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 1-9 of 9 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

A198242 G.f.: q-Cosh(x) evaluated at q=-x.

Original entry on oeis.org

1, 0, 0, -1, -1, -1, -1, -1, -1, -1, 0, 2, 3, 2, 1, 2, 4, 5, 5, 5, 5, 4, 1, -3, -5, -4, -2, -1, -3, -9, -15, -16, -14, -15, -21, -29, -33, -26, -7, 12, 14, -3, -21, -22, -7, 9, 16, 17, 20, 31, 52, 75, 84, 76, 72, 92, 124, 131, 91, 27, -8, 18, 83, 132, 127, 81, 46, 55
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2012

Keywords

Comments

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

Examples

			G.f.: 1 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8 - x^9 + 2*x^11 + 3*x^12 +...

Links

Crossrefs

Cf. A152398 (e_q), A198197 (E_q), A198243 (q-Sinh), A198201 (q-cosh), A198202 (q-sinh).
Cf. A198199, A198200.

Programs

  • PARI
    {a(n)=local(Cosh_q=sum(k=0, sqrtint(n+4), (-x)^(k*(2*k-1))*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) * (-x)^(n*(2*n-1)) / Product_{k=1..2*n} (1-(-x)^k)/(1+x).

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).

A306691 Expansion of Product_{k>=1} (1 + x^k * (1 - x)).

Original entry on oeis.org

1, 1, 0, 1, -1, 1, 0, -1, 1, 0, 0, -2, 4, -4, 3, -2, 0, 2, -1, -2, 3, -1, -3, 8, -11, 10, -8, 9, -13, 15, -9, -2, 6, 2, -14, 21, -20, 10, 8, -25, 29, -22, 20, -31, 46, -53, 48, -33, 15, -9, 27, -57, 71, -62, 53, -63, 83, -87, 53, 16, -80, 91, -47, 3, -19, 99, -186, 219, -184
Offset: 0

Views

Author

Seiichi Manyama, Apr 16 2019

Keywords

Links

Crossrefs

Cf. A160571, A198197, A306565, A306749.

Programs

  • Mathematica
    With[{nn=70},CoefficientList[Series[Product[1+x^k (1-x),{k,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Aug 03 2020 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1+x^k*(1-x)))

Formula

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1)*(1 - x)^d/d). - Ilya Gutkovskiy, Apr 16 2019

A198262 Logarithmic derivative of the q-exponential of x, E_q(x,q), evaluated at q=-x.

Original entry on oeis.org

1, -1, -2, -1, -4, -10, -13, -17, -20, -16, -21, -46, -77, -99, -97, -81, -101, -172, -265, -376, -499, -595, -666, -806, -1129, -1639, -2234, -2871, -3624, -4615, -6044, -8177, -11178, -15063, -19793, -25444, -32633, -42751, -57410, -77712, -104221, -137679, -180126, -235775, -311020, -413748
Offset: 0

Views

Author

Paul D. Hanna, Oct 23 2011

Keywords

Examples

			L.g.f.: log(E_q(x,-x)) = x - x^2/2 - 2*x^3/3 - x^4/4 - 4*x^5/5 - 10*x^6/6 - 13*x^7/7 - 17*x^8/8 - 20*x^9/9 - 16*x^10/10 +...
The logarithm of E_q(x,-x) equals the series:
log(E_q(x,-x)) = x - (1+x)^2/(1-x^2)*x^2/2 + (1+x)^3/(1+x^3)*x^3/3 - (1+x)^4/(1-x^4)*x^4/4 + (1+x)^5/(1+x^5)*x^5/5 - (1+x)^6/(1-x^6)*x^6/6 +...
where
E_q(x,-x) = 1 + x - x^3/(1-x) - x^6/((1-x)*(1-x+x^2)) + x^10/((1-x)*(1-x+x^2)*(1-x+x^2-x^3)) + x^15/((1-x)*(1-x+x^2)*(1-x+x^2-x^3)*(1-x+x^2-x^3+x^4)) +...
more explicitly,
E_q(x,-x) = 1 + x - x^3 - x^4 - x^5 - 2*x^6 - 3*x^7 - 3*x^8 +...+ A198197(n)*x^n +...

Crossrefs

Cf. A198197.

Programs

  • PARI
    {a(n)=local(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))));n*polcoeff(log(E_q), n)}
  • PARI
    {a(n)=local(Lgf=sum(k=1, n, -(1+x)^k/(1-(-x)^k) * (-x)^k/k)+x*O(x^n));n*polcoeff(Lgf, n)}
  • PARI
    {a(n)=local(Lgf=sum(k=1, n, log(1-(1+x)*(-x)^k+x*O(x^n))));n*polcoeff(Lgf, n)}

Formula

L.g.f.: log(E_q(x,-x)) = Sum_{n>=1} -(1+x)^n/(1-(-x)^n) * (-x)^n/n.
L.g.f.: log(E_q(x,-x)) = Sum_{n>=1} log(1 - (1+x)*(-x)^n).
L.g.f.: log(E_q(x,-x)), where E_q(x,-x) = Sum_{n>=0} (-x)^(n*(n-1)/2) * x^n/Product_{k=1..n} (1 - (-x)^k)/(1+x).

A307676 Expansion of Product_{k>=1} (1 - x^k*(1 - x))/(1 - x^k*(1 + x)).

Original entry on oeis.org

1, 0, 2, 4, 6, 14, 22, 46, 74, 138, 236, 406, 698, 1182, 1994, 3342, 5590, 9274, 15386, 25380, 41818, 68670, 112586, 184210, 300940, 490962, 800026, 1302278, 2118008, 3442042, 5590092, 9073632, 14720738, 23872776, 38700910, 62720726, 101622398, 164617032
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2019

Keywords

Links

Crossrefs

Cf. A227681, A198197.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^k*(1 - x))/(1 - x^k*(1 + x)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k*(1-x))/(1-x^k*(1+x))))
  • PARI
    N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, ((1+x)^d-(1-x)^d)/d))))

Formula

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} ((1+x)^d - (1-x)^d)/d).
a(n) ~ phi^(n+4) / sqrt(5), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 31 2021
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.