A370444 -id:A370444 - cp's OEIS frontend

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

Paul D. Hanna, Dec 28 2008

nonn

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

			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 +...

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

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

(* 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 *)

/* 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

/* 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

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)

A370442 Expansion of g.f. A(q) satisfying -2 = Product_{n>=0} (1 - 3q^nA(q)).

Original entry on oeis.org

1, 2, 12, 78, 570, 4434, 36174, 305142, 2640612, 23311068, 209111736, 1900666896, 17466522690, 162014855658, 1514885838582, 14263411673472, 135117683341050, 1286880634334490, 12315286940334942, 118362499698060384, 1141990331203349562, 11056838563337857548, 107394670044059002968

Offset: 0

Paul D. Hanna, Feb 18 2024

nonn

			G.f.: A(q) = 1 + 2*q + 12*q^2 + 78*q^3 + 570*q^4 + 4434*q^5 + 36174*q^6 + 305142*q^7 + 2640612*q^8 + 23311068*q^9 + 209111736*q^10 + ...
where A(q) satisfies the infinite product
-2 = (1 - 3*A(q)) * (1 - 3*q*A(q)) * (1 - 3*q^2*A(q)) * (1 - 3*q^3*A(q)) * (1 - 3*q^4*A(q)) * (1 - 3*q^5*A(q)) * ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A152807, A370443, A370444, A370445.

(* Calculation of constants {d,c}: *) With[{m = 3}, {1/r, -s*Log[r] * Sqrt[r*Derivative[0, 1][QPochhammer][m*s, r] / (2*Pi*(m - 1)*QPolyGamma[1, Log[m*s]/Log[r], r])]} /. FindRoot[{m + QPochhammer[m*s, r] == 1, Log[1 - r] + QPolyGamma[0, Log[m*s]/Log[r], r] == 0}, {r, 1/m^2}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Feb 18 2024 *)

/* A(q) satisfies -2 = Product_{n>=0} (1 - 3*q^n*A(q)) */
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( 2 + prod(k=0,#A, 1 - 3*x^k*Ser(A)) /3, #A-1, x) ); H=A; A[n+1]}
for(n=0,30, print1(a(n),", "))

/* limit_{n->oo} R_{n}(q) / R_{n+1}(q) */
{faq(n,q) = prod(j=1, n, (q^j-1)/(q-1))} \\ q-factorial of n
{R(n) = faq(n,q) * polcoeff( 3/(1 + 2*sum(m=0, n, (3*x)^m/faq(m,q) + x*O(x^(n+2)))), n, x)}
{a(n) = polcoeff(R(n+2)/R(n+3) + q*O(q^n), n, q)}
for(n=0,30, print1(a(n),", "))

G.f. A(q) = Sum_{n>=0} a(n)*q^n satisfies the following formulas.
(1) -2 = Product_{n>=0} (1 - 3*q^n*A(q)).
(2) -2 = Sum_{n>=0} (-3)^n * q^(n*(n-1)/2) * A(q)^n / Product_{k=1..n} (1 - q^k).
(3) A(q) = lim_{n->oo} R_{n}(q) / R_{n+1}(q) where R_{n}(q) = faq(n,q) * [x^n] 3/(1 + 2*e_q(3*x,q)), e_q(x,q) is the q-exponential of x and faq(n,q) is the q-factorial of n.
a(n) ~ c * d^n / n^(3/2), where d = 10.3914965886269147720490605009350781702243358825286425537327254915874... and c = 0.49970395101356434785108820969954986510927554236884857759717688447784... - Vaclav Kotesovec, Feb 18 2024

A370443 Expansion of g.f. A(q) satisfying -3 = Product_{n>=0} (1 - 4q^nA(q)).

Original entry on oeis.org

1, 3, 24, 216, 2184, 23592, 267144, 3128472, 37582680, 460564632, 5735093832, 72359126376, 923021734344, 11884281689688, 154243249784856, 2015831498613720, 26506024201097352, 350404606655241768, 4654501489433893512, 62092356103141330584, 831534637662059617368

Offset: 0

Paul D. Hanna, Feb 18 2024

nonn

			G.f.: A(q) = 1 + 3*q + 24*q^2 + 216*q^3 + 2184*q^4 + 23592*q^5 + 267144*q^6 + 3128472*q^7 + 37582680*q^8 + 460564632*q^9 + 5735093832*q^10 + ...
where A(q) satisfies the infinite product
-3 = (1 - 4*A(q)) * (1 - 4*q*A(q)) * (1 - 4*q^2*A(q)) * (1 - 4*q^3*A(q)) * (1 - 4*q^4*A(q)) * (1 - 4*q^5*A(q)) * ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A152807, A370442, A370444, A370445.

(* Calculation of constants {d,c}: *) With[{m = 4}, {1/r, -s*Log[r] * Sqrt[r*Derivative[0, 1][QPochhammer][m*s, r] / (2*Pi*(m - 1)*QPolyGamma[1, Log[m*s]/Log[r], r])]} /. FindRoot[{m + QPochhammer[m*s, r] == 1, Log[1 - r] + QPolyGamma[0, Log[m*s]/Log[r], r] == 0}, {r, 1/m^2}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Feb 18 2024 *)

/* A(q) satisfies -3 = Product_{n>=0} (1 - 4*q^n*A(q)) */
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( 3 + prod(k=0,#A, 1 - 4*x^k*Ser(A)) /4, #A-1, x) ); H=A; A[n+1]}
for(n=0,30, print1(a(n),", "))

/* limit_{n->oo} R_{n}(q) / R_{n+1}(q) */
{faq(n,q) = prod(j=1, n, (q^j-1)/(q-1))} \\ q-factorial of n
{R(n) = faq(n,q) * polcoeff( 4/(1 + 3*sum(m=0, n, (4*x)^m/faq(m,q) + x*O(x^(n+2)))), n, x)}
{a(n) = polcoeff(R(n+1)/R(n+2) + q*O(q^n), n, q)}
for(n=0,30, print1(a(n),", "))

G.f. A(q) = Sum_{n>=0} a(n)*q^n satisfies the following formulas.
(1) -3 = Product_{n>=0} (1 - 4*q^n*A(q)).
(2) -3 = Sum_{n>=0} (-4)^n * q^(n*(n-1)/2) * A(q)^n / Product_{k=1..n} (1 - q^k).
(3) A(q) = lim_{n->oo} R_{n}(q) / R_{n+1}(q) where R_{n}(q) = faq(n,q) * [x^n] 4/(1 + 3*e_q(4*x,q)), e_q(x,q) is the q-exponential of x and faq(n,q) is the q-factorial of n.
a(n) ~ c * d^n / n^(3/2), where d = 14.4231123176639449630408542507057843543532473958120624505916750124669... and c = 0.51707658317945675859732872615636765308571079799394608380170536716694... - Vaclav Kotesovec, Feb 18 2024

A370445 Expansion of g.f. A(q) satisfying -5 = Product_{n>=0} (1 - 6q^nA(q)).

Original entry on oeis.org

1, 5, 60, 840, 13200, 222060, 3914880, 71382000, 1335069060, 25471344000, 493782003960, 9698680513380, 192596799930360, 3860313616039080, 77994531433117500, 1586781765309289080, 32479537197876474480, 668397905439059302860, 13820909295920888641200, 287010766542642877455600

Offset: 0

Paul D. Hanna, Feb 18 2024

nonn

			G.f.: A(q) = 1 + 5*q + 60*q^2 + 840*q^3 + 13200*q^4 + 222060*q^5 + 3914880*q^6 + 71382000*q^7 + 1335069060*q^8 + 25471344000*q^9 + 493782003960*q^10 + ...
where A(q) satisfies the infinite product
-5 = (1 - 6*A(q)) * (1 - 6*q*A(q)) * (1 - 6*q^2*A(q)) * (1 - 6*q^3*A(q)) * (1 - 6*q^4*A(q)) * (1 - 6*q^5*A(q)) * ...

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

Cf. A152807, A370442, A370443, A370444.

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

/* A(q) satisfies -5 = Product_{n>=0} (1 - 6*q^n*A(q)) */
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( 5 + prod(k=0,#A, 1 - 6*x^k*Ser(A)) /6, #A-1, x) ); H=A; A[n+1]}
for(n=0,30, print1(a(n),", "))

/* limit_{n->oo} R_{n}(q) / R_{n+1}(q) */
{faq(n,q) = prod(j=1, n, (q^j-1)/(q-1))} \\ q-factorial of n
{R(n) = faq(n,q) * polcoeff( 6/(1 + 5*sum(m=0, n, (6*x)^m/faq(m,q) + x*O(x^(n+2)))), n, x)}
{a(n) = polcoeff(R(n+1)/R(n+2) + q*O(q^n), n, q)}
for(n=0,30, print1(a(n),", "))

G.f. A(q) = Sum_{n>=0} a(n)*q^n satisfies the following formulas.
(1) -5 = Product_{n>=0} (1 - 6*q^n*A(q)).
(2) -5 = Sum_{n>=0} (-6)^n * q^(n*(n-1)/2) * A(q)^n / Product_{k=1..n} (1 - q^k).
(3) A(q) = lim_{n->oo} R_{n}(q) / R_{n+1}(q) where R_{n}(q) = faq(n,q) * [x^n] 6/(1 + 5*e_q(6*x,q)), e_q(x,q) is the q-exponential of x and faq(n,q) is the q-factorial of n.
a(n) ~ c * d^n / n^(3/2), where d = 22.4513560224097075964072006016077322187366268210808011681398076264064... and c = 0.53352515179176712675216292204161595268714373520357919375218426836012... - Vaclav Kotesovec, Feb 18 2024

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

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A370442 Expansion of g.f. A(q) satisfying -2 = Product_{n>=0} (1 - 3*q^n*A(q)).

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A370443 Expansion of g.f. A(q) satisfying -3 = Product_{n>=0} (1 - 4*q^n*A(q)).

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A370445 Expansion of g.f. A(q) satisfying -5 = Product_{n>=0} (1 - 6*q^n*A(q)).

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A370442 Expansion of g.f. A(q) satisfying -2 = Product_{n>=0} (1 - 3q^nA(q)).

A370443 Expansion of g.f. A(q) satisfying -3 = Product_{n>=0} (1 - 4q^nA(q)).

A370445 Expansion of g.f. A(q) satisfying -5 = Product_{n>=0} (1 - 6q^nA(q)).