A272467 -id:A272467 - cp's OEIS frontend

A272158 Expansion of e.g.f.: (sin(x) + sin(4x)) / sin(5x), even-indexed terms only.

1, 4, 116, 8764, 1242356, 283202524, 94690800596, 43653497804284, 26538141745926836, 20569900661155862044, 19799583458238177373076, 23170654021955185224223804, 32397957659053038859810291316, 53342240536065395589518876137564, 102148810140776173440241789042633556, 225108984136852617968906778958292851324, 565646056287498262815832721506444163551796

Offset: 0

Paul D. Hanna, Apr 30 2016

nonn

			E.g.f.: A(x) = 1 + 4*x^2/2! + 116*x^4/4! + 8764*x^6/6! + 1242356*x^8/8! + 283202524*x^10/10! + 94690800596*x^12/12! +...
such that A(x) = (sin(x) + sin(4*x)) / sin(5*x).
O.g.f.: F(x) = 1 + 4*x + 116*x^2 + 8764*x^3 + 1242356*x^4 + 283202524*x^5 + 94690800596*x^6 + 43653497804284*x^7 + 26538141745926836*x^8 +...
such that the o.g.f. can be expressed as the continued fraction:
F(x) = 1/(1 - 1*4*x/(1 - 5^2*x/(1 - 6*9*x/(1 - 10^2*x/(1 - 11*14*x/(1 - 15^2*x/(1 - 16*19*x/(1 - 20^2*x/(1 - 21*24*x/(1 - 25^2*x/(1 - 26*29*x/(1 - ...)))))))))))).

G. C. Greubel, Table of n, a(n) for n = 0..208

Cf. A272467.

seq((-25)^n*euler(2*n, 1/5), n = 0..16); # Peter Luschny, Nov 26 2020

Table[(CoefficientList[Series[(Sin[x] + Sin[4*x]) / Sin[5*x], {x, 0, 40}], x]*Range[0, 40]!)[[2*n + 1]], {n, 0, 20}] (* Vaclav Kotesovec, Apr 30 2016 *)
With[{nmax = 60}, CoefficientList[Series[Cos[3*x/2]/Cos[5*x/2], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; ;; 2]] (* G. C. Greubel, Oct 11 2018 *)

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (sin(X) + sin(4*X))/sin(5*X), 2*n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (cos(X) + cos(4*X))/(1 + cos(5*X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (exp(I*X) + exp(4*I*X))/(1 + exp(5*I*X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: cos(3*x/2) / cos(5*x/2).
E.g.f.: (cos(x) + cos(4*x)) / (1 + cos(5*x)).
E.g.f.: (exp(i*x) + exp(4*i*x)) / (1 + exp(5*i*x)), where i^2 = -1.
E.g.f.: exp(i*x)/(1 + exp(5*i*x)) + exp(-i*x)/(1 + exp(-5*i*x)), where i^2 = -1.
O.g.f.: 1/(1 - 1*4*x/(1 - 5^2*x/(1 - 6*9*x/(1 - 10^2*x/(1 - ... - (5*n+1)*(5*n+4)*x/(1 - (5*n+5)^2*x/(1 - ...))))))), a continued fraction.
a(n) ~ (2*n)! * sqrt(2*(5 - sqrt(5))) * 5^(2*n) / Pi^(2*n+1). - Vaclav Kotesovec, Apr 30 2016
a(n) = (-25)^n*Euler(2*n, 1/5). - Peter Luschny, Nov 26 2020

A273031 Expansion of e.g.f.: (sin(x) + sin(6x)) / sin(7x), even-indexed terms only.

Original entry on oeis.org

1, 6, 330, 48726, 13534410, 6046913046, 3962771924490, 3580686141374166, 4266519857080266570, 6481738795978992136086, 12228451239686387772736650, 28048508112504152087554462806, 76867928701091608252297826870730, 248058932215537567368765344245378326, 931049990613171839116868739409352364810, 4021504762182514582910341826029900914866646

Offset: 0

Paul D. Hanna, May 13 2016

nonn

			E.g.f.: A(x) = 1 + 6*x^2/2! + 330*x^4/4! + 48726*x^6/6! + 13534410*x^8/8! + 6046913046*x^10/10! + 3962771924490*x^12/12! + 3580686141374166*x^14/14! +...
such that A(x) = (sin(x) + sin(6*x)) / sin(7*x).
O.g.f.: F(x) = 1 + 6*x + 330*x^2 + 48726*x^3 + 13534410*x^4 + 6046913046*x^5 + 3962771924490*x^6 + 3580686141374166*x^7 +...
such that the o.g.f. can be expressed as the continued fraction:
F(x) = 1/(1 - 1*6*x/(1 - 7^2*x/(1 - 8*13*x/(1 - 14^2*x/(1 - 15*20*x/(1 - 21^2*x/(1 - 22*27*x/(1 - 28^2*x/(1 - 29*34*x/(1 - 35^2*x/(1 - 36*41*x/(1 - ...)))))))))))).

Cf. A272158, A272467, A273032, A273033, A156191.

seq((-49)^n*euler(2*n, 1/7), n = 0..15); # Peter Luschny, Nov 26 2020

With[{nn=30},Take[CoefficientList[Series[(Sin[x]+Sin[6x])/Sin[7x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Jul 08 2018 *)

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (sin(1*X) + sin(6*X))/sin(7*X), 2*n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (cos(1*X) + cos(6*X))/(1 + cos(7*X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (exp(1*I*X) + exp(6*I*X))/(1 + exp(7*I*X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: cos(5*x/2) / cos(7*x/2).
E.g.f.: (cos(x) + cos(6*x)) / (1 + cos(7*x)).
E.g.f.: (exp(i*x) + exp(6*i*x)) / (1 + exp(7*i*x)), where i^2 = -1.
E.g.f.: exp(i*x)/(1 + exp(7*i*x)) + exp(-i*x)/(1 + exp(-7*i*x)), where i^2 = -1.
O.g.f.: 1/(1 - 1*6*x/(1 - 7^2*x/(1 - 8*13*x/(1 - 14^2*x/(1 - ... - (7*n+1)*(7*n+6)*x/(1 - (7*n+7)^2*x/(1 - ...))))))), a continued fraction.
a(n) ~ (2*n)! * 4*cos(5*Pi/14) * 7^(2*n) / Pi^(2*n+1). - Vaclav Kotesovec, May 14 2016
a(n) = (-49)^n*Euler(2*n, 1/7). - Peter Luschny, Nov 26 2020

A272468 E.g.f.: ( (sin(2x) + sin(3x)) / sin(5*x) )^(1/6).

Original entry on oeis.org

1, 1, 16, 861, 96151, 18222146, 5239250961, 2125867405481, 1156996954702696, 813362896424049741, 717389213154874345231, 775695142663748111834426, 1009031532010773852853587441, 1554520965241408817492939532161, 2799176143277347317623990785312576, 5825065298299069164298296125454811821, 13872866932424152546975929055708940259511, 37490505378893715802821349609594581921197906

Offset: 0

Paul D. Hanna, Apr 30 2016

nonn

Conjecture: Given positive integers a and b, then the coefficient of x^(2*n)/(2*n)! is integral for n>=0 in the power series expansion of ( (sin(a*x) + sin(b*x)) / sin((a+b)*x) )^(1/(a*b)).

			G.f.: A(x) = 1 + x^2/2! + 16*x^4/4! + 861*x^6/6! + 96151*x^8/8! + 18222146*x^10/10! + 5239250961*x^12/12! + 2125867405481*x^14/14! +...
RELATED SERIES.
The logarithm of the e.g.f. begins:
log(A(x)) = x^2/2! + 13*x^4/4! + 651*x^6/6! + 69173*x^8/8! + 12613931*x^10/10! + 3514607733*x^12/12! + 1388804117611*x^14/14! + 738755067184693*x^16/16! + 508990446726347691*x^18/18! + 440936448176697240053*x^20/20! +...
such that the coefficients of x^(2*n)/(2*n)! consist entirely of odd integers.

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A272467.

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( ((sin(2*X) + sin(3*X))/sin(5*X))^(1/6), 2*n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( ( (cos(2*X) + cos(3*X))/(1 + cos(5*X)) )^(1/6), 2*n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( ((exp(2*I*X) + exp(3*I*X))/(1 + exp(5*I*X)))^(1/6), 2*n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: ( (cos(2*x) + cos(3*x)) / (1 + cos(5*x)) )^(1/6).
E.g.f.: ( (exp(2*i*x) + exp(3*i*x)) / (1 + exp(5*i*x)) )^(1/6), where i^2 = -1.
a(n) = 1 (mod 5) for n>0.
a(n) ~ (2*n)! * (2*(5 + sqrt(5)))^(1/12) * 5^(2*n) / (Gamma(1/6) * Pi^(2*n + 1/6) * n^(5/6)). - Vaclav Kotesovec, Apr 30 2016

A272481 E.g.f. A(x,y) = cos((x - xy)/2) / cos((x + xy)/2) represented as a triangle, read by rows, where row n lists of coefficients T(n,k) of x^(2n)y^k/n! in A(x,y), for k=0..2*n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 3, 1, 0, 0, 3, 15, 25, 15, 3, 0, 0, 17, 119, 329, 455, 329, 119, 17, 0, 0, 155, 1395, 5325, 11235, 14301, 11235, 5325, 1395, 155, 0, 0, 2073, 22803, 110605, 311355, 563013, 683067, 563013, 311355, 110605, 22803, 2073, 0, 0, 38227, 496951, 2918825, 10241231, 23904881, 39102063, 45956625, 39102063, 23904881, 10241231, 2918825, 496951, 38227, 0, 0, 929569, 13943535, 96075665, 403075855, 1150348017, 2362479119, 3600524785, 4136759055, 3600524785, 2362479119, 1150348017, 403075855, 96075665, 13943535, 929569, 0

Offset: 0

Paul D. Hanna, May 01 2016

Row sums equal the Euler numbers, A000364.
Column 1 equals A110501, the unsigned Genocchi numbers of first kind.
Main diagonal equals A272482, where A272482(n) = A005799(n)/2^n * (2*n)!/(n!)^2.
Sum_{k=0..2*n} (-1)^k*T(n,k) = (-1)^n.
Sum_{k=0..2*n} (-2)^k*T(n,k) = 2*(-1)^n for n>0.
Sum_{k=0..2*n} 2^k*T(n,k) = (-1)^n*A210657(n).
Sum_{k=0..2*n} 3^k*T(n,k) = A000281(n).
Sum_{k=0..2*n} 4^k*T(n,k) = A272158(n).
Sum_{k=0..2*n} 2^k*3^(2*n-k)*T(n,k) = A272467(n).

			E.g.f.: A(x,y) = 1 + x^2*(y)/2! + x^4*(y + 3*y^2 + y^3)/4! +
x^6*(3*y + 15*y^2 + 25*y^3 + 15*y^4 + 3*y^5)/6! +
x^8*(17*y + 119*y^2 + 329*y^3 + 455*y^4 + 329*y^5 + 119*y^6 + 17*y^7)/8! +
x^10*(155*y + 1395*y^2 + 5325*y^3 + 11235*y^4 + 14301*y^5 + 11235*y^6 + 5325*y^7 + 1395*y^8 + 155*y^9)/10! +
x^12*(2073*y + 22803*y^2 + 110605*y^3 + 311355*y^4 + 563013*y^5 + 683067*y^6 + 563013*y^7 + 311355*y^8 + 110605*y^9 + 22803*y^10 + 2073*y^11)/12! +...
where A(x,y) = cos((x - x*y)/2) / cos((x + x*y)/2).
This triangle of coefficients of x^(2*n)*y^k/(2*n)!, k=0..2*n, begins:
[1];
[0, 1, 0];
[0, 1, 3, 1, 0];
[0, 3, 15, 25, 15, 3, 0];
[0, 17, 119, 329, 455, 329, 119, 17, 0];
[0, 155, 1395, 5325, 11235, 14301, 11235, 5325, 1395, 155, 0];
[0, 2073, 22803, 110605, 311355, 563013, 683067, 563013, 311355, 110605, 22803, 2073, 0];
[0, 38227, 496951, 2918825, 10241231, 23904881, 39102063, 45956625, 39102063, 23904881, 10241231, 2918825, 496951, 38227, 0];
[0, 929569, 13943535, 96075665, 403075855, 1150348017, 2362479119, 3600524785, 4136759055, 3600524785, 2362479119, 1150348017, 403075855, 96075665, 13943535, 929569, 0]; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1088, of flattened triangle for rows 0..32.

Cf. A000364, A110501, A272482, A210657, A000281, A272158, A272467.

{T(n,k) = my(X=x+x*O(x^(2*n))); (2*n)!*polcoeff(polcoeff( cos((X-x*y)/2)/cos((X+x*y)/2), 2*n,x), k,y)}
for(n=0,10, for(k=0,2*n, print1(T(n,k),", "));print(""))

E.g.f.: A(x,y) = (cos(x) + cos(x*y)) / (1 + cos(x + x*y)).
E.g.f.: A(x,y) = (sin(x) + sin(x*y)) / sin(x + x*y).
E.g.f.: A(x,y) = (exp(i*x) + exp(i*x*y)) / (1 + exp(i*(x + x*y))), where i^2 = -1.
O.g.f.: 1/(1 - 1*y*x/(1 - (1+y)^2*x/(1 - (1+2*y)*(2+1*y)*x/(1 - (2+2*y)^2*x/(1 - (2+3*y)*(3+2*y)*x/(1 - (3+3*y)^2*x/(1 - (3+4*y)*(4+3*y)*x/(1 - (4+4*y)^2*x/(1 - (4+5*y)*(5+4*y)*x/(1 - (5+5*y)^2*x/(1 - ...))))))))))), a continued fraction.

A273032 E.g.f.: (sin(2x) + sin(5x)) / sin(7*x).

Original entry on oeis.org

1, 10, 590, 87730, 24386030, 10896056050, 7140660673070, 6452172716731570, 7688003030273049710, 11679689713099591922290, 22034907735675944799243950, 50541665200040978421599836210, 138511221399376147951707017623790, 446986750662532432703671725548281330, 1677694112006573410256120810193681597230, 7246501185695514998554969680297128881865650

Offset: 0

Paul D. Hanna, May 13 2016

nonn

			E.g.f.: A(x) = 1 + 10*x^2/2! + 590*x^4/4! + 87730*x^6/6! + 24386030*x^8/8! + 10896056050*x^10/10! + 7140660673070*x^12/12! +...
such that A(x) = (sin(2*x) + sin(5*x)) / sin(7*x).
O.g.f.: F(x) = 1 + 10*x + 590*x^2 + 87730*x^3 + 24386030*x^4 + 10896056050*x^5 + 7140660673070*x^6 + 6452172716731570*x^7 +...
such that the o.g.f. can be expressed as the continued fraction:
F(x) = 1/(1 - 2*5*x/(1 - 7^2*x/(1 - 9*12*x/(1 - 14^2*x/(1 - 16*19*x/(1 - 21^2*x/(1 - 23*26*x/(1 - 28^2*x/(1 - 30*33*x/(1 - 35^2*x/(1 - 37*40*x/(1 - ...)))))))))))).

Cf. A272158, A272467, A273031, A273033, A156193.

With[{nn=30},Take[CoefficientList[Series[(Sin[2x]+Sin[5x])/Sin[7x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Jul 20 2018 *)

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (sin(2*X) + sin(5*X))/sin(7*X), 2*n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (cos(2*X) + cos(5*X))/(1 + cos(7*X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (exp(2*I*X) + exp(5*I*X))/(1 + exp(7*I*X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: cos(3*x/2) / cos(7*x/2).
E.g.f.: (cos(2*x) + cos(5*x)) / (1 + cos(7*x)).
E.g.f.: (exp(2*i*x) + exp(5*i*x)) / (1 + exp(7*i*x)), where i^2 = -1.
E.g.f.: exp(2*i*x)/(1 + exp(7*i*x)) + exp(-2*i*x)/(1 + exp(-7*i*x)), where i^2 = -1.
O.g.f.: 1/(1 - 2*5*x/(1 - 7^2*x/(1 - 9*12*x/(1 - 14^2*x/(1 - ... - (7*n+2)*(7*n+5)*x/(1 - (7*n+7)^2*x/(1 - ...))))))), a continued fraction.
a(n) ~ (2*n)! * 4*cos(3*Pi/14) * 7^(2*n) / Pi^(2*n+1). - Vaclav Kotesovec, May 14 2016

A273033 E.g.f.: (sin(3x) + sin(4x)) / sin(7*x).

Original entry on oeis.org

1, 12, 732, 109332, 30406812, 13587056052, 8904250650492, 8045727017033172, 9586782871360007772, 14564334832981893064692, 27477080512619965247054652, 63024425641459625896776174612, 172720667970739808701108304367132, 557383361208023769780400587942586932, 2092050338949043346342979863638489321212, 9036239176876728629700436615577988154925652

Offset: 0

Paul D. Hanna, May 13 2016

nonn

			E.g.f.: A(x) = 1 + 12*x^2/2! + 732*x^4/4! + 109332*x^6/6! + 30406812*x^8/8! + 13587056052*x^10/10! + 8904250650492*x^12/12! +...
such that A(x) = (sin(3*x) + sin(4*x)) / sin(7*x).
O.g.f.: F(x) = 1 + 12*x + 732*x^2 + 109332*x^3 + 30406812*x^4 + 13587056052*x^5 + 8904250650492*x^6 + 8045727017033172*x^7 +...
such that the o.g.f. can be expressed as the continued fraction:
F(x) = 1/(1 - 3*4*x/(1 - 7^2*x/(1 - 10*11*x/(1 - 14^2*x/(1 - 17*18*x/(1 - 21^2*x/(1 - 24*25*x/(1 - 28^2*x/(1 - 31*32*x/(1 - 35^2*x/(1 - 38*39*x/(1 - ...)))))))))))).

P. Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)

Cf. A272158, A272467, A273031, A273032, A156196.

With[{nn=40},Take[CoefficientList[Series[(Sin[3x]+Sin[4x])/Sin[7x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Sep 23 2019 *)

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (sin(3*X) + sin(4*X))/sin(7*X), 2*n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (cos(3*X) + cos(4*X))/(1 + cos(7*X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1, X=x+x*O(x^(2*n+1))); (2*n)! * polcoeff( (exp(3*I*X) + exp(4*I*X))/(1 + exp(7*I*X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: cos(x/2) / cos(7*x/2).
E.g.f.: (cos(3*x) + cos(4*x)) / (1 + cos(7*x)).
E.g.f.: (exp(3*i*x) + exp(4*i*x)) / (1 + exp(7*i*x)), where i^2 = -1.
E.g.f.: exp(3*i*x)/(1 + exp(7*i*x)) + exp(-3*i*x)/(1 + exp(-7*i*x)), where i^2 = -1.
O.g.f.: 1/(1 - 3*4*x/(1 - 7^2*x/(1 - 10*11*x/(1 - 14^2*x/(1 - ... - (7*n+3)*(7*n+4)*x/(1 - (7*n+7)^2*x/(1 - ...))))))), a continued fraction.
a(n) ~ (2*n)! * 4*cos(Pi/14) * 7^(2*n) / Pi^(2*n+1). - Vaclav Kotesovec, May 14 2016
From Peter Bala, May 13 2017: (Start)
G.f.: 1/(1 + 9*x - 21*x/(1 - 28*x/(1 + 9*x - 140*x/(1 - 154*x/(1 + 9*x - ... - 7*n*(7*n-4)*x/(1 - 7*n*(7*n-3)*x/(1 + 9*x - ...
G.f.: 1/(1 + 16*x - 28*x/(1 - 21*x/(1 + 16*x - 154*x/(1 - 140*x/(1 + 16*x - ... - 7*n*(7*n-3)*x/(1 - 7*n*(7*n-4)*x/(1 + 16*x - .... (End)

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A272158 Expansion of e.g.f.: (sin(x) + sin(4*x)) / sin(5*x), even-indexed terms only.

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A273031 Expansion of e.g.f.: (sin(x) + sin(6*x)) / sin(7*x), even-indexed terms only.

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A272468 E.g.f.: ( (sin(2*x) + sin(3*x)) / sin(5*x) )^(1/6).

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A272481 E.g.f. A(x,y) = cos((x - x*y)/2) / cos((x + x*y)/2) represented as a triangle, read by rows, where row n lists of coefficients T(n,k) of x^(2*n)*y^k/n! in A(x,y), for k=0..2*n.

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A273032 E.g.f.: (sin(2*x) + sin(5*x)) / sin(7*x).

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A273033 E.g.f.: (sin(3*x) + sin(4*x)) / sin(7*x).

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A272158 Expansion of e.g.f.: (sin(x) + sin(4x)) / sin(5x), even-indexed terms only.

A273031 Expansion of e.g.f.: (sin(x) + sin(6x)) / sin(7x), even-indexed terms only.

A272468 E.g.f.: ( (sin(2x) + sin(3x)) / sin(5*x) )^(1/6).

A272481 E.g.f. A(x,y) = cos((x - xy)/2) / cos((x + xy)/2) represented as a triangle, read by rows, where row n lists of coefficients T(n,k) of x^(2n)y^k/n! in A(x,y), for k=0..2*n.

A273032 E.g.f.: (sin(2x) + sin(5x)) / sin(7*x).

A273033 E.g.f.: (sin(3x) + sin(4x)) / sin(7*x).