This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A276912 #11 Sep 24 2016 03:21:43
%S A276912 1,1,1,4,13,116,661,8632,70617,1247248,13329001,285675776,3782734693,
%T A276912 107823153088,1685127882621,28683829833856,574020572798641,
%U A276912 133507199865641216,2477747434090344913,-832289494713919714304,-16453576647394853560899,11260772482520581109810176,246622016098219255086463333,-219530418791080092679815129088,-5247252347909156791432867741559,6177525915951437030555334153342976,160073955175697692672876432040185401
%N A276912 E.g.f.: A(x) satisfies: Series_Reversion( log(A(x)) * A(x) ) = log(A(x)) / A(x).
%H A276912 Paul D. Hanna, <a href="/A276912/b276912.txt">Table of n, a(n) for n = 0..300</a>
%F A276912 E.g.f. A(x) also satisfies:
%F A276912 (1) A( log(A(x)) / A(x) ) = x / LambertW(x).
%F A276912 (2) A( log(A(x)) * A(x) ) = LambertW(-x) / (-x).
%F A276912 (3) A(x) * A(-x) = 1.
%e A276912 E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 116*x^5/5! + 661*x^6/6! + 8632*x^7/7! + 70617*x^8/8! + 1247248*x^9/9! + 13329001*x^10/10! + 285675776*x^11/11! + 3782734693*x^12/12! + 107823153088*x^13/13! + 1685127882621*x^14/14! + 28683829833856*x^15/15! +...
%e A276912 such that Series_Reversion( log(A(x)) * A(x) ) = log(A(x)) / A(x).
%e A276912 RELATED SERIES.
%e A276912 The logarithm of the e.g.f. is an odd function:
%e A276912 log(A(x)) = x + 3*x^3/3! + 85*x^5/5! + 6111*x^7/7! + 872649*x^9/9! + 195062395*x^11/11! + 76208072733*x^13/13! + 12330526252695*x^15/15! + 125980697776559377*x^17/17! - 857710566759117989133*x^19/19! + 11428318296234746748941925*x^21/21! +...+ i^(n-1)*A276910(n)*x^n/n! +...
%e A276912 and thus A(x) = 1/A(-x).
%e A276912 log(A(x)) * A(x) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1446*x^6/6! + 13888*x^7/7! + 156472*x^8/8! + 2034000*x^9/9! + 29724490*x^10/10! + 476806176*x^11/11! + 8502508884*x^12/12! + 174802753216*x^13/13! + 3768345692398*x^14/14! +...+ A276911(n)*x^n/n! +...
%e A276912 log(A(x)) / A(x) = x - 2*x^2/2! + 6*x^3/3! - 28*x^4/4! + 180*x^5/5! - 1446*x^6/6! + 13888*x^7/7! - 156472*x^8/8! +...+ (-1)^(n-1)*A276911(n)*x^n/n! +...
%e A276912 RELATION TO LambertW(x):
%e A276912 A( log(A(x)) * A(x) ) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! +...+ (n+1)^(n-1)*x^n/n! +...
%e A276912 which equals LambertW(-x) / (-x).
%e A276912 A( log(A(x)) / A(x) ) = 1 + x - x^2/2! + 4*x^3/3! - 27*x^4/4! + 256*x^5/5! - 3125*x^6/6! + 46656*x^7/7! +...+ (n-1)^(n-1)*(-x)^n/n! +...
%e A276912 which equals x / LambertW(x).
%o A276912 (PARI) {a(n) = my(A=1+x,L); for(i=1,n, L = log(A +x*O(x^n)); A = exp( sqrt( L*A* serreverse(L*A) ) ) ); n!*polcoeff(A,n)}
%o A276912 for(n=0,30, print1(a(n),", "))
%Y A276912 Cf. A276910, A276911.
%K A276912 sign
%O A276912 0,4
%A A276912 _Paul D. Hanna_, Sep 24 2016