A383375 G.f. A(x) satisfies [x^n] 1/A(x)^(n+1) = [x^n] 1/A(x)^(2*n+2) for n > 1, with A'(0) = 1.

1, 1, 5, 40, 414, 5100, 71678, 1121273, 19216748, 356943612, 7130028364, 152267876318, 3460605407367, 83386349441711, 2123571541190759, 57000879370143239, 1608746374389534964, 47636112766991357023, 1476931395095225314527, 47858488423054347510410, 1618037571915550646760348, 56984337381224407981871465

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n along with related series B(x) and C(x) satisfy the following formulas.

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

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

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

(2.a) B(x*A(x)) = x.

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

(3.a) B(x) = C(x) + x^2.

(3.b) B(x*A(x)^2) = x + x^2*A(x)^4.

(3.c) C(x*A(x)) = x - x^2*A(x)^2.

(4.a) B(x) = x / A(B(x)).

(4.b) C(x) = x / A(C(x))^2.

(5.a) B(x) = C( x^2/B(x) ).

(5.b) C(x) = B( sqrt(x*C(x)) ).

(6.a) B(x) = B( x^2/B(x) ) - x^4/B(x)^2 where x/B(x) is the g.f. of A383374.

(6.b) C(x) = C( sqrt(x*C(x)) ) / (1 - x) where sqrt(x/C(x)) is the g.f. of A383373.

a(n) ~ c * n^(n + 3/2 + 3*log(2)) / (exp(n) * log(2)^n), where c = 0.105405526766033561788392... = 4*c_A383373. - Vaclav Kotesovec, Apr 25 2025

Example

G.f.: A(x) = 1 + x + 5*x^2 + 40*x^3 + 414*x^4 + 5100*x^5 + 71678*x^6 + 1121273*x^7 + 19216748*x^8 + 356943612*x^9 + ...
RELATED SERIES.
1/A(x) = 1 - x - 4*x^2 - 31*x^3 - 323*x^4 - 4048*x^5 - 58019*x^6 - 925182*x^7 - ...
B(x) = x - x^2 - 3*x^3 - 20*x^4 - 190*x^5 - 2244*x^6 - 30940*x^7 - 481040*x^8 - ...
 where B(x*A(x)) = x.
C(x) = x - 2*x^2 - 3*x^3 - 20*x^4 - 190*x^5 - 2244*x^6 - 30940*x^7 - 481040*x^8 - ...
 where C(x*A(x)^2) = x and C(x) = B(x) - x^2.
The g.f. of related sequence A383374 begins
x/B(x) = 1 + x + 4*x^2 + 27*x^3 + 249*x^4 + 2844*x^5 + 38075*x^6 + 577673*x^7 + ... + A383374(n)*x^n + ...
Another related series begins
x/C(x) = 1 + 2*x + 7*x^2 + 40*x^3 + 331*x^4 + 3546*x^5 + 45643*x^6 + 674772*x^7 + ...
the square root of which equals the g.f. of A383373,
sqrt(x/C(x)) = 1 + x + 3*x^2 + 17*x^3 + 144*x^4 + 1578*x^5 + 20667*x^6 + 309537*x^7 + ... + A383373(n)*x^n + ...
RELATED TABLE.
The table of coefficients of x^k in 1/A(x)^n begins
  n = 1: [1,  -1,  -4, -31,  -323,  -4048,  -58019, ...];
  n = 2: [1, (-2), -7, -54,  -568,  -7202, -104397, ...];
  n = 3: [1,  -3, (-9),-70,  -747,  -9603, -140911, ...];
  n = 4: [1, (-4),-10,(-80), -871, -11376, -169118, ...];
  n = 5: [1,  -5, -10, -85, (-950),-12631, -190375, ...];
  n = 6: [1,  -6, (-9),-86,  -993,(-13464),-205858, ...];
  n = 7: [1,  -7,  -7, -84, -1008, -13958,(-216580),...];
  n = 8: [1,  -8,  -4,(-80),-1002, -14184, -223408, ...];
  n = 9: [1,  -9,   0, -75,  -981, -14202, -227079, ...];
  n =10: [1, -10,   5, -70, (-950),-14062, -228215, ...];
  n =11: [1, -11,  11, -66,  -913, -13805, -227337, ...];
  n =12: [1, -12,  18, -64,  -873,(-13464),-224878, ...];
  n =13: [1, -13,  26, -65,  -832, -13065, -221195, ...];
  n =14: [1, -14,  35, -70,  -791, -12628,(-216580),...];
  ...
in which we see [x^n] 1/A(x)^(n+1) = [x^n] 1/A(x)^(2*n+2) for n > 1.
From the above table we also obtain the related series B(x) and C(x).
The coefficients in B(x) equal (1/n) * [x^n] x/A(x)^n:
[1, (-2)/2, (-9)/3, (-80)/4, (-950)/5, (-13464)/6, (-216580)/7, ...].
The coefficients in C(x) equal (1/n) * [x^n] x/A(x)^(2*n):
[1, (-4)/2, (-9)/3, (-80)/4, (-950)/5, (-13464)/6, (-216580)/7, ...]
so that C(x) = B(x) - x^2.

Prog

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

Crossrefs

Cf. A383373, A383374.

Sequence in context: A306029 A243671 A375456 * A083304 A394772 A375954

Adjacent sequences: A383372 A383373 A383374 * A383376 A383377 A383378

Keyword

nonn

Author

Paul D. Hanna, Apr 24 2025

Status

approved