A290580 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].
1, 1, 3, 0, 16, 1, 125, 20, 0, 1296, 364, 1, 16807, 7028, 112, 0, 262144, 148752, 5868, 1, 4782969, 3471192, 250128, 576, 0, 100000000, 89097664, 10020912, 82408, 1, 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
Offset: 0
Examples
E.g.f. W(x,m) = 1 + (1)*x + (3)*x^2/2! + (16 + m)*x^3/3! +
(125 + 20*m)*x^4/4! + (1296 + 364*m + m^2)*x^5/5! +
(16807 + 7028*m + 112*m^2)*x^6/6! +
(262144 + 148752*m + 5868*m^2 + m^3)*x^7/7! +
(4782969 + 3471192*m + 250128*m^2 + 576*m^3)*x^8/8! +
(100000000 + 89097664*m + 10020912*m^2 + 82408*m^3 + m^4)*x^9/9! +
(2357947691 + 2503362488*m + 399379728*m^2 + 7354688*m^3 + 2816*m^4)*x^10/10! +...
such that W = W(x,m) satisfies:
W = E(x*W,m)
where E(x,m) is an elliptic analog to the exponential function, defined by
E(x,m) = cn(i*x,m) - i*sn(i*x,m).
By Jacobi's imaginary transformation, we have
E(x,m) = (1 + sn(x,1-m)) / cn(x,1-m),
where
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! +...
Explicitly,
W(x,m) = (1/x) Series_Reversion( x/E(x,m) ).
As a series of row polynomial coefficients of powers of x,
W(x,m) = Sum_{n>=0} x^n/n! * { [x^n/n!] E(x,m)^(n+1) / (n+1) }.
IRREGULAR TRIANGLE.
This triangle of coefficients in e.g.f. W(x,m) begins:
1 ;
1 ;
3, 0 ;
16, 1 ;
125, 20, 0 ;
1296, 364, 1 ;
16807, 7028, 112, 0 ;
262144, 148752, 5868, 1 ;
4782969, 3471192, 250128, 576, 0 ;
100000000, 89097664, 10020912, 82408, 1 ;
2357947691, 2503362488, 399379728, 7354688, 2816, 0 ;
61917364224, 76575071488, 16255733440, 533661360, 1066552, 1 ;
1792160394037, 2536513162508, 684615750832, 35063521792, 194025728, 13312, 0 ; ...
Programs
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PARI
/* By definition: */ { T(n,k) = my(W=1,E=1, S=x,C=1,D=1); for(i=0,n, S = intformal(C*D +x*O(x^n)) ; C = 1 - intformal(S*D) ; D = 1 - m*intformal(S*C) ; E = subst(C - I*S,x,I*x) ) ; for(i=0,n, W = subst(E,x,x*W)); n!*polcoeff(polcoeff(W, n,x), k,m) } for(n=0,10, for(k=0,n\2, print1( T(n,k), ", ")); print("")) -
PARI
/* Using Jacobi's imaginary transformation: */ { T(n,k) = my(W=1,E=1, S=x,C=1,D=1); for(i=0,n, S = intformal(C*D +x*O(x^n)) ; C = 1 - intformal(S*D) ; D = 1 - m*intformal(S*C) ; E = subst( (1 + S)/C,m,1-m) ) ; for(i=0,n, W = subst(E,x,x*W)); n!*polcoeff(polcoeff(W, n,x), k,m) } for(n=0,10, for(k=0,n\2, print1( T(n,k), ", ")); print(""))
Formula
E.g.f.: W(x,m) = (1/x) * Series_Reversion( x*cn(x,1-m)/(1 + sn(x,1-m)) ).
Define E(x,m) = (1 + sn(x,1-m)) / cn(x,1-m), then
(1) W(x,m) = (1/x) Series_Reversion( x/E(x,m) ).
Further, the n-th row polynomial in m, R(n,m), is given by
(2) R(n,m) = [x^n/n!] E(x,m)^(n+1) / (n+1) for n>=0, where
W(x,m) = Sum_{n>=0} R(n,m) * x^n/n!.
Comments