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%I #28 Apr 14 2023 09:50:05
%S 1,2,0,40,120,0,3680,37440,21600,0,880000,20592000,38966400,8553600,0,
%T 435776000,19269888000,79491456000,57708288000,6329664000,0,
%U 386949376000,28748332800000,213892766208000,335872728576000,123646051584000,7852204800000,0,560034421760000,64544356546560000,774705298498560000,2169194182594560000,1730103155573760000,374841224017920000,15132769090560000,0
%N Coefficients T(n,k) of x^(3*n)*r^(3*k)/(3*n)! in power series C(x,r) = 1 + Integral S(x,r)^2 * D(x,r)^2 dx such that C(x,r)^3 - S(x,r)^3 = 1 and D(x,r)^3 - r^3*S(x,r)^3 = 1, as a triangle read by rows.
%C Related to Dixon elliptic function cm(x,0) (cf. A104134).
%C Equals a row reversal of triangle A357542, which describes the related function D(x,r).
%H Paul D. Hanna, <a href="/A357541/b357541.txt">Table of n, a(n) for n = 0..2555</a>
%F Generating function C(x,r) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(3*n) * r^(3*k) / (3*n)! and related functions S(x,r) and D(x,r) satisfy the following formulas.
%F For brevity, some formulas here will use S = S(x,r), C = C(x,r), and D = D(x,r).
%F (1.a) C(x,r)^3 - S(x,r)^3 = 1.
%F (1.b) D(x,r)^3 - r^3 * S(x,r)^3 = 1.
%F (1.c) D(x,r)^3 - r^3 * C(x,r)^3 = 1 - r^3.
%F Integral formulas.
%F (2.a) S(x,r) = Integral C(x,r)^2 * D(x,r)^2 dx.
%F (2.b) C(x,r) = 1 + Integral S(x,r)^2 * D(x,r)^2 dx.
%F (2.c) D(x,r) = 1 + r^3 * Integral S(x,r)^2 * C(x,r)^2 dx.
%F (2.d) C(x,r)^3 = 1 + 3 * Integral S(x,r)^2 * C(x,r)^2 * D(x,r)^2 dx.
%F Derivatives.
%F (3.a) d/dx S(x,r) = C(x,r)^2 * D(x,r)^2.
%F (3.b) d/dx C(x,r) = S(x,r)^2 * D(x,r)^2.
%F (3.c) d/dx D(x,r) = r^3 * S(x,r)^2 * C(x,r)^2.
%F Exponential formulas.
%F (4.a) C - S = exp( -Integral (C + S) * D^2 dx ).
%F (4.b) D - r*S = exp( -r * Integral (D + r*S) * C^2 dx ).
%F (4.c) C + S = sqrt(C^2 - S^2) * exp( Integral D^2/(C^2 - S^2) dx ).
%F (4.d) D + r*S = sqrt(D^2 - r^2*S^2) * exp( r * Integral C^2/(D^2 - r^2*S^2) dx ).
%F (5.a) C^2 - S^2 = exp( -2 * Integral S*C/(C + S) * D^2 dx ).
%F (5.b) D^2 - r^2*S^2 = exp( -2*r^2 * Integral S*D/(D + r*S) * C^2 dx ).
%F (5.c) C^2 + S^2 = exp( 2 * Integral S*C*(C + S)/(C^2 + S^2) * D^2 dx ).
%F (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 ).
%F Hyperbolic functions.
%F (6.a) C = sqrt(C^2 - S^2) * cosh( Integral D^2/(C^2 - S^2) dx ).
%F (6.b) S = sqrt(C^2 - S^2) * sinh( Integral D^2/(C^2 - S^2) dx ).
%F (6.c) D = sqrt(D^2 - r^2*S^2) * cosh( r * Integral C^2/(D^2 - r^2*S^2) dx ).
%F (6.d) r*S = sqrt(D^2 - r^2*S^2) * sinh( r * Integral C^2/(D^2 - r^2*S^2) dx ).
%F Other formulas.
%F (7) S(x,r) = Series_Reversion( Integral 1/((1 + x^3)*(1 + r^3*x^3))^(2/3) dx ).
%F (8.a) T(n,0) = (-1)^n * A104134(n).
%F (8.b) Sum_{k=0..n} T(n,k) = (3*n)!/(3^n*n!) * Product_{k=1..n} (3*k - 2) = A178575(n), for n >= 0.
%F From _Paul D. Hanna_, Apr 14 2023 (Start):
%F Let F(x,r) = Integral 1/((1 + x^3)*(1 + r^3*x^3))^(2/3) dx, then
%F (9.a) S( F(x,r), r) = x,
%F (9.b) C( F(x,r), r) = (1 + x^3)^(1/3),
%F (9.c) D( F(x,r), r) = (1 + r^3*x^3)^(1/3). (End)
%e E.g.f.: C(x,r) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(3*n) * r^(3*k) / (3*n)! begins:
%e 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! + ...
%e This table of coefficients T(n,k) of x^(3*n) * r^(3*k) / (3*n)! in C(x,r) for n >= 0, k = 0..n, begins:
%e n = 0: [1];
%e n = 1: [2, 0];
%e n = 2: [40, 120, 0];
%e n = 3: [3680, 37440, 21600, 0];
%e n = 4: [880000, 20592000, 38966400, 8553600, 0];
%e n = 5: [435776000, 19269888000, 79491456000, 57708288000, 6329664000, 0];
%e n = 6: [386949376000, 28748332800000, 213892766208000, 335872728576000, 123646051584000, 7852204800000, 0];
%e n = 7: [560034421760000, 64544356546560000, 774705298498560000, 2169194182594560000, 1730103155573760000, 374841224017920000, 15132769090560000, 0];
%e n = 8: [1233482823823360000, 208114576947425280000, 3741268129758720000000, 16693947940315852800000, 23676862831649280000000, 11169319418477383680000, 1563368171330211840000, 42815371615948800000, 0];
%e ...
%e in which column 0 gives the unsigned coefficients in the Dixon elliptic function cm(x,0) (cf. A104134).
%e RELATED SERIES.
%e 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! + ...
%e where C(x,r)^3 - S(x,r)^3 = 1.
%e 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! + ...
%e where D(x,r)^3 - r^3 * C(x,r)^3 = (1 - r^3).
%o (PARI) {T(n,k) = my(S=x,C=1,D=1); for(i=0,n,
%o S = intformal( C^2*D^2 +O(x^(3*n+3)));
%o C = 1 + intformal( S^2*D^2);
%o D = 1 + r^3*intformal( S^2*C^2); );
%o (3*n)!*polcoeff( polcoeff(C,3*n,x),3*k,r)}
%o for(n=0,10, for(k=0,n, print1( T(n,k),", "));print(""))
%o (PARI) /* Using Series Reversion for S(x,r) (faster) */
%o {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) )) );
%o (3*n)!*polcoeff( polcoeff((1 + S^3)^(1/3),3*n,x),3*k,r)}
%o for(n=0,10, for(k=0,n, print1( T(n,k),", "));print(""))
%Y Cf. A104134 (cm(x,0)), A357540 (S(x,r)), A357542 (D(x,r)), A178575 (row sums), A357545 (central terms).
%Y Cf. A357801.
%K nonn,tabl
%O 0,2
%A _Paul D. Hanna_, Oct 09 2022
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