cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Paul D. Hanna, Feb 18 2024

Keywords

Examples

			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)) * ...

Links

Crossrefs

Cf. A152807, A370443, A370444, A370445.

Programs

  • Mathematica
    (* 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 *)
  • PARI
    /* 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),", "))
  • PARI
    /* 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),", "))

Formula

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 - 4*q^n*A(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

Views

Author

Paul D. Hanna, Feb 18 2024

Keywords

Examples

			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)) * ...

Links

Crossrefs

Cf. A152807, A370442, A370444, A370445.

Programs

  • Mathematica
    (* 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 *)
  • PARI
    /* 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),", "))
  • PARI
    /* 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),", "))

Formula

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

A370444 Expansion of g.f. A(q) satisfying -4 = Product_{n>=0} (1 - 5*q^n*A(q)).

Original entry on oeis.org

1, 4, 40, 460, 5940, 82060, 1188140, 17792060, 273299640, 4282438360, 68184114040, 1099949668960, 17940069922740, 295334808497460, 4900888478007740, 81893191113037760, 1376770060020516140, 23270650287508521860, 395214289798485048340, 6740892510015481994360, 115419341703097589417340
Offset: 0

Views

Author

Paul D. Hanna, Feb 18 2024

Keywords

Examples

			G.f.: A(q) = 1 + 4*q + 40*q^2 + 460*q^3 + 5940*q^4 + 82060*q^5 + 1188140*q^6 + 17792060*q^7 + 273299640*q^8 + 4282438360*q^9 + 68184114040*q^10 + ...
where A(q) satisfies the infinite product
-4 = (1 - 5*A(q)) * (1 - 5*q*A(q)) * (1 - 5*q^2*A(q)) * (1 - 5*q^3*A(q)) * (1 - 5*q^4*A(q)) * (1 - 5*q^5*A(q)) * ...

Links

Crossrefs

Cf. A152807, A370442, A370443, A370445.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) With[{m = 5}, {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 *)
  • PARI
    /* A(q) satisfies -4 = Product_{n>=0} (1 - 5*q^n*A(q)) */
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( 4 + prod(k=0,#A, 1 - 5*x^k*Ser(A)) /5, #A-1, x) ); H=A; A[n+1]}
for(n=0,30, print1(a(n),", "))
  • PARI
    /* 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( 5/(1 + 4*sum(m=0, n, (5*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),", "))

Formula

G.f. A(q) = Sum_{n>=0} a(n)*q^n satisfies the following formulas.
(1) -4 = Product_{n>=0} (1 - 5*q^n*A(q)).
(2) -4 = Sum_{n>=0} (-5)^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] 5/(1 + 4*e_q(5*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 = 18.4404204270365344730849662390340654154532966962429860615573702674131... and c = 0.52704660567512847547508143537941958515989557138934496501237493733513... - Vaclav Kotesovec, Feb 18 2024

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

Views

Author

Paul D. Hanna, Feb 18 2024

Keywords

Examples

			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)) * ...

Links

Crossrefs

Cf. A152807, A370442, A370443, A370444.

Programs

  • Mathematica
    (* 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 *)
  • PARI
    /* 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),", "))
  • PARI
    /* 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),", "))

Formula

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

A152806 Unsigned row sums of triangle A152805.

Original entry on oeis.org

1, 1, 2, 6, 14, 44, 148, 544, 2400, 13268, 73016, 519252, 3391406, 30119132, 226113492, 2402746648, 20167617670, 252482755188, 2329941443632, 33731346482920, 336826566536064, 5581686585375024, 59708797929080240
Offset: 0

Views

Author

Paul D. Hanna, Dec 28 2008

Keywords

Comments

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

Links

Crossrefs

Cf. A152805, A152807.

Programs

  • PARI
    {a(n)=sum(k=0,n*(n-1)/2,abs(polcoeff(polcoeff(2/(1+sum(m=0,n,(2*x)^m/prod(j=1,m,(q^j-1)/(q-1))+x*O(x^(n+2)))),n,x)*prod(j=1,n,(q^j-1)/(q-1)),k,q)))}

A152805 Expansion of 2/(1 + e_q(2x,q)) where e_q(2x,q) is the q-exponential of 2x, as a triangle of coefficients read by rows.

Original entry on oeis.org

1, -1, -1, 1, -1, 2, 2, -1, -1, 3, 3, 0, -3, -3, 1, -1, 4, 3, -3, -4, -14, -4, -3, 3, 4, -1, -1, 5, 2, -9, -11, -19, -16, -11, 11, 16, 19, 11, 9, -2, -5, 1, -1, 6, 0, -17, -18, -25, 1, -7, 41, 73, 83, 83, 73, 41, -7, 1, -25, -18, -17, 0, 6, -1, -1, 7, -3, -26, -20, -17, 38, 67
Offset: 0

Views

Author

Paul D. Hanna, Dec 28 2008

Keywords

Comments

May be considered as 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.

Examples

			Row n lists coefficients of powers of q ranging from q^0 to q^(n(n-1)/2).
Triangle begins:
1;
-1;
-1,1;
-1,2,2,-1;
-1,3,3,0,-3,-3,1;
-1,4,3,-3,-4,-14,-4,-3,3,4,-1;
-1,5,2,-9,-11,-19,-16,-11,11,16,19,11,9,-2,-5,1;
-1,6,0,-17,-18,-25,1,-7,41,73,83,83,73,41,-7,1,-25,-18,-17,0,6,-1;
-1,7,-3,-26,-20,-17,38,67,115,184,223,217,198,84,0,-84,-198,-217,-223,-184,-115,-67,-38,17,20,26,3,-7,1;
-1,8,-7,-35,-13,12,110,161,258,271,261,219,33,-257,-638,-876,-1269,-1423,-1564,-1423,-1269,-876,-638,-257,33,219,261,271,258,161,110,12,-13,-35,-7,8,-1;
...
EXPLICIT EXPANSION OF G.F.:
1 - x + x^2*(-1 + q)/faq(2,q) + x^3*(-1 + 2*q + 2*q^2 - q^3)/faq(3,q) +
x^4*(-1 + 3*q + 3*q^2 - 3*q^4 - 3*q^5 + q^6)/faq(4,q) +
x^5*(-1 + 4*q + 3*q^2 - 3*q^3 - 4*q^4 - 14*q^5 - 4*q^6 - 3*q^7 + 3*q^8 + 4*q^9 - q^10)/faq(5,q) +...

Crossrefs

Cf. A000182 (row sums=signed tangent numbers); A152806 (unsigned row sums); A152807; A152800.

Programs

  • PARI
    {T(n,k)=polcoeff(polcoeff(2/(1+sum(m=0,n,(2*x)^m/prod(j=1,m,(q^j-1)/(q-1))+x*O(x^(n+2)))),n,x)*prod(j=1,n,(q^j-1)/(q-1)),k,q)}
for(n=0,8,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))

Formula

Sum_{k=0..n(n-1)/2} T(n,k) * exp(2*Pi*I*k/n) = -2^(n-1) for n>0.
Showing 1-6 of 6 results.

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