A381360 E.g.f. satisfies A(x) = exp( Integral abs(1/A(x)) dx ), where abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x).
1, 1, 2, 4, 12, 40, 160, 720, 3680, 20800, 129600, 880000, 6476800, 51321600, 435776000, 3946624000, 37977984000, 386949376000, 4161608704000, 47113228800000, 560034421760000, 6974121256960000, 90796614543360000, 1233482823823360000, 17455222222028800000, 256892229695692800000
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 12*x^4/4! + 40*x^5/5! + 160*x^6/6! + 720*x^7/7! + 3680*x^8/8! + 20800*x^9/9! + 129600*x^10/10! + 880000*x^11/11! + ... RELATED SERIES. Compare the expansion 1/A(x) = 1 - x + 2*x^3/3! - 4*x^4/4! + 40*x^6/6! - 160*x^7/7! + 3680*x^9/9! - 20800*x^10/10! + 880000*x^12/12! -+ ... to the logarithmic derivative of A(x), which starts as A'(x)/A(x) = 1 + x + 2*x^3/3! + 4*x^4/4! + 40*x^6/6! + 160*x^7/7! + 3680*x^9/9! + 20800*x^10/10! + 880000*x^12/12! + ... Compare the expansion of A(x)^2, A(x)^2 = 1 + 2*x + 6*x^2/2! + 20*x^3/3! + 80*x^4/4! + 360*x^5/5! + 1840*x^6/6! + 10400*x^7/7! + 64800*x^8/8! + ... to the second derivative A''(x), A''(x) = 2 + 4*x + 12*x^2/2! + 40*x^3/3! + 160*x^4/4! + 720*x^5/5! + 3680*x^6/6! + 20800*x^7/7! + 129600*x^8/8! + ... to see that A(x)^2 = A''(x)/2. The trisections of A(x) = T0(x) + T1(x) + T2(x) begin T0(x) = 1 + 4*x^3/3! + 160*x^6/6! + 20800*x^9/9! + 6476800*x^12/12! + 3946624000*x^15/15! + ... + (-1)^n*A104133(n)*x^(3*n)/(3*n)! + ... T1(x) = x + 12*x^4/4! + 720*x^7/7! + 129600*x^10/10! + 51321600*x^13/13! + 37977984000*x^16/16! + ... + A381359(n)*x^(3*n+1)/(3*n+1)! + ... T2(x) = 2*x^2/2! + 40*x^5/5! + 3680*x^8/8! + 880000*x^11/11! + 435776000*x^14/14! + 386949376000*x^17/17! + ... + (-1)^(n+1)*A104134(n+1)*x^(3*n+2)/(3*n+2)! + ... where T1(x)^2 = T0(x) * T2(x). SPECIFIC VALUES. A(t) = 2 at t = 0.539124944413127749680177459133394238743062994868860... A(1/2) = 1.88010715382441610819143840438161395486393689452572... A(1/3) = 1.47706812099920711982922832653100254037067385119264... A(1/4) = 1.32525993686962199738143806921665823348728083964591... A(1/5) = 1.24625631796359603424461055708975063363800998581855... A(1/6) = 1.19796486760762119976757848551541621821975995313722... A'(1/2) = 2.92023018229509494330357714872624246635412675769217... where A'(1/2) = sqrt((4*A(1/2)^3 - 1)/3). A'(1/3) = 1.99083372061994133511782552227672484323798669077638... A'(1/4) = 1.66436071755184885803761485601436009074671346271447... A'(1/5) = 1.49916821151732137928429531008042817934393824335471... A'(1/6) = 1.39963001300856269549100477471424098446045000735981... A''(1/2) = 7.0696058197234933075959047886381506980186062076077... where A''(1/2) = 2*A(1/2)^2.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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PARI
{a(n) = my(A = 1 + serreverse( intformal( 1/sqrt(1 + 4*x + 4*x^2 + 4*x^3/3 + x*O(x^n)) ) )); n!*polcoef(A,n)} for(n=0,30, print1(a(n),", "))
Formula
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + Series_Reversion( Integral 1/sqrt(1 + 4*x + 4*x^2 + 4*x^3/3) dx ).
(2) A(x) = 1 + Integral sqrt( (4*A(x)^3 - 1)/3 ) dx.
(3) A(x) = 1 + x + Integral Integral 2*A(x)^2 dx dx.
(4) A(x) = exp( x + Integral Integral (2*A(x)^3 + 1)/(3*A(x)^2) dx dx ).
(5) A(x)^2 = A''(x)/2.
(6) 0 = 1 + 3*A'(x)^2 - 4*A(x)^3.
Define the series trisections of A(x) = T0 + T1 + T2 by
T0 = Sum{n>=0} a(3*n)*x^(3*n)/(3*n)!,
T1 = Sum{n>=0} a(3*n+1)*x^(3*n+1)/(3*n+1)!,
T2 = Sum{n>=0} a(3*n+2)*x^(3*n+2)/(3*n+2)!,
then these series obey the following formulas.
(7.a) T1^2 = T0 * T2.
(7.b) T0 = (T0^2 - T1*T2)^2.
(7.c) T2 = (T2^2 - T0*T1)^2.
(7.d) T1 = -(T0^2 - T1*T2) * (T2^2 - T0*T1).
(7.e) T0 = (sqrt(T2))' = -(T2^2 - T0*T1)'.
(7.f) T2 = (sqrt(T0))' = (T0^2 - T1*T2)'.
(7.g) T0^(3/2) = 1 + T2^(3/2).
(7.h) T0^3 + T1^3 + T2^3 = 1 + 3*T0*T1*T2.
(7.i) T1' = sqrt(1 + 4*T1^3) = T0^3 - T2^3.
(7.j) T0' = T0 * (sqrt(1 + 4*T1^3) - 1)/T1.
(7.k) T2' = T2 * (sqrt(1 + 4*T1^3) + 1)/T1.
(8.a) T1 = Series_Reversion( Integral 1/(1 + 4*x^3)^(1/2) dx ).
(8.b) T2 = Series_Reversion( Integral 1/(1 + x^3)^(2/3) dx )^2.
(8.c) T0 = d/dx Series_Reversion( Integral 1/(1 + x^3)^(2/3) dx ).
(8.d) T0 = exp( Integral (sqrt(1 + 4*T1^3) - 1)/T1 dx ).
(9.a) A(x) = 1/(1 - Integral T0 - T2 dx).
(9.b) A(x) = exp( x + Integral Integral T0 + T2 dx dx ).
(10.a) T0 = ((A'(x) - 1)/A(x))'/2.
(10.b) T1 = A(x) - (A'(x)/A(x))'.
(10.c) T2 = ((A'(x) + 1)/A(x))'/2.
(11.a) T0 = A(x)/3 + (1 + 3*A'(x))/(6*A(x)^2).
(11.b) T1 = A(x)/3 - 1/(3*A(x)^2).
(11.c) T2 = A(x)/3 + (1 - 3*A'(x))/(6*A(x)^2).
(12.a) T1'' = 6*T1^2.
(12.b) T0'' = 2*(T2^2 + 2*T0*T1).
(12.c) T2'' = 2*(T0^2 + 2*T1*T2).
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