OFFSET
0,2
COMMENTS
Related to Dixon elliptic function sm(x,0) (cf. A104133).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..2555
FORMULA
Generating function S(x,r) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(3*n+1) * r^(3*k) / (3*n+1)! and related functions C(x,r) and D(x,r) satisfy the following formulas.
For brevity, some formulas here will use S = S(x,r), C = C(x,r), and D = D(x,r).
(1.a) C(x,r)^3 - S(x,r)^3 = 1.
(1.b) D(x,r)^3 - r^3 * S(x,r)^3 = 1.
(1.c) D(x,r)^3 - r^3 * C(x,r)^3 = 1 - r^3.
Integral formulas.
(2.a) S(x,r) = Integral C(x,r)^2 * D(x,r)^2 dx.
(2.b) C(x,r) = 1 + Integral S(x,r)^2 * D(x,r)^2 dx.
(2.c) D(x,r) = 1 + r^3 * Integral S(x,r)^2 * C(x,r)^2 dx.
(2.d) S(x,r)^3 = Integral 3 * S(x,r)^2 * C(x,r)^2 * D(x,r)^2 dx.
Derivatives.
(3.a) d/dx S(x,r) = C(x,r)^2 * D(x,r)^2.
(3.b) d/dx C(x,r) = S(x,r)^2 * D(x,r)^2.
(3.c) d/dx D(x,r) = r^3 * S(x,r)^2 * C(x,r)^2.
Exponential formulas.
(4.a) C - S = exp( -Integral (C + S) * D^2 dx ).
(4.b) D - r*S = exp( -r * Integral (D + r*S) * C^2 dx ).
(4.c) C + S = sqrt(C^2 - S^2) * exp( Integral D^2/(C^2 - S^2) dx ).
(4.d) D + r*S = sqrt(D^2 - r^2*S^2) * exp( r * Integral C^2/(D^2 - r^2*S^2) dx ).
(5.a) C^2 - S^2 = exp( -2 * Integral S*C/(C + S) * D^2 dx ).
(5.b) D^2 - r^2*S^2 = exp( -2*r^2 * Integral S*D/(D + r*S) * C^2 dx ).
(5.c) C^2 + S^2 = exp( 2 * Integral S*C*(C + S)/(C^2 + S^2) * D^2 dx ).
(5.d) D^2 + r^2*S^2 = exp( 2*r^2 * Integral S*D*(D + r*S)/(D^2 + r^2*S^2) * C^2 dx ).
Hyperbolic functions.
(6.a) C = sqrt(C^2 - S^2) * cosh( Integral D^2/(C^2 - S^2) dx ).
(6.b) S = sqrt(C^2 - S^2) * sinh( Integral D^2/(C^2 - S^2) dx ).
(6.c) D = sqrt(D^2 - r^2*S^2) * cosh( r * Integral C^2/(D^2 - r^2*S^2) dx ).
(6.d) r*S = sqrt(D^2 - r^2*S^2) * sinh( r * Integral C^2/(D^2 - r^2*S^2) dx ).
Other formulas.
(7) S(x,r) = Series_Reversion( Integral ( (1 + x^3)^2 * (1 + r^3*x^3)^2 )^(-1/3) dx ).
(8.a) T(n,0) = T(n,n) = (-1)^n * A104133(n).
(8.b) Sum_{k=0..n} T(n,k) = (3*n+1)!/(3^n*n!) * Product_{k=1..n} (3*k - 2) = A357543(n), for n >= 0.
EXAMPLE
E.g.f.: S(x,r) = Sum_{n>=0} T(n,k) * x^(3*n+1) * r^(3*k) / (3*n+1)! begins:
S(x,r) = Integral C(x,r)^2 * D(x,r)^2 dx = x + (4 + 4*r^3)*x^4/4! + (160 + 800*r^3 + 160*r^6)*x^7/7! + (20800 + 292800*r^3 + 292800*r^6 + 20800*r^9)*x^10/10! + (6476800 + 191910400*r^3 + 500121600*r^6 + 191910400*r^9 + 6476800*r^12)*x^13/13! + (3946624000 + 210590336000*r^3 + 1091343616000*r^6 + 1091343616000*r^9 + 210590336000*r^12 + 3946624000*r^15)*x^16/16! + (4161608704000 + 361556726784000*r^3 + 3216369361920000*r^6 + 6333406238720000*r^9 + 3216369361920000*r^12 + 361556726784000*r^15 + 4161608704000*r^18)*x^19/19! + (6974121256960000 + 919365914368000000*r^3 + 12789764316088320000*r^6 + 42703786876467200000*r^9 + 42703786876467200000*r^12 + 12789764316088320000*r^15 + 919365914368000000*r^18 + 6974121256960000*r^21)*x^22/22! + ...
This table of coefficients T(n,k) of x^(3*n+1) * r^(3*k) / (3*n+1)! in S(x,r) for n >= 0, k = 0..n, begins:
n = 0: [1];
n = 1: [4, 4];
n = 2: [160, 800, 160];
n = 3: [20800, 292800, 292800, 20800];
n = 4: [6476800, 191910400, 500121600, 191910400, 6476800];
n = 5: [3946624000, 210590336000, 1091343616000, 1091343616000, 210590336000, 3946624000];
n = 6: [4161608704000, 361556726784000, 3216369361920000, 6333406238720000, 3216369361920000, 361556726784000, 4161608704000];
n = 7: [6974121256960000, 919365914368000000, 12789764316088320000, 42703786876467200000, 42703786876467200000, 12789764316088320000, 919365914368000000, 6974121256960000];
n = 8: [17455222222028800000, 3313522085749145600000, 67574136526308966400000, 348431220691544883200000, 588750579021316096000000, 348431220691544883200000, 67574136526308966400000, 3313522085749145600000, 17455222222028800000];
...
in which both column 0 and the main diagoal give the unsigned coefficients in the Dixon elliptic function sm(x,0) (cf. A104133).
RELATED SERIES.
C(x,r) = 1 + Integral S(x,r)^2 * D(x,r)^2 dx = 1 + 2*x^3/3! + (40 + 120*r^3)*x^6/6! + (3680 + 37440*r^3 + 21600*r^6)*x^9/9! + (880000 + 20592000*r^3 + 38966400*r^6 + 8553600*r^9)*x^12/12! + (435776000 + 19269888000*r^3 + 79491456000*r^6 + 57708288000*r^9 + 6329664000*r^12)*x^15/15! + (386949376000 + 28748332800000*r^3 + 213892766208000*r^6 + 335872728576000*r^9 + 123646051584000*r^12 + 7852204800000*r^15)*x^18/18! + (560034421760000 + 64544356546560000*r^3 + 774705298498560000*r^6 + 2169194182594560000*r^9 + 1730103155573760000*r^12 + 374841224017920000*r^15 + 15132769090560000*r^18)*x^21/21! + ...
where C(x,r)^3 - S(x,r)^3 = 1.
D(x,r) = 1 + r^3 * Integral S(x,r)^2 * C(x,r)^2 dx = 1 + 2*r^3*x^3/3! + (120*r^3 + 40*r^6)*x^6/6! + (21600*r^3 + 37440*r^6 + 3680*r^9)*x^9/9! + (8553600*r^3 + 38966400*r^6 + 20592000*r^9 + 880000*r^12)*x^12/12! + (6329664000*r^3 + 57708288000*r^6 + 79491456000*r^9 + 19269888000*r^12 + 435776000*r^15)*x^15/15! + (7852204800000*r^3 + 123646051584000*r^6 + 335872728576000*r^9 + 213892766208000*r^12 + 28748332800000*r^15 + 386949376000*r^18)*x^18/18! + (15132769090560000*r^3 + 374841224017920000*r^6 + 1730103155573760000*r^9 + 2169194182594560000*r^12 + 774705298498560000*r^15 + 64544356546560000*r^18 + 560034421760000*r^21)*x^21/21! + ...
where D(x,r)^3 - r^3 * S(x,r)^3 = 1.
PROG
(PARI) {T(n, k) = my(S=x, C=1, D=1); for(i=0, n,
S = intformal( C^2*D^2 +O(x^(3*n+3)));
C = 1 + intformal( S^2*D^2);
D = 1 + r^3*intformal( S^2*C^2); );
(3*n+1)!*polcoeff( polcoeff(S, 3*n+1, x), 3*k, r)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))
(PARI) /* Using Series Reversion (faster) */
{T(n, k) = my(S = serreverse( intformal( 1/((1 + x^3)^2*(1 + r^3*x^3)^2 +O(x^(3*n+3)) )^(1/3) )) );
(3*n+1)!*polcoeff( polcoeff(S, 3*n+1, x), 3*k, r)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Oct 09 2022
STATUS
approved