A357805 a(n) = coefficient of x^(4*n)/(4*n)! in power series C(x) = 1 + Integral S(x)^3 * C(x)^3 dx such that C(x)^4 - S(x)^4 = 1.

1, 6, 8316, 98843976, 4698140798736, 623259279912288096, 186936162949832833285056, 110352751044119383032310847616, 116215132158682166284921510741483776, 202905498509713715271588290261091671041536, 554890365215965228675768455367962915432839248896

Offset

0,2

Comments

Equals row sums of triangles A357801 and A357802.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

Generating function C(x) = Sum_{n>=0} a(n)*x^(4*n)/(4*n)! and related function S(x) satisfies the following formulas.

For brevity, some formulas here will use C = C(x) and S = S(x), where S(x) = (C(x)^4 - 1)^(1/4) is the e.g.f. of A357804.

(1) C(x)^4 - S(x)^4 = 1.

Integral formulas.

(2.a) S(x) = Integral C(x)^6 dx.

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

(2.c) S(x)^4 = Integral 4 * S(x)^3 * C(x)^6 dx.

(2.d) C(x)^4 = 1 + Integral 4 * S(x)^3 * C(x)^6 dx.

Derivatives.

(3.a) d/dx S(x) = C(x)^6.

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

Exponential formulas.

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

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

(5.a) C^2 + S^2 = exp( 2 * Integral S*C^4 dx ).

(5.b) C^2 - S^2 = exp( -2 * Integral S*C^4 dx ).

Hyperbolic functions.

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

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

(7.a) C^2 = cosh( 2 * Integral S*C^4 dx ).

(7.b) S^2 = sinh( 2 * Integral S*C^4 dx ).

Explicit formulas.

(8.a) S(x) = Series_Reversion( Integral 1/(1 + x^4)^(3/2) dx ).

(8.b) C( Integral 1/(1 + x^4)^(3/2) dx ) = (1 + x^4)^(1/4).

Example

E.g.f.: C(x) = 1 + 6*x^4/4! + 8316*x^8/8! + 98843976*x^12/12! + 4698140798736*x^16/16! + 623259279912288096*x^20/20! + 186936162949832833285056*x^24/24! + 110352751044119383032310847616*x^28/28! + ...
such that
C( Integral 1/(1 + x^4)^(3/2) dx ) = (1 + x^4)^(1/4)
also
C(x)^4 - S(x)^4 = 1,
where
S(x) = x + 36*x^5/5! + 87696*x^9/9! + 1483707456*x^13/13! + 91329084354816*x^17/17! + 14862901723860427776*x^21/21! + 5279211177231308343054336*x^25/25! + ... + A357804(n)*x^(4*n+1)/(4*n+1)! + ...

Prog

(PARI) /* Using Series Reversion (faster) */
{a(n) = my(S = serreverse( intformal( 1/(1 + x^4 +O(x^(4*n+4)))^(3/2) )) );
(4*n)!*polcoeff( (1 + S^4)^(1/4), 4*n)}
for(n=0, 10, print1( a(n), ", "))
(PARI) {a(n) = my(S=x, C=1); for(i=0, n,
S = intformal( C^6 +O(x^(4*n+4)));
C = 1 + intformal( S^3*C^3 ) );
(4*n)!*polcoeff( C, 4*n)}
for(n=0, 10, print1( a(n), ", "))

Crossrefs

Cf. A357804 (S(x)), A357801, A357802, A153300.

Sequence in context: A158880 A341874 A137040 * A172733 A361901 A329911

Adjacent sequences: A357802 A357803 A357804 * A357806 A357807 A357808

Keyword

nonn

Author

Paul D. Hanna, Oct 14 2022

Status

approved