A278745 G.f. satisfies: A(x) = x*(1 - x^2*A(x)^2)/(1 + x^2*A(x)^2)^2.

1, -3, 23, -232, 2671, -33247, 435732, -5923596, 82761455, -1181085841, 17143012047, -252288796800, 3755832135428, -56459641712052, 855828940166728, -13066760979482436, 200764834403473647, -3101861571115286485, 48161808069368073765, -751107354803633628504, 11760546724914570170423, -184805245095048170080367, 2913533082844307942651984, -46070266558711138024672784, 730480047034266200626268676

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 8.

Formula

G.f. A(x) satisfies:

(1) A(x) = x/( G(A(x)^2) * G(-A(x)^2) ) where G(x) = 1 + x*G(x)^2.

(2) A(x) = x - x^3*A(x)^2 - 2*x^2*A(x)^3 - x^4*A(x)^5.

(3) A(x) = Series_Reversion( x*(1 + x^2)^2 / (1 - x^2) ).

(4) A(x) = x*C(x)*D(x) where

(4.a) C(x) = C(x)^2 - A(x)^2.

(4.b) D(x) = D(x)^2 + A(x)^2.

(4.c) C(x) = (1 + x*A(x))/(1 + x^2*A(x)^2).

(4.d) D(x) = (1 - x*A(x))/(1 + x^2*A(x)^2).

(4.e) C(x) = (1 + sqrt(1 + 4*A(x)^2))/2.

(4.f) D(x) = (1 + sqrt(1 - 4*A(x)^2))/2.

(4.g) C(x) = 1/G(-A(x)^2) where G(x) = 1 + x*G(x)^2.

(4.h) D(x) = 1/G(A(x)^2) where G(x) = 1 + x*G(x)^2 is the g.f. of Catalan numbers (A000108).

a(n) = Sum_{k=0..2*n-2} (-1)^k * A278880(2*n-1,k) for n>=1.

a(n) = Sum_{k=0..2*n-2} (-1)^k * (4*n-3)/((4*n-2*k-3)*(2*k+1)) * binomial(4*n-k-4, k) * binomial(2*n+k-2, 2*n-k-2). - Paul D. Hanna, Dec 08 2016

D-finite with recurrence -256*(n-1)*(4*n-5)*(2*n-1)*(142049551*n -178081473) *(4*n-3)*a(n) +16*(-1155885932064*n^5 +6748253449456*n^4 -14295401330216*n^3 +11571204221621*n^2 +77734459403*n -3289778607450)*a(n-1) +6*(3234453621264*n^5 -46690598461608*n^4 +268825512890063*n^3 -771308050258028*n^2 +1102485156931319*n -627947169605910)*a(n-2) -3*(n-3) *(2231943393*n -5530565638)*(3*n-10) *(2*n-7)*(3*n-11)*a(n-3)=0. - R. J. Mathar, Nov 22 2024

Example

G.f.: A(x) = x - 3*x^5 + 23*x^9 - 232*x^13 + 2671*x^17 - 33247*x^21 + 435732*x^25 - 5923596*x^29 + 82761455*x^33 - 1181085841*x^37 + 17143012047*x^41 +...
such that A(x) = x*(1 - x^2*A(x)^2)/(1 + x^2*A(x)^2)^2.
RELATED SERIES.
A(x)^2 = x^2 - 6*x^6 + 55*x^10 - 602*x^14 + 7263*x^18 - 93192*x^22 + 1247636*x^26 - 17230290*x^30 + 243669007*x^34 - 3511010950*x^38 + 51361157967*x^42 +...
G.f. A(x) = x*C(x)*D(x) where
C(x) = (1 + x*A(x))/(1 + x^2*A(x)^2) = 1 + x^2 - x^4 - 4*x^6 + 7*x^8 + 33*x^10 - 68*x^12 - 344*x^14 + 767*x^16 + 4035*x^18 +...+ A243863(n)*x^(2*n) +...
D(x) = (1 - x*A(x))/(1 + x^2*A(x)^2) = 1 - x^2 - x^4 + 4*x^6 + 7*x^8 - 33*x^10 - 68*x^12 + 344*x^14 + 767*x^16 - 4035*x^18 +...+ (-1)^n*A243863(n)*x^(2*n) +...
such that C(x)^2 - A(x)^2 = C(x) and D(x)^2 + A(x)^2 = D(x).

Prog

(PARI) {a(n) = my(A=x); for(i=0, 4*n, A = x*(1 - x^2*A^2)/(1 + x^2*A^2 +x*O(x^(4*n)))^2 ); polcoeff(A, 4*n-3)}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* Explicit formula from triangle A278880 */
{a(n) = sum(k=0, 2*n-2, (-1)^k * (4*n-3)/((4*n-2*k-3)*(2*k+1)) * binomial(4*n-k-4, k) * binomial(2*n+k-2, 2*n-k-2) )}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A278880, A243863.

Sequence in context: A007781 A068146 A162591 * A122009 A360987 A151433

Adjacent sequences: A278742 A278743 A278744 * A278746 A278747 A278748

Keyword

sign

Author

Paul D. Hanna, Dec 01 2016

Status

approved