A290580 - OEIS

%I #26 Aug 21 2017 17:24:19

%S 1,1,3,0,16,1,125,20,0,1296,364,1,16807,7028,112,0,262144,148752,5868,

%T 1,4782969,3471192,250128,576,0,100000000,89097664,10020912,82408,1,

%U 2357947691,2503362488,399379728,7354688,2816,0,61917364224,76575071488,16255733440,533661360,1066552,1,1792160394037,2536513162508,684615750832,35063521792,194025728,13312,0,56693912375296,90532686154752,30031767680256,2200207121408,24852054816,13053492,1,1946195068359375,3465845396598540,1376568893633760,135791393602560,2630843800320,4759188480,61440,0

%N E.g.f. W = W(x,m) satisfies: W = E(x*W,m) where E(x,m) = cn(i*x,m) - i*sn(i*x,m), with sn(x,m) and cn(x,m) being Jacobi elliptic functions, read as an irregular triangle of coefficients T(n,k) of x^n*m^k for n>=0 and k=0..[n/2].

%C An elliptic analog of the function W = LambertW(-x)/(-x) where W = exp(x*W).

%F E.g.f.: W(x,m) = (1/x) * Series_Reversion( x*cn(x,1-m)/(1 + sn(x,1-m)) ).

%F Define E(x,m) = (1 + sn(x,1-m)) / cn(x,1-m), then

%F (1) W(x,m) = (1/x) Series_Reversion( x/E(x,m) ).

%F Further, the n-th row polynomial in m, R(n,m), is given by

%F (2) R(n,m) = [x^n/n!] E(x,m)^(n+1) / (n+1) for n>=0, where

%F W(x,m) = Sum_{n>=0} R(n,m) * x^n/n!.

%e E.g.f. W(x,m) = 1 + (1)*x + (3)*x^2/2! + (16 + m)*x^3/3! +

%e (125 + 20*m)*x^4/4! + (1296 + 364*m + m^2)*x^5/5! +

%e (16807 + 7028*m + 112*m^2)*x^6/6! +

%e (262144 + 148752*m + 5868*m^2 + m^3)*x^7/7! +

%e (4782969 + 3471192*m + 250128*m^2 + 576*m^3)*x^8/8! +

%e (100000000 + 89097664*m + 10020912*m^2 + 82408*m^3 + m^4)*x^9/9! +

%e (2357947691 + 2503362488*m + 399379728*m^2 + 7354688*m^3 + 2816*m^4)*x^10/10! +...

%e such that W = W(x,m) satisfies:

%e W = E(x*W,m)

%e where E(x,m) is an elliptic analog to the exponential function, defined by

%e E(x,m) = cn(i*x,m) - i*sn(i*x,m).

%e By Jacobi's imaginary transformation, we have

%e E(x,m) = (1 + sn(x,1-m)) / cn(x,1-m),

%e where

%e E(x,m) = 1 + x + x^2/2! + (m + 1)*x^3/3! + (4*m + 1)*x^4/4! + (m^2 + 14*m + 1)*x^5/5! + (16*m^2 + 44*m + 1)*x^6/6! + (m^3 + 135*m^2 + 135*m + 1)*x^7/7! + (64*m^3 + 912*m^2 + 408*m + 1)*x^8/8! + (m^4 + 1228*m^3 + 5478*m^2 + 1228*m + 1)*x^9/9! + (256*m^4 + 15808*m^3 + 30768*m^2 + 3688*m + 1)*x^10/10! +...

%e Explicitly,

%e W(x,m) = (1/x) Series_Reversion( x/E(x,m) ).

%e As a series of row polynomial coefficients of powers of x,

%e W(x,m) = Sum_{n>=0} x^n/n! * { [x^n/n!] E(x,m)^(n+1) / (n+1) }.

%e IRREGULAR TRIANGLE.

%e This triangle of coefficients in e.g.f. W(x,m) begins:

%e 1 ;

%e 1 ;

%e 3, 0 ;

%e 16, 1 ;

%e 125, 20, 0 ;

%e 1296, 364, 1 ;

%e 16807, 7028, 112, 0 ;

%e 262144, 148752, 5868, 1 ;

%e 4782969, 3471192, 250128, 576, 0 ;

%e 100000000, 89097664, 10020912, 82408, 1 ;

%e 2357947691, 2503362488, 399379728, 7354688, 2816, 0 ;

%e 61917364224, 76575071488, 16255733440, 533661360, 1066552, 1 ;

%e 1792160394037, 2536513162508, 684615750832, 35063521792, 194025728, 13312, 0 ; ...

%o (PARI) /* By definition: */

%o { T(n,k) = my(W=1,E=1, S=x,C=1,D=1); for(i=0,n,

%o S = intformal(C*D +x*O(x^n)) ;

%o C = 1 - intformal(S*D) ; D = 1 - m*intformal(S*C) ;

%o E = subst(C - I*S,x,I*x) ) ;

%o for(i=0,n, W = subst(E,x,x*W));

%o n!*polcoeff(polcoeff(W, n,x), k,m) }

%o for(n=0,10, for(k=0,n\2, print1( T(n,k), ", ")); print(""))

%o (PARI) /* Using Jacobi's imaginary transformation: */

%o { T(n,k) = my(W=1,E=1, S=x,C=1,D=1); for(i=0,n,

%o S = intformal(C*D +x*O(x^n)) ;

%o C = 1 - intformal(S*D) ; D = 1 - m*intformal(S*C) ;

%o E = subst( (1 + S)/C,m,1-m) ) ;

%o for(i=0,n, W = subst(E,x,x*W));

%o n!*polcoeff(polcoeff(W, n,x), k,m) }

%o for(n=0,10, for(k=0,n\2, print1( T(n,k), ", ")); print(""))

%Y Cf. A290579 (row sums), A000272 (column 0), A290581 (column 1), A291214 (column 2).

%K nonn,tabf

%O 0,3

%A _Paul D. Hanna_, Aug 07 2017