A357802 - OEIS

A357802

Coefficients T(n,k) of x^(4*n)*r^(4*k)/(4*n)! in power series D(x,r) = 1 + r^4 * Integral S(x,r)^3 * C(x,r)^3 dx such that C(x,r)^4 - S(x,r)^4 = 1 and D(x,r)^4 - r^4*S(x,r)^4 = 1, as a triangle read by rows.

1, 0, 6, 0, 6048, 2268, 0, 35126784, 56282688, 7434504, 0, 679185948672, 2514356038656, 1409371197696, 95227613712, 0, 33022883487154176, 220415417637617664, 284770675495950336, 81696140755536384, 3354162536029536, 0, 3393656235362623684608, 35701050229143616880640, 83079959422282198548480, 54913653475645427527680, 9583398717725834749440, 264444869673131894208

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OFFSET

0,3

COMMENTS

Equals a row reversal of triangle A357801.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..2925

FORMULA

Generating function D(x,r) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(4*n) * r^(4*k) / (4*n)! and related functions S(x,r) and C(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)^4 - S(x,r)^4 = 1.

(1.b) D(x,r)^4 - r^4 * S(x,r)^4 = 1.

(1.c) D(x,r)^4 - r^4 * C(x,r)^4 = 1 - r^4.

Integral formulas.

(2.a) S(x,r) = Integral C(x,r)^3 * D(x,r)^3 dx.

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

(2.c) D(x,r) = 1 + r^4 * Integral S(x,r)^3 * C(x,r)^3 dx.

(2.d) D(x,r)^4 = 1 + r^4 * Integral 4 * S(x,r)^3 * C(x,r)^3 * D(x,r)^3 dx.

Derivatives.

(3.a) d/dx S(x,r) = C(x,r)^3 * D(x,r)^3.

(3.b) d/dx C(x,r) = S(x,r)^3 * D(x,r)^3.

(3.c) d/dx D(x,r) = r^4 * S(x,r)^3 * C(x,r)^3.

Exponential formulas.

(4.a) C + S = exp( Integral (C^2 - C*S + S^2) * D^3 dx ).

(4.b) D + r*S = exp( r * Integral (D^2 - r*D*S + r^2*S^2) * C^3 dx ).

(4.c) C - S = exp( -Integral (C^2 + C*S + S^2) * D^3 dx ).

(4.d) D - r*S = exp( -r * Integral (D^2 + r*D*S + r^2*S^2) * C^3 dx ).

(5.a) C^2 + S^2 = exp( 2 * Integral S*C * D^3 dx ).

(5.b) D^2 + r^2*S^2 = exp( 2*r^2 * Integral S*D * C^3 dx ).

(5.c) C^2 - S^2 = exp( -2 * Integral S*C * D^3 dx ).

(5.d) D^2 - r^2*S^2 = exp( -2*r^2 * Integral S*D * C^3 dx ).

Hyperbolic functions.

(6.a) C = sqrt(C^2 - S^2) * cosh( Integral (C^2 + S^2) * D^3 dx ).

(6.b) S = sqrt(C^2 - S^2) * sinh( Integral (C^2 + S^2) * D^3 dx ).

(6.c) D = sqrt(D^2 - r^2*S^2) * cosh( r * Integral (D^2 + r^2*S^2) * C^3 dx ).

(6.d) r*S = sqrt(D^2 - r^2*S^2) * sinh( r * Integral (D^2 + r^2*S^2) * C^3 dx ).

(7.a) C^2 = cosh( 2 * Integral S*C * D^3 dx ).

(7.b) S^2 = sinh( 2 * Integral S*C * D^3 dx ).

(7.c) D^2 = cosh( 2*r^2 * Integral S*D * C^3 dx ).

(7.d) r^2*S^2 = sinh( 2*r^2 * Integral S*D * C^3 dx ).

Other formulas.

(8) S(x,r) = Series_Reversion( Integral 1/((1 + x^4)*(1 + r^4*x^4))^(3/4) dx ).

(9.a) T(n,0) = T(n,n) = A153301(n).

(9.b) Sum_{k=0..n} T(n,k) = A357805(n), for n >= 0.

From Paul D. Hanna, Apr 12 2023 (Start):

Let F(x,r) = Integral 1/((1 + x^4)*(1 + r^4*x^4))^(3/4) dx, then

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

(10.b) C( F(x,r), r) = (1 + x^4)^(1/4),

(10.c) D( F(x,r), r) = (1 + r^4*x^4)^(1/4). (End)

EXAMPLE

E.g.f.: D(x,r) = Sum_{n>=0} T(n,k) * x^(4*n) * r^(4*k) / (4*n)! begins:

D(x,r) = 1 + 6*r^4*x^4/4! + (6048*r^4 + 2268*r^8)*x^8/8! + (35126784*r^4 + 56282688*r^8 + 7434504*r^12)*x^12/12! + (679185948672*r^4 + 2514356038656*r^8 + 1409371197696*r^12 + 95227613712*r^16)*x^16/16! + (33022883487154176*r^4 + 220415417637617664*r^8 + 284770675495950336*r^12 + 81696140755536384*r^16 + 3354162536029536*r^20)*x^20/20! + (3393656235362623684608*r^4 + 35701050229143616880640*r^8 + 83079959422282198548480*r^12 + 54913653475645427527680*r^16 + 9583398717725834749440*r^20 + 264444869673131894208*r^24)*x^24/24! +

where D(x,r) = 1 + r^4 * Integral S(x,r)^3 * D(x,r)^3 dx.

TRIANGLE.

This triangle of coefficients T(n,k) of x^(4*n) * r^(4*k) / (4*n)! in D(x,r) for n >= 0, k = 0..n, begins:

n = 0: [1];

n = 1: [0, 6];

n = 2: [0, 6048, 2268];

n = 3: [0, 35126784, 56282688, 7434504];

n = 4: [0, 679185948672, 2514356038656, 1409371197696, 95227613712];

n = 5: [0, 33022883487154176, 220415417637617664, 284770675495950336, 81696140755536384, 3354162536029536];

n = 6: [0, 3393656235362623684608, 35701050229143616880640, 83079959422282198548480, 54913653475645427527680, 9583398717725834749440, 264444869673131894208]; ...

in which the main diagonal equals A153300.

RELATED SERIES.

S(x,r) = x + (18 + 18*r^4)*x^5/5! + (14364 + 58968*r^4 + 14364*r^8)*x^9/9! + (70203672 + 671650056*r^4 + 671650056*r^8 + 70203672*r^12)*x^13/13! + (1192064637456 + 20707300240704*r^4 + 47530354598496*r^8 + 20707300240704*r^12 + 1192064637456*r^16)*x^17/17! + (52269828456672288 + 1437626817559769760*r^4 + 5941554215913771840*r^8 + 5941554215913771840*r^12 + 1437626817559769760*r^16 + 52269828456672288*r^20)*x^21/21! + (4930307288899134335424 + 197041019249105562351744*r^4 + 1283341580573615116868160*r^8 + 2308585363008068715943680*r^12 + 1283341580573615116868160*r^16 + 197041019249105562351744*r^20 + 4930307288899134335424*r^24)*x^25/25! + ...

where D(x,r)^4 - r^4*S(x,r)^4 = 1.

C(x,r) = 1 + 6*x^4/4! + (2268 + 6048*r^4)*x^8/8! + (7434504 + 56282688*r^4 + 35126784*r^8)*x^12/12! + (95227613712 + 1409371197696*r^4 + 2514356038656*r^8 + 679185948672*r^12)*x^16/16! + (3354162536029536 + 81696140755536384*r^4 + 284770675495950336*r^8 + 220415417637617664*r^12 + 33022883487154176*r^16)*x^20/20! + (264444869673131894208 + 9583398717725834749440*r^4 + 54913653475645427527680*r^8 + 83079959422282198548480*r^12 + 35701050229143616880640*r^16 + 3393656235362623684608*r^20)*x^24/24! + ...

where D(x,r)^4 - r^4 * C(x,r)^4 = 1 - r^4.

PROG

(PARI) {T(n, k) = my(S=x, C=1, D=1); for(i=0, n,

S = intformal( C^3*D^3 +O(x^(4*n+4)));

C = 1 + intformal( S^3*D^3);

D = 1 + r^4*intformal( S^3*C^3); );

(4*n)!*polcoeff( polcoeff(D, 4*n, x), 4*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^4)^3*(1 + r^4*x^4)^3 +O(x^(4*n+4)) )^(1/4) )) );

(4*n)!*polcoeff( polcoeff( (1 + r^4*S^4)^(1/4), 4*n, x), 4*k, r)}

for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))