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 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. 4
 1, 6, 8316, 98843976, 4698140798736, 623259279912288096, 186936162949832833285056, 110352751044119383032310847616, 116215132158682166284921510741483776, 202905498509713715271588290261091671041536, 554890365215965228675768455367962915432839248896 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 6 00:15 EDT 2024. Contains 374957 sequences. (Running on oeis4.)