A152807 - OEIS

%I #25 Feb 18 2024 15:47:50

%S 1,1,4,16,72,340,1688,8648,45468,243832,1328800,7337500,40965984,

%T 230864496,1311526532,7502799104,43183861352,249897858164,

%U 1453076715976,8485587751112,49745923115916,292655237446616,1727193226708608

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

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

%C 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

%H Paul D. Hanna, <a href="/A152807/b152807.txt">Table of n, a(n) for n = 0..300</a>

%F From _Paul D. Hanna_, Feb 18 2024: (Start)

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

%F (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.

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

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

%F (End)

%F From _Vaclav Kotesovec_, Feb 18 2024: (Start)

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

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

%e G.f.: 1 + q + 4*q^2 + 16*q^3 + 72*q^4 + 340*q^5 + 1688*q^6 + 8648*q^7 +...

%e SAMPLE ROW G.F.s OF TRIANGLE A152805:

%e R_4(q) = -1 + 3*q + 3*q^2 - 3*q^4 - 3*q^5 + q^6;

%e R_5(q) = -1 + 4*q + 3*q^2 - 3*q^3 - 4*q^4 - 14*q^5 - 4*q^6 - 3*q^7 +...;

%e R_6(q) = -1 + 5*q + 2*q^2 - 9*q^3 - 11*q^4 - 19*q^5 - 16*q^6 - 11*q^7 +...

%e RATIO OF ROW G.F.s approach the g.f. of this sequence:

%e R_4(q)/R_5(q) = 1 + q + 4*q^2 + 16*q^3 + 72*q^4 + 309*q^5 +...

%e R_5(q)/R_6(q) = 1 + q + 4*q^2 + 16*q^3 + 72*q^4 + 340*q^5 +...

%e Limit_{n->infty} R_n(q)/R_{n+1}(q) = 1 + q + 4*q^2 + 16*q^3 + 72*q^4 +...

%t (* 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 *)

%o (PARI) /* Limit of n-th row polynomial in q of triangle A152805 */

%o {faq(n,q) = prod(j=1, n, (q^j-1)/(q-1))} \\ q-factorial of n

%o {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)}

%o {a(n) = polcoeff(R(n)/R(n+1) + q*O(q^n), n, q)}

%o for(n=0,30, print1(a(n),", ")) \\ _Paul D. Hanna_, Feb 18 2024

%o (PARI) /* A(q) satisfies -1 = Product_{n>=0} (1 - 2*q^n*A(q)) */

%o {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff( 1 + prod(k=0,#A, 1 - 2*x^k*Ser(A))/2, #A-1, x) );A[n+1]}

%o for(n=0,30, print1(a(n),", ")) \\ _Paul D. Hanna_, Feb 18 2024

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 28 2008