G(x) of A300870 satisfies: [x^n] G(x)^(n*(n+1)) = n*(n+1) * [x^(n-1)] G(x)^(n*(n+1)) for n>=1.

It is conjectured that this sequence consists entirely of integers.

a(n) is divisible by n*(n+1)/2 (conjecture); a(n) = n*(n+1)/2 * A300872(n).

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..200

EXAMPLE

O.g.f.: A(x) = x + 3*x^2 + 48*x^3 + 1510*x^4 + 71280*x^5 + 4511808*x^6 + 361640832*x^7 + 35516910960*x^8 + 4184770003200*x^9 + ...

where

exp(A(x)) = 1 + x + 7*x^2/2! + 307*x^3/3! + 37537*x^4/4! + 8755561*x^5/5! + 3304572391*x^6/6! + 1847063377867*x^7/7! + 1447456397632897*x^8/8! + ... + A300870(n)*x^n/n! + ...

ILLUSTRATION OF DEFINITION.

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

n=1: [(1), (2), 8, 328/3, 9728/3, 2241184/15, 420248704/45, ...];

n=2: [1, (6), (36), 432, 11328, 2470464/5, 150254784/5, ...];

n=3: [1, 12, (108), (1296), 29136, 5776128/5, 335166336/5, ...];

n=4: [1, 20, 260, (10480/3), (209600/3), 7265600/3, 1173400640/9, ...];

n=5: [1, 30, 540, 8640, (166800), (5004000), 241367040, 116509893120/7...];

n=6: [1, 42, 1008, 19656, 396816, (53339328/5), (2240251776/5), ...];

n=7: [1, 56, 1736, 124096/3, 2767184/3, 355355392/15, (38932329856/45), (2180210471936/45), ...]; ...

in which the coefficients in parenthesis are related by

2 = 1*2*(1); 36 = 2*3*(6); 1296 = 3*4*(108); 209600/3 = 4*5*(10480/3); 5004000 = 5*6*(166800); 2240251776/5 = 6*7*(53339328/5); ...

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

The values A300872(n) = a(n) / (n*(n+1)/2) begin:

[1, 1, 8, 151, 4752, 214848, 12915744, 986580860, 92994888960, ...]

and appear to consist entirely of integers.

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] = ((#A-1)*(#A)*V[#A-1] - V[#A])/(#A-1)/(#A) ); polcoeff( log(Ser(A)), n)}

for(n=1, 20, print1(a(n), ", "))

CROSSREFS

Cf. A300870, A300872, A300591, A296171, A300874.

Sequence in context: A319732 A199012 A304208 * A341498 A201698 A295813

Adjacent sequences:  A300868 A300869 A300870 * A300872 A300873 A300874

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 14 2018

STATUS

approved

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Last modified May 12 15:33 EDT 2021. Contains 343825 sequences. (Running on oeis4.)