A392212 G.f. A(x) satisfies A(x) - A(x)^2 - x is an even function and A(x) - 2*A(x)^2 + x^2 is an odd function.

1, 1, 2, 10, 24, 144, 376, 2472, 6752, 46816, 131456, 943552, 2699264, 19850368, 57553280, 430913152, 1262056960, 9580742144, 28282194944, 217067031552, 644844589056, 4993494147072, 14911395631104, 116325175037952, 348873398829056, 2738566237405184, 8243483715649536, 65053575212630016

Offset

1,3

Comments

Let F(x) be the g.f. of A369082, then G(x) = F(2*x) satisfies G(x) - G(x)^2 - x is an even function and G(x) - 2*G(x)^2 is an odd function. The g.f. of this sequence satisfies a similar condition. See also A392211.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1.a) (A(x) + A(-x))/2 = 2*A(x) - 2*A(x)^2 - 2*x + x^2, the even bisection of A(x).

(1.b) (A(x) - A(-x))/2 = -A(x) + 2*A(x)^2 + 2*x - x^2, the odd bisection of A(x).

(2.a) (A(x)^2 + A(-x)^2)/2 = (A(x) + A(-x))/4 - x^2/2, the even bisection of A(x)^2.

(2.b) (A(x)^2 - A(-x)^2)/2 = (A(x) - A(-x))/2 - x, the odd bisection of A(x)^2.

(3.a) A(x) = 3*A(-x) - 4*A(-x)^2 + 4*x + 2*x^2.

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

(4) 0 = 8*A(x)^4 - 12*A(x)^3 + (6 + 16*x - 8*x^2)*A(x)^2 - (1 + 12*x - 6*x^2)*A(x) + (x + 7*x^2 - 8*x^3 + 2*x^4).

D-finite with recurrence: -4096*n*(n - 1)*(n + 1)*a(n) + 12288*n*(n + 3)*(n + 1)*a(n + 1) + 1536*(n + 1)*(61*n^2 + 322*n + 464)*a(n + 2) - 1536*(77*n^3 + 609*n^2 + 1562*n + 1352)*a(n + 3) - 512*(1241*n^3 + 14211*n^2 + 54277*n + 68628)*a(n + 4) + 384*(693*n^3 + 7975*n^2 + 28422*n + 30840)*a(n + 5) + 64*(24503*n^3 + 413007*n^2 + 2320018*n + 4347192)*a(n + 6) + 384*(239*n^3 + 8767*n^2 + 86690*n + 258132)*a(n + 7) - 192*(7933*n^3 + 181606*n^2 + 1380223*n + 3482168)*a(n + 8) - 96*(4706*n^3 + 120151*n^2 + 1018587*n + 2865478)*a(n + 9) + 8*(60937*n^3 + 1809075*n^2 + 17867606*n + 58704600)*a(n + 10) - 24*(2345*n^3 + 83645*n^2 + 986434*n + 3851072)*a(n + 11) - 24*(1485*n^3 + 53145*n^2 + 634397*n + 2525692)*a(n + 12) + 18*(153*n^3 + 5661*n^2 + 69700*n + 285556)*a(n + 13) + 81*(3*n + 40)*(n + 14)*(3*n + 41)*a(n + 14) = 0. - Robert Israel, Mar 16 2026

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 10*x^4 + 24*x^5 + 144*x^6 + 376*x^7 + 2472*x^8 + 6752*x^9 + 46816*x^10 + 131456*x^11 + 943552*x^12 + ...
RELATED SERIES.
The coefficients of A(x) and A(x)^2 are closely related, as we can see from
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 24*x^5 + 72*x^6 + 376*x^7 + 1236*x^8 + 6752*x^9 + 23408*x^10 + 131456*x^11 + 471776*x^12 + ...
Observe that A(x) - A(x)^2 - x is an even function
A(x) - A(x)^2 = x + 5*x^4 + 72*x^6 + 1236*x^8 + 23408*x^10 + 471776*x^12 + 9925184*x^14 + 215456576*x^16 + ...
and that A(x) - 2*A(x)^2 + x^2 is an odd function
A(x) - 2*A(x)^2 = x - x^2 - 2*x^3 - 24*x^5 - 376*x^7 - 6752*x^9 - 131456*x^11 - 2699264*x^13 - 57553280*x^15 - ...
SPECIFIC VALUES.
A(t) = 1/4 at t = 0.1770149854180355857751046132098215445590...
A(-t) = -0.1453913315470476409691255249675470351060...
  where 1/4 = 3*A(-t) - 4*A(-t)^2 + 4*t + 2*t^2
  and A(-t) = 1/2 - 4*t + 2*t^2
  thus 0 = 64*t^4 - 256*t^3 + 256*t^2 - 32*t - 1.
A(t) = 1/5 at t = 0.1552810300281760021355969486572537096825...
A(-t) = -0.1328997235394814140092092450269026190639...
  where 1/5 = 3*A(-t) - 4*A(-t)^2 + 4*t + 2*t^2
  and A(-t) = 11/25 - 4*t + 2*t^2
  thus 0 = 1250*t^4 - 5000*t^3 + 4925*t^2 - 475*t - 27.
A(1/6) = 0.2238532524030680339946234698220511681105969004946...
A(-1/6) = -0.13999246834763374727264017563964149352654248605...
  where A(1/6) = 3*A(-1/6) - 4*A(-1/6)^2 + 13/18
  and 0 = 5184*y^4 - 7776*y^3 + 5472*y^2 - 1836*y + 211 at y = A(1/6).
A(1/7) = 0.1773695635963163643313341429370831665439817106271...
A(-1/7) = -0.12434340247040095171141997753905411866239042329...
  where A(1/7) = 3*A(-1/7) - 4*A(-1/7)^2 + 30/49
  and 0 = 19208*y^4 - 28812*y^3 + 19502*y^2 - 6223*y + 632 at y = A(1/7).
A(1/10) = 0.113460609478171216640433705839319878164162599991...
A(-1/10) = -0.0911114111781186545636460704848594184352181449...
  where A(1/10) = 3*A(-1/10) - 4*A(-1/10)^2 + 42/100
  and 0 = 40000*y^4 - 60000*y^3 + 37600*y^2 - 10700*y + 811 at y = A(1/10).

Maple

f:= gfun:-rectoproc({-4096*n*(n - 1)*(n + 1)*a(n) + 12288*n*(n + 3)*(n + 1)*a(n + 1) + 1536*(n + 1)*(61*n^2 + 322*n + 464)*a(n + 2) - 1536*(77*n^3 + 609*n^2 + 1562*n + 1352)*a(n + 3) - 512*(1241*n^3 + 14211*n^2 + 54277*n + 68628)*a(n + 4) + 384*(693*n^3 + 7975*n^2 + 28422*n + 30840)*a(n + 5) + 64*(24503*n^3 + 413007*n^2 + 2320018*n + 4347192)*a(n + 6) + 384*(239*n^3 + 8767*n^2 + 86690*n + 258132)*a(n + 7) - 192*(7933*n^3 + 181606*n^2 + 1380223*n + 3482168)*a(n + 8) - 96*(4706*n^3 + 120151*n^2 + 1018587*n + 2865478)*a(n + 9) + 8*(60937*n^3 + 1809075*n^2 + 17867606*n + 58704600)*a(n + 10) - 24*(2345*n^3 + 83645*n^2 + 986434*n + 3851072)*a(n + 11) - 24*(1485*n^3 + 53145*n^2 + 634397*n + 2525692)*a(n + 12) + 18*(153*n^3 + 5661*n^2 + 69700*n + 285556)*a(n + 13) + 81*(3*n + 40)*(n + 14)*(3*n + 41)*a(n + 14), a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 10, a(5) = 24, a(6) = 144, a(7) = 376, a(8) = 2472, a(9) = 6752, a(10) = 46816, a(11) = 131456, a(12) = 943552, a(13) = 2699264}, a(n), remember):
map(f, [$1..30]); # Robert Israel, Mar 16 2026

Prog

(PARI) \\ By definition
{a(n) = my(V=[0, 1, 1], A=x); for(i=1, n, V = concat(V, 0); A=Ser(V);
if((#V)%2==0, V[#V] = -polcoef(A - A^2 - x, #V-1); ,
V[#V] = -polcoef(A - 2*A^2 + x^2, #V-1); )); V[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A369082, A392211.

Sequence in context: A270822 A248117 A345695 * A336958 A305600 A396714

Adjacent sequences: A392209 A392210 A392211 * A392213 A392214 A392215

Keyword

nonn

Author

Paul D. Hanna, Feb 05 2026

Status

approved