A383560 E.g.f. satisfies A(x) = exp(-x*A(x)) * A( x*exp(-x*A(x)) )^2.

1, 1, 7, 118, 3517, 160086, 10224319, 867305622, 94034404377, 12665879397046, 2073375227292691, 405303998554127718, 93253207746857953381, 24948396475041099571590, 7680545481780577676806215, 2696379400609033981226573206, 1070917017856110681841654586929, 477809258637756188291409832547094

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! along with related series B(x) (A383561) and C(x) (A383562) satisfy the following formulas.

(1.a) A(x) = exp(-x*A(x)) * A( x*exp(-x*A(x)) )^2.

(1.b) A(x)^2 = exp(x*A(x)^2) * A( x*exp(x*A(x)^2) ).

(2.a) [x^n] 1/A(x)^n = 0^(n-1) + [x^n] 1/A(x)^(2*n) for n >= 1.

(2.b) [x^n] x/B(x) = (1/n) * [x^n] x/A(x)^n for n >= 1.

(2.c) [x^n] x/C(x)^2 = (1/n) * [x^n] x/A(x)^(2*n) for n >= 1.

(3.a) C(x)^2 = exp(x) * B(x).

(3.b) A(x) = B(x*A(x)) = C( x*A(x)^2 ).

(3.c) B(x) = C(x*B(x)) = A( x/B(x) ).

(3.d) C(x) = B(x/C(x)) = A( x/C(x)^2 ).

(4.a) B(x) = A( x*exp(x)/C(x)^2 ).

(4.b) C(x) = A( x*exp(-x)/B(x) ).

(5.a) B(x*A(x)^2) = A( x*exp(x*A(x)^2) ).

(5.b) C(x*A(x)) = A( x*exp(-x*A(x)) ).

(5.c) B(x*A(x)^2) = A(x)^2 * exp(-x*A(x)^2).

(5.d) C(x*A(x))^2 = A(x) * exp(x*A(x)).

(6.a) B(x*B(x)) = A( x*exp(x*B(x)) / B(x) ).

(6.b) C(x/C(x)) = A( x*exp(-x/C(x)) / C(x)^2 ).

(6.c) B(x*B(x)) = B(x)^2 * exp(-x*B(x)).

(6.d) C(x/C(x))^2 = C(x) * exp(x/C(x)).

(7.a) A(x) = (1/x) * Series_Reversion( x/B(x) ).

(7.b) A(x) = sqrt( (1/x) * Series_Reversion( x/C(x)^2 ) ).

(7.c) B(x) = x / Series_Reversion( x*A(x) ).

(7.d) C(x) = sqrt( x / Series_Reversion( x*A(x)^2 ) ).

Example

E.g.f.: A(x) = 1 + x + 7*x^2/2! + 118*x^3/3! + 3517*x^4/4! + 160086*x^5/5! + 10224319*x^6/6! + 867305622*x^7/7! + 94034404377*x^8/8! + ...
 where A(x) = exp(-x*A(x)) * A( x*exp(-x*A(x)) )^2
 also, A(x)^2 = exp(x*A(x)^2) * A( x*exp(x*A(x)^2) ).
RELATED SERIES.
A(x)^2 = 1 + 2*x + 16*x^2/2! + 278*x^3/3! + 8272*x^4/4! + 371862*x^5/5! + 23386720*x^6/6! + 1953867414*x^7/7! + 208935178560*x^8/8! + ...
B(x) = 1 + x + 5*x^2/2! + 67*x^3/3! + 1761*x^4/4! + 75291*x^5/5! + 4676833*x^6/6! + 393156443*x^7/7! + ... + A383561(n)*x^n/n! + ...
 where B(x) = A( x/B(x) )
 also, B(x)^2 = exp(x*B(x)) * B(x*B(x)).
C(x) = 1 + x + 3*x^2/2! + 34*x^3/3! + 869*x^4/4! + 37046*x^5/5! + 2305267*x^6/6! + 194264862*x^7/7! + ... + A383562(n)*x^n/n! + ...
 where C(x) = A( x/C(x)^2 )
 also, C(x) = exp(-x/C(x)) * C(x/C(x))^2.
C(x)^2 = 1 + 2*x + 8*x^2/2! + 86*x^3/3! + 2064*x^4/4! + 84822*x^5/5! + 5156416*x^6/6! + 427539478*x^7/7! + ...
 where C(x)^2 = exp(x) * B(x).
log(A(x)) = x + 6*x^2/2! + 99*x^3/3! + 2976*x^4/4! + 137675*x^5/5! + 8945208*x^6/6! + 770944979*x^7/7! + ...
 where log(A(x)) = -x*A(x) + 2*log( A( x*exp(-x*A(x)) ) ).
log(B(x)) = x + 4*x^2/2! + 54*x^3/3! + 1472*x^4/4! + 64950*x^5/5! + 4131792*x^6/6! + 353611734*x^7/7! + ...
log(C(x)) = x + 2*x^2/2! + 27*x^3/3! + 736*x^4/4! + 32475*x^5/5! + 2065896*x^6/6! + 176805867*x^7/7! + ...
 where log(B(x)) = 2*log(C(x)) - x.
1/A(x) = 1 - x - 5*x^2/2! - 82*x^3/3! - 2507*x^4/4! - 118326*x^5/5! - 7833317*x^6/6! - 686251518*x^7/7! + ...
RELATED TABLE.
The table of coefficients of x^k/k! in 1/A(x)^n begins
  n = 1: [1, (-1), -5,  -82, -2507, -118326,  -7833317, ...];
  n = 2: [1, (-2),(-8),-134, -4208, -203382, -13736192, ...];
  n = 3: [1,  -3,  -9,(-162),-5283, -262338, -18093105, ...];
  n = 4: [1,  -4, (-8),-172,(-5888),-301164, -21222176, ...];
  n = 5: [1,  -5,  -5, -170, -6155,(-324750),-23384165, ...];
  n = 6: [1,  -6,   0,(-162),-6192, -337026,(-24790752),...];
  n = 7: [1,  -7,   7, -154, -6083, -341082, -25612097, ...];
  n = 8: [1,  -8,  16, -152,(-5888),-339288, -25983680, ...];
  n = 9: [1,  -9,  27, -162, -5643, -333414, -26012421, ...];
  n =10: [1, -10,  40, -190, -5360,(-324750),-25782080, ...];
  n =11: [1, -11,  55, -242, -5027, -314226, -25357937, ...];
  n =12: [1, -12,  72, -324, -4608, -302532,(-24790752),...];
  n =13: [1, -13,  91, -442, -4043, -290238, -24120005, ...];
  ...
in which we see [x^n] 1/A(x)^n = 0^(n-1) + [x^n] 1/A(x)^(2*n) for n >= 1.
From the above table we also obtain the series 1/B(x) and 1/C(x)^2:
1/B(x) = exp((-1)*x + (-8/2)*x^2/2! + (-162/3)*x^3/3! + (-5888/4)*x^4/4! + (-324750/5)*x^5/5! + ...) = 1 + (-2/2)*x + (-9/3)*x^2/2! + (-172/4)*x^3/3! + (-6155/5)*x^4/4! + (-337026/6)*x^5/5! + ...
1/C(x)^2 = exp((-2)*x + (-8/2)*x^2/2! + (-162/3)*x^3/3! + (-5888/4)*x^4/4! + (-324750/5)*x^5/5! + ...) = 1 + (-4/2)*x + (0/3)*x^2/2! + (-152/4)*x^3/3! + (-5360/5)*x^4/4! + (-302532/6)*x^5/5! + ...
 where 1/B(x) = exp(x)/C(x)^2.

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); m=#A-1;
A[#A] = (0^(m-1) + polcoef( 1/Ser(A)^(2*m) - 1/Ser(A)^m, m))/m; ); H=A; n!*A[n+1]}
for(n=0, 21, print1(a(n), ", "))

Crossrefs

Cf. A383561 (B(x)), A383562 (C(x)).

Sequence in context: A266482 A384720 A368631 * A076283 A057769 A221031

Adjacent sequences: A383557 A383558 A383559 * A383561 A383562 A383563

Keyword

nonn

Author

Paul D. Hanna, May 01 2025

Status

approved