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 1-5 of 5 results.

A198199 G.f.: q-Cosh(x,q)^2 - q-Sinh(x,q)^2 at q=-x.

Original entry on oeis.org

1, 0, -1, -2, -2, -2, -1, 2, 5, 6, 7, 10, 12, 8, -1, -8, -11, -16, -26, -38, -50, -58, -52, -30, -8, 0, 4, 24, 63, 112, 170, 232, 268, 252, 208, 186, 193, 190, 154, 92, -5, -164, -383, -630, -873, -1062, -1128, -1080, -1055, -1172, -1374, -1508, -1508, -1392, -1139, -700, -85, 684, 1627, 2652, 3455
Offset: 0

Views

Author

Paul D. Hanna, Oct 22 2011

Keywords

Comments

This sequence illustrates in part the identities:
* q-Cosh(x,q)^2 - q-Sinh(x,q)^2 = E_q(x,q) / e_q(x,q),
* q-cosh(x,q)^2 - q-sinh(x,q)^2 = e_q(x,q) / E_q(x,q).
Here the following q-analogs are employed (see MathWorld links):
q-cosh(x,q) = Sum_{n>=0} x^(2*n)/faq(2*n,q),
q-sinh(x,q) = Sum_{n>=0} x^(2*n+1)/faq(2*n+1,q),
and the dual expressions:
q-Cosh(x,q) = Sum_{n>=0} q^(n*(2*n-1))*x^(2*n)/faq(2*n,q),
q-Sinh(x,q) = Sum_{n>=0} q^(n*(2*n+1))*x^(2*n+1)/faq(2*n+1,q),
along with the dual q-exponential functions of x:
e_q(x,q) = Sum_{n>=0} x^n/faq(n,q),
E_q(x,q) = Sum_{n>=0} q^(n*(n-1)/2) * 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

			G.f.: A(x) = 1 - x^2 - 2*x^3 - 2*x^4 - 2*x^5 - x^6 + 2*x^7 + 5*x^8 +...
The g.f. may be expressed by:
(0) A(x) = q-Cosh(x,q)^2 - q-Sinh(x,q)^2 at q=-x, where
q-Cosh(x,-x) = 1 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8 - x^9 + 2*x^11 + 3*x^12 + 2*x^13 + x^14 + 2*x^15 +...
q-Sinh(x,-x) = x - x^6 - 2*x^7 - 2*x^8 - x^9 - x^12 - 2*x^13 - 2*x^14 + 4*x^16 +...
q-Cosh(x,-x)^2 = 1 - 2*x^3 - 2*x^4 - 2*x^5 - x^6 + x^8 + 2*x^9 + 5*x^10 + 10*x^11 + 13*x^12 +...
q-Sinh(x,-x)^2 = x^2 - 2*x^7 - 4*x^8 - 4*x^9 - 2*x^10 + x^12 + 2*x^13 + 4*x^14 + 6*x^15 +...
(1) A(x) = E_q(x,q) / e_q(x,q) at q=-x, where
e_q(x,-x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 11*x^6 + 17*x^7 + 28*x^8 + 48*x^9 + 80*x^10 + 128*x^11 + 204*x^12 +...
E_q(x,-x) = 1 + x - x^3 - x^4 - x^5 - 2*x^6 - 3*x^7 - 3*x^8 - 2*x^9 + 2*x^11 + 2*x^12 +...
(2) -log(A(x)) = (1+x)^2/(1-x^2)*x^2 + (1+x)^4/(1-x^4)*x^4/2 + (1+x)^6/(1-x^6)*x^6/3 + (1+x)^8/(1-x^8)*x^8/4 +...
(3) A(x) = (1 - x^2*(1+x)^2) * (1 - x^4*(1+x)^2) * (1 - x^6*(1+x)^2) * (1 - x^8*(1+x)^2) * (1 - x^10*(1+x)^2) *...

Links

Crossrefs

Cf. A198200 (dual), A152398 (e_q), A198197 (E_q), A198242 (q-Cosh), A198243 (q-Sinh), A198201 (q-cosh), A198202 (q-sinh).

Programs

  • PARI
    /* (0) G.f. q-Cosh(x,q)^2 - q-Sinh(x,q)^2 at q=-x: */
{a(n)=local(Cosh_q=sum(k=0, n, (-x)^(k*(2*k-1))*x^(2*k)/(prod(j=1, 2*k, (1-(-x)^j)/(1+x))+x*O(x^n))),Sinh_q=sum(k=0, n, (-x)^(k*(2*k+1))*x^(2*k+1)/(prod(j=1, 2*k+1, (1-(-x)^j)/(1+x))+x*O(x^n))));polcoeff(Cosh_q^2-Sinh_q^2, n)}
  • PARI
    /* (1) G.f. E_q(x,q) / e_q(x,q) at q=-x: */
{a(n)=local(e_q=sum(k=0, n, x^k/(prod(j=1, k, (1-(-x)^j)/(1+x))+x*O(x^n))),E_q=sum(k=0, n, (-x)^(k*(k-1)/2)*x^k/(prod(j=1, k, (1-(-x)^j)/(1+x))+x*O(x^n))));polcoeff(E_q/e_q, n)}
  • PARI
    /* (1) G.f. E_q(x,q) / e_q(x,q) at q=-x: */
{a(n)=local(e_q=exp(sum(k=1, n, x^k*(1+x)^k/(1-(-x)^k)/k)+x*O(x^n)), E_q=exp(sum(k=1, n, -(-x)^k*(1+x)^k/(1-(-x)^k)/k)+x*O(x^n)));polcoeff(E_q/e_q, n)}
  • PARI
    /* (2) G.f. exp( -Sum_{n>=1} (1+x)^(2*n)/(1-x^(2*n)) * x^(2*n)/n): */
{a(n)=polcoeff( exp( -sum(m=1,n\2+1,(1+x)^(2*m)/(1-x^(2*m)+x*O(x^n))*x^(2*m)/m)),n)}
  • PARI
    /* (3) G.f. Product_{n>=1} (1 - x^(2*n)*(1+x)^2): */
{a(n)=polcoeff(prod(k=1, n, 1-(1+x)^2*x^(2*k)+x*O(x^n)), n)}

Formula

(1) G.f.: E_q(x,q) / e_q(x,q) at q=-x, where
e_q(x,-x) = Sum_{n>=0} x^n/Product(k=1, n, (1 - (-x)^k)/(1+x)),
E_q(x,-x) = Sum_{n>=0} (-x)^(n*(n-1)/2) * x^n/Product(k=1, n, (1 - (-x)^k)/(1+x)).
(2) G.f.: exp( -Sum_{n>=1} (1+x)^(2*n)/(1-x^(2*n)) * x^(2*n)/n ).
(3) G.f.: Product_{n>=1} (1 - x^(2*n)*(1+x)^2).
(4) Given g.f. A(x), A( (sqrt(5)-1)/2 ) = 0.

A198200 G.f.: q-cosh(x,q)^2 - q-sinh(x,q)^2 at q=-x.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 10, 16, 28, 48, 79, 130, 215, 356, 587, 960, 1566, 2558, 4176, 6804, 11066, 17978, 29198, 47406, 76916, 124716, 202152, 327600, 530775, 859734, 1392265, 2254336, 3649840, 5908632, 9564377, 15480706, 25055322, 40549980, 65624224, 106199306, 171856555, 278099872
Offset: 0

Views

Author

Paul D. Hanna, Oct 22 2011

Keywords

Comments

This sequence illustrates in part the identities:
* q-cosh(x,q)^2 - q-sinh(x,q)^2 = e_q(x,q) / E_q(x,q),
* q-Cosh(x,q)^2 - q-Sinh(x,q)^2 = E_q(x,q) / e_q(x,q).
Here the following q-analogs are employed (see MathWorld links):
q-cosh(x,q) = Sum_{n>=0} x^(2*n)/faq(2*n,q),
q-sinh(x,q) = Sum_{n>=0} x^(2*n+1)/faq(2*n+1,q),
and the dual expressions:
q-Cosh(x,q) = Sum_{n>=0} q^(n*(2*n-1))*x^(2*n)/faq(2*n,q),
q-Sinh(x,q) = Sum_{n>=0} q^(n*(2*n+1))*x^(2*n+1)/faq(2*n+1,q),
along with the dual q-exponential functions of x:
e_q(x,q) = Sum_{n>=0} x^n/faq(n,q),
E_q(x,q) = Sum_{n>=0} q^(n*(n-1)/2) * 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

			G.f.: A(x) = 1 + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 16*x^7 + 28*x^8 +...
The g.f. may be expressed by:
(0) A(x) = q-cosh(x,q)^2 - q-sinh(x,q)^2 at q=-x, where
q-cosh(x,-x) = 1 + x^2 + x^3 + 2*x^4 + 4*x^5 + 6*x^6 + 9*x^7 + 15*x^8 + 25*x^9 + 41*x^10 + 66*x^11 + 105*x^12 +...
q-sinh(x,-x) = x + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 8*x^7 + 13*x^8 + 23*x^9 + 39*x^10 + 62*x^11 + 99*x^12 +...
q-cosh(x,-x)^2 = 1 + 2*x^2 + 2*x^3 + 5*x^4 + 10*x^5 + 17*x^6 + 30*x^7 + 54*x^8 + 96*x^9 + 170*x^10 + 296*x^11 + 510*x^12 +...
q-sinh(x,-x)^2 = x^2 + 2*x^4 + 4*x^5 + 7*x^6 + 14*x^7 + 26*x^8 + 48*x^9 + 91*x^10 + 166*x^11 + 295*x^12 +...
(1) A(x) = e_q(x,q) / E_q(x,q) at q=-x, where
e_q(x,-x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 11*x^6 + 17*x^7 + 28*x^8 + 48*x^9 + 80*x^10 + 128*x^11 + 204*x^12 +...
E_q(x,-x) = 1 + x - x^3 - x^4 - x^5 - 2*x^6 - 3*x^7 - 3*x^8 - 2*x^9 + 2*x^11 + 2*x^12 +...
(2) log(A(x)) = (1+x)^2/(1-x^2)*x^2 + (1+x)^4/(1-x^4)*x^4/2 + (1+x)^6/(1-x^6)*x^6/3 + (1+x)^8/(1-x^8)*x^8/4 +...
(3) A(x) = 1/((1 - x^2*(1+x)^2) * (1 - x^4*(1+x)^2) * (1 - x^6*(1+x)^2) * (1 - x^8*(1+x)^2) * (1 - x^10*(1+x)^2) *...).

Links

Crossrefs

Cf. A198199 (dual), A152398 (e_q), A198197 (E_q), A198201 (q-cosh), A198202 (q-sinh), A198242 (q-Cosh), A198243 (q-Sinh).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/(1 - x^(2*k)*(1+x)^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 22 2020 *)
  • PARI
    /* (0) G.f. q-cosh(x,q)^2 - q-sinh(x,q)^2 at q=-x: */
{a(n)=local(cosh_q=sum(k=0, n, x^(2*k)/(prod(j=1, 2*k, (1-(-x)^j)/(1+x))+x*O(x^n))),sinh_q=sum(k=0, n, x^(2*k+1)/(prod(j=1, 2*k+1, (1-(-x)^j)/(1+x))+x*O(x^n))));polcoeff(cosh_q^2-sinh_q^2, n)}
  • PARI
    /* (1) G.f. e_q(x,q) / E_q(x,q) at q=-x: */
{a(n)=local(e_q=sum(k=0, n, x^k/(prod(j=1, k, (1-(-x)^j)/(1+x))+x*O(x^n))),E_q=sum(k=0, n, (-x)^(k*(k-1)/2)*x^k/(prod(j=1, k, (1-(-x)^j)/(1+x))+x*O(x^n))));polcoeff(e_q/E_q, n)}
  • PARI
    /* (1) G.f. e_q(x,q) / E_q(x,q) at q=-x: */
{a(n)=local(e_q=exp(sum(k=1, n, x^k*(1+x)^k/(1-(-x)^k)/k)+x*O(x^n)), E_q=exp(sum(k=1, n, -(-x)^k*(1+x)^k/(1-(-x)^k)/k)+x*O(x^n)));polcoeff(e_q/E_q, n)}
  • PARI
    /* (2) G.f. exp( Sum_{n>=1} (1+x)^(2*n)/(1-x^(2*n)) * x^(2*n)/n): */
{a(n)=polcoeff( exp( sum(m=1,n\2+1,(1+x)^(2*m)/(1-x^(2*m)+x*O(x^n))*x^(2*m)/m)),n)}
  • PARI
    /* (3) G.f. Product_{n>=1} 1/(1 - x^(2*n)*(1+x)^2): */
{a(n)=polcoeff(1/prod(k=1, n, 1-(1+x)^2*x^(2*k)+x*O(x^n)), n)}

Formula

(1) G.f.: e_q(x,q) / E_q(x,q) at q=-x, where
e_q(x,-x) = Sum_{n>=0} x^n/Product_{k=1..n} (1-(-x)^k)/(1+x),
E_q(x,-x) = Sum_{n>=0} (-x)^(n*(n-1)/2) * x^n/Product_{k=1..n} (1-(-x)^k)/(1+x).
(2) G.f.: exp( Sum_{n>=1} (1+x)^(2*n)/(1-x^(2*n)) * x^(2*n)/n ).
(3) G.f.: Product_{n>=1} 1/(1 - x^(2*n)*(1+x)^2).
(4) Limit a(n+1)/a(n) = phi = (sqrt(5)+1)/2 with Limit a(n)/phi^n = 0.75149846280232258786564518960536101986114488526276981847216113150440...
Limit a(n)/phi^n = phi / (2*sqrt(5)) * Product_{k>=2} 1/(1 - phi^(2 - 2*k)). - Vaclav Kotesovec, Oct 22 2020

A198201 G.f.: q-cosh(x) evaluated at q=-x.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 6, 9, 14, 18, 18, 16, 67, 67, 66, 72, 84, 99, 117, 138, 159, 468, 516, 578, 679, 810, 933, 1018, 1072, 1138, 1262, 1448, 4745, 5196, 5851, 6630, 7382, 8031, 8649, 9405, 10409, 11569, 12649, 13530, 14378, 51022, 55567, 60439, 65906, 71953, 78283
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2012

Keywords

Comments

Note: q-cosh(x) = Sum_{n>=0} x^(2*n) / Product_{k=1..2*n} (1-q^k)/(1-q).

Examples

			 G.f.: 1 + x^2 + x^3 + 2*x^4 + 4*x^5 + 6*x^6 + 9*x^7 + 14*x^8 + 18*x^9 +...

Crossrefs

Cf. A152398 (e_q), A198197 (E_q), A198242 (q-Cosh), A198243 (q-Sinh), A198202 (q-sinh).
Cf. A198199, A198200.

Programs

  • PARI
    {a(n)=local(Cosh_q=sum(k=0, sqrtint(n+4), x^(2*k)/(prod(j=1, 2*k, (1-(-x)^j)/(1+x)+x*O(x^n))))); polcoeff(Cosh_q, n)}
for(n=0,81,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} x^(2*n) / Product_{k=1..2*n} (1-(-x)^k)/(1+x).

A198202 G.f.: q-sinh(x) evaluated at q=-x.

Original entry on oeis.org

1, 0, 1, 2, 3, 5, 8, 13, 22, 31, 32, 89, 115, 126, 122, 118, 127, 150, 178, 198, 653, 747, 835, 921, 1011, 1113, 1236, 1377, 1520, 1661, 1823, 6483, 6894, 7584, 8646, 9906, 11074, 11995, 12787, 13752, 15105, 16742, 18316, 19608, 71188, 78144, 84610, 90794, 97895
Offset: 1

Views

Author

Paul D. Hanna, Aug 07 2012

Keywords

Comments

Note: q-sinh(x) = Sum_{n>=0} x^(2*n+1) / Product_{k=1..2*n+1} (1-q^k)/(1-q).

Examples

			G.f.: x + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 8*x^7 + 13*x^8 + 22*x^9 + 31*x^10 +...

Crossrefs

Cf. A152398 (e_q), A198197 (E_q), A198242 (q-Cosh), A198243 (q-Sinh), A198201 (q-cosh).
Cf. A198199, A198200.

Programs

  • PARI
    {a(n)=local(Sinh_q=sum(k=0, sqrtint(n+4), x^(2*k+1)/(prod(j=1, 2*k+1, (1-(-x)^j)/(1+x))+x*O(x^n)))); polcoeff(Sinh_q, n)}
for(n=0,81,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} x^(2*n+1) / Product_{k=1..2*n+1} (1-(-x)^k)/(1+x).

A198243 G.f.: q-Sinh(x) evaluated at q=-x.

Original entry on oeis.org

1, 0, 0, 0, 0, -1, -2, -2, -1, 0, 0, -1, -2, -2, 0, 4, 7, 6, 3, 2, 3, 4, 5, 6, 6, 6, 8, 10, 6, -6, -18, -20, -13, -7, -8, -13, -16, -15, -13, -15, -25, -41, -53, -53, -44, -32, -16, 5, 22, 25, 18, 13, 14, 19, 29, 41, 44, 38, 43, 72, 109, 130, 135, 146, 180, 232, 274
Offset: 1

Views

Author

Paul D. Hanna, Aug 07 2012

Keywords

Comments

Note: q-Sinh(x) = Sum_{n>=0} x^(2*n+1) * q^(n*(2*n+1)) / Product_{k=1..2*n+1} (1-q^k)/(1-q).

Examples

			G.f.: x - x^6 - 2*x^7 - 2*x^8 - x^9 - x^12 - 2*x^13 - 2*x^14 + 4*x^16 + 7*x^17 +...

Crossrefs

Cf. A152398 (e_q), A198197 (E_q), A198242 (q-Cosh), A198201 (q-cosh), A198202 (q-sinh).
Cf. A198199, A198200.

Programs

  • PARI
    {a(n)=local(Sinh_q=sum(k=0, sqrtint(n+4), (-x)^(k*(2*k+1))*x^(2*k+1)/(prod(j=1, 2*k+1, (1-(-x)^j)/(1+x))+x*O(x^n)))); polcoeff(Sinh_q, n)}
for(n=0,81,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} x^(2*n+1) * (-x)^(n*(2*n+1)) / Product_{k=1..2*n+1} (1-(-x)^k)/(1+x).
Showing 1-5 of 5 results.

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