A357541 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.

1, 2, 0, 40, 120, 0, 3680, 37440, 21600, 0, 880000, 20592000, 38966400, 8553600, 0, 435776000, 19269888000, 79491456000, 57708288000, 6329664000, 0, 386949376000, 28748332800000, 213892766208000, 335872728576000, 123646051584000, 7852204800000, 0, 560034421760000, 64544356546560000, 774705298498560000, 2169194182594560000, 1730103155573760000, 374841224017920000, 15132769090560000, 0

Offset

0,2

Comments

Related to Dixon elliptic function cm(x,0) (cf. A104134).

Equals a row reversal of triangle A357542, which describes the related function D(x,r).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..2555

Formula

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.

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) C(x,r)^3 = 1 + 3 * Integral 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/((1 + x^3)*(1 + r^3*x^3))^(2/3) dx ).

(8.a) T(n,0) = (-1)^n * A104134(n).

(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.

From Paul D. Hanna, Apr 14 2023 (Start):

Let F(x,r) = Integral 1/((1 + x^3)*(1 + r^3*x^3))^(2/3) dx, then

(9.a) S( F(x,r), r) = x,

(9.b) C( F(x,r), r) = (1 + x^3)^(1/3),

(9.c) D( F(x,r), r) = (1 + r^3*x^3)^(1/3). (End)

Example

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:
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! + ...
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:
n = 0: [1];
n = 1: [2, 0];
n = 2: [40, 120, 0];
n = 3: [3680, 37440, 21600, 0];
n = 4: [880000, 20592000, 38966400, 8553600, 0];
n = 5: [435776000, 19269888000, 79491456000, 57708288000, 6329664000, 0];
n = 6: [386949376000, 28748332800000, 213892766208000, 335872728576000, 123646051584000, 7852204800000, 0];
n = 7: [560034421760000, 64544356546560000, 774705298498560000, 2169194182594560000, 1730103155573760000, 374841224017920000, 15132769090560000, 0];
n = 8: [1233482823823360000, 208114576947425280000, 3741268129758720000000, 16693947940315852800000, 23676862831649280000000, 11169319418477383680000, 1563368171330211840000, 42815371615948800000, 0];
...
in which column 0 gives the unsigned coefficients in the Dixon elliptic function cm(x,0) (cf. A104134).
RELATED SERIES.
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! + ...
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 * C(x,r)^3 = (1 - r^3).

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)!*polcoeff( polcoeff(C, 3*n, x), 3*k, r)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))
(PARI) /* Using Series Reversion for S(x, r) (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)!*polcoeff( polcoeff((1 + S^3)^(1/3), 3*n, x), 3*k, r)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))

Crossrefs

Cf. A104134 (cm(x,0)), A357540 (S(x,r)), A357542 (D(x,r)), A178575 (row sums), A357545 (central terms).

Cf. A357801.

Sequence in context: A145576 A178235 A214447 * A230888 A185281 A012445

Adjacent sequences: A357538 A357539 A357540 * A357542 A357543 A357544

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 09 2022

Status

approved