A319145 E.g.f. A = A(x,m) satisfies: cn(A + x, m) + sn(A - x, m) = 1, where sn(x,m) and cn(x,m) are Jacobi elliptic functions with parameter m, as an irregular triangle of coefficients read by rows.

1, 4, 24, 0, 224, -64, 2880, -1920, 0, 48064, -49984, 1024, 989184, -1365504, 129024, 0, 24218624, -40854528, 8583168, -16384, 687083520, -1352540160, 471859200, -8355840, 0, 22151148544, -49507063808, 24589796352, -1331806208, 262144, 799546834944, -1993321955328, 1286051069952, -141582532608, 536346624, 0, 31934834253824, -87721489006592, 69349000355840, -12549922078720, 198078103552, -4194304

Offset: 1

Author

Paul D. Hanna, Sep 11 2018

Keywords

• sign
• tabf

Examples

			E.g.f.: A(x,m) = x + 4*x^2/2! + 24*x^3/3! + (-64*m + 224)*x^4/4! + (-1920*m + 2880)*x^5/5! + (1024*m^2 - 49984*m + 48064)*x^6/6! + (129024*m^2 - 1365504*m + 989184)*x^7/7! + (-16384*m^3 + 8583168*m^2 - 40854528*m + 24218624)*x^8/8! + (-8355840*m^3 + 471859200*m^2 - 1352540160*m + 687083520)*x^9/9! + (262144*m^4 - 1331806208*m^3 + 24589796352*m^2 - 49507063808*m + 22151148544)*x^10/10! + (536346624*m^4 - 141582532608*m^3 + 1286051069952*m^2 - 1993321955328*m + 799546834944)*x^11/11! + (-4194304*m^5 + 198078103552*m^4 - 12549922078720*m^3 + 69349000355840*m^2 - 87721489006592*m + 31934834253824)*x^12/12! + ...
such that cn(A + x, m)  +  sn(A - x, m)  =  1.
This triangle of coefficients of x^n*m^k/n! in A(x,m) begins
1;
4;
24, 0;
224, -64;
2880, -1920, 0;
48064, -49984, 1024;
989184, -1365504, 129024, 0;
24218624, -40854528, 8583168, -16384;
687083520, -1352540160, 471859200, -8355840, 0;
22151148544, -49507063808, 24589796352, -1331806208, 262144;
799546834944, -1993321955328, 1286051069952, -141582532608, 536346624, 0;
31934834253824, -87721489006592, 69349000355840, -12549922078720, 198078103552, -4194304; ...
RELATED SERIES.
cn(A(x,m) + x, m) = 1 - 4*x^2/2! - 24*x^3/3! + (64*m - 224)*x^4/4! + (1920*m - 2880)*x^5/5! + (-1024*m^2 + 50944*m - 47104)*x^6/6! + (-129024*m^2 + 1405824*m - 948864)*x^7/7! + (16384*m^3 - 8798208*m^2 + 42037248*m - 22820864)*x^8/8! + (8355840*m^3 - 491212800*m^2 + 1381570560*m - 638699520)*x^9/9! + (-262144*m^4 + 1367932928*m^3 - 25781010432*m^2 + 50035417088*m - 20383842304)*x^10 + ...
sn(A(x,m) - x, m) = 1 - cn(A(x,m) + x, m) = 4*x^2/2! + 24*x^3/3! + (-64*m + 224)*x^4/4! + (-1920*m + 2880)*x^5/5! + (1024*m^2 - 50944*m + 47104)*x^6/6! + (129024*m^2 - 1405824*m + 948864)*x^7/7! + ...
Related Jacobi elliptic functions with parameter m begin:
sn(x,m) = x + (-m - 1)*x^3/3! + (m^2 + 14*m + 1)*x^5/5! + (-m^3 - 135*m^2 - 135*m - 1)*x^7/7! + (m^4 + 1228*m^3 + 5478*m^2 + 1228*m + 1)*x^9/9! + (-m^5 - 11069*m^4 - 165826*m^3 - 165826*m^2 - 11069*m - 1)*x^11/11! + ...
cn(x,m) = 1 - x^2/2! + (4*m + 1)*x^4/4! + (-16*m^2 - 44*m - 1)*x^6/6! + (64*m^3 + 912*m^2 + 408*m + 1)*x^8/8! + (-256*m^4 - 15808*m^3 - 30768*m^2 - 3688*m - 1)*x^10/10! + (1024*m^5 + 259328*m^4 + 1538560*m^3 + 870640*m^2 + 33212*m + 1)*x^12/12! + ...
dn(x,m) = 1 - m*x^2/2! + (m^2 + 4*m)*x^4/4! + (-m^3 - 44*m^2 - 16*m)*x^6/6! + (m^4 + 408*m^3 + 912*m^2 + 64*m)*x^8/8! + (-m^5 - 3688*m^4 - 30768*m^3 - 15808*m^2 - 256*m)*x^10/10! + (m^6 + 33212*m^5 + 870640*m^4 + 1538560*m^3 + 259328*m^2 + 1024*m)*x^12/12! + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..930, for rows 1..60 of this triangle in flattened form.

Crossrefs

Cf. A318005 (column 0).

Programs

PARI

{T(n,k) = my(A=[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); );
for(i=1,n, A=concat(A,0);
A[#A] = -Vec( subst(C,x,x*Ser(A) + x) + subst(S,x,x*Ser(A) - x) )[#A+1] );
n!*polcoeff(polcoeff(A,n,x),k,m)}
/* Print as a triangle: */
for(n=1,12,for(k=0,(n-1)\2,print1(T(n,k),", "));print(""))

Formula

E.g.f. A = A(x,m) = Sum_{n>=1} Sum_{k=0..floor((n-1)/2)} T(n,k)*x^n*m^k/n! satisfies:

(1) A(-A(-x, m), m) = x.

(2) 1 = cn(A + x, m) + sn(A - x, m).

(3) (cn(A) + sn(A)*dn(x)) * (cn(x) - sn(x)*dn(A)) = 1 - m*sn(x)^2*sn(A)^2, where parameter m is implicit.