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A152807
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G.f.: limit of the ratio of the g.f.s of adjacent rows in triangle A152805.
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7
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1, 1, 4, 16, 72, 340, 1688, 8648, 45468, 243832, 1328800, 7337500, 40965984, 230864496, 1311526532, 7502799104, 43183861352, 249897858164, 1453076715976, 8485587751112, 49745923115916, 292655237446616, 1727193226708608
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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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)
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)
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EXAMPLE
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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 +...
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MATHEMATICA
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(* 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 *)
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PROG
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(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)}
(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]}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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