A152807 G.f.: limit of the ratio of the g.f.s of adjacent rows in triangle A152805.

1, 1, 4, 16, 72, 340, 1688, 8648, 45468, 243832, 1328800, 7337500, 40965984, 230864496, 1311526532, 7502799104, 43183861352, 249897858164, 1453076715976, 8485587751112, 49745923115916, 292655237446616, 1727193226708608

Offset: 0

Author

Paul D. Hanna, Dec 28 2008

Keywords

• nonn

Comments

Triangle A152805 lists coefficients of a q-analog of the tangent numbers (A000182).

The q-exponential of x is e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) where faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n. - Paul D. Hanna, Feb 18 2024

Examples

			G.f.: 1 + q + 4*q^2 + 16*q^3 + 72*q^4 + 340*q^5 + 1688*q^6 + 8648*q^7 +...
SAMPLE ROW G.F.s OF TRIANGLE A152805:
R_4(q) = -1 + 3*q + 3*q^2 - 3*q^4 - 3*q^5 + q^6;
R_5(q) = -1 + 4*q + 3*q^2 - 3*q^3 - 4*q^4 - 14*q^5 - 4*q^6 - 3*q^7 +...;
R_6(q) = -1 + 5*q + 2*q^2 - 9*q^3 - 11*q^4 - 19*q^5 - 16*q^6 - 11*q^7 +...
RATIO OF ROW G.F.s approach the g.f. of this sequence:
R_4(q)/R_5(q) = 1 + q + 4*q^2 + 16*q^3 + 72*q^4 + 309*q^5 +...
R_5(q)/R_6(q) = 1 + q + 4*q^2 + 16*q^3 + 72*q^4 + 340*q^5 +...
Limit_{n->infty} R_n(q)/R_{n+1}(q) = 1 + q + 4*q^2 + 16*q^3 + 72*q^4 +...
		

Links

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

Crossrefs

Cf. A152805, A152806, A370442, A370443, A370444, A370445.

Programs

Mathematica

(* Calculation of constants {d,c}: *) {1/r, -s*Log[r]*Sqrt[r*Derivative[0, 1][QPochhammer][2*s, r] / (2*Pi*QPolyGamma[1, Log[2*s]/Log[r], r])]} /. FindRoot[{1 + QPochhammer[2*s, r] == 0, Log[1 - r] + QPolyGamma[0, Log[2*s]/Log[r], r] == 0}, {r, 1/6}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Feb 18 2024 *)

PARI

/* Limit of n-th row polynomial in q of triangle A152805 */
{faq(n,q) = prod(j=1, n, (q^j-1)/(q-1))} \\ q-factorial of n
{R(n) = faq(n,q) * polcoeff( 2/(1 + sum(m=0, n, (2*x)^m/faq(m,q) + x*O(x^(n+2)))), n, x)}
{a(n) = polcoeff(R(n)/R(n+1) + q*O(q^n), n, q)}
for(n=0,30, print1(a(n),", ")) \\ Paul D. Hanna, Feb 18 2024

PARI

/* A(q) satisfies -1 = Product_{n>=0} (1 - 2*q^n*A(q)) */
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( 1 + prod(k=0,#A, 1 - 2*x^k*Ser(A))/2, #A-1, x) );A[n+1]}
for(n=0,30, print1(a(n),", ")) \\ Paul D. Hanna, Feb 18 2024

Formula

From Paul D. Hanna, Feb 18 2024: (Start)

G.f. A(q) = Sum_{n>=0} a(n)*q^n satisfies the following formulas.

(1) A(q) = limit_{n->oo} R_{n}(q) / R_{n+1}(q) where R_{n}(q) = faq(n,q) * [x^n] 2/(1 + e_q(2*x,q)) is the n-th row polynomial in q of triangle A152805, e_q(x,q) is the q-exponential of x and faq(n,q) is the q-factorial of n.

(2) -1 = Product_{n>=0} (1 - 2*q^n*A(q)).

(3) -1 = Sum_{n>=0} (-2)^n * q^(n*(n-1)/2) * A(q)^n / Product_{k=1..n} (1 - q^k).

(End)

From Vaclav Kotesovec, Feb 18 2024: (Start)

Formula (2) can be rewritten as the functional equation QPochhammer(2*y,x) = -1.

a(n) ~ c * d^n / n^(3/2), where d = 6.3144728881807672285224679191139428... and c = 0.461350895847503384343179658336... (End)