A000281 -id:A000281 - cp's OEIS frontend

A000825 Expansion of cos x (1 + sin x ) /cos 2x.

1, 1, 3, 8, 57, 256, 2763, 17408, 250737, 2031616, 36581523, 362283008, 7828053417, 91620376576, 2309644635483, 31191159799808, 898621108880097, 13753735117275136, 445777636063460643, 7625476699018231808

Offset: 0

N. J. A. Sloane

nonn

R. J. Mathar, Table of n, a(n) for n = 0..200

Cf. A000828, A000831.

Bisections are A000281 and (1/2) * A012393.

With[{nn=20},CoefficientList[Series[Cos[x] (1+Sin[x])/Cos[2x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 08 2013 *)

a(n) ~ n! * (sqrt(2) + 1 + (sqrt(2)-1)*(-1)^n) * 4^n / Pi^(n+1). - Vaclav Kotesovec, Jun 01 2015

A156134 Q_2n(sqrt(2)) (see A104035).

Original entry on oeis.org

1, 5, 157, 12425, 1836697, 436366445, 152053957237, 73053601590065, 46283414838553777, 37386890114969267285, 37503815980582784378317, 45739346519434253222582105, 66650214918099514832427062857, 114363498315755726948758209518525, 228234739109951323288351261455519397

Offset: 0

N. J. A. Sloane, Nov 06 2009

G. C. Greubel, Table of n, a(n) for n = 0..207

Cf. A012494, A001209, A000364, A000281, A002437.

Cf. other sequences with a g.f. of the form cos(x)/(1 - k*sin^2(x)): A012494 (k=-1), A001209 (k=1/2), A000364(k=1), A000281 (k=2), A002437 (k=4).

with(gfun):
series(cos(x)/(1-3*sin(x)^2), x, 30):
L := seriestolist(%):
seq(op(2*i-1,L)*(2*i-2)!, i = 1..floor((1/2)*nops(L)));
# Peter Bala, Feb 06 2017

With[{nmax = 50}, CoefficientList[Series[Cos[x]/(1 - 3*Sin[x]^2), {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; ;; 2]] (* G. C. Greubel, Mar 29 2018 *)

x='x+O('x^50); v=Vec(serlaplace(cos(x)/(1 - 3*sin(x)^2))); vector(#v\2,n,v[2*n-1]) \\ G. C. Greubel, Mar 29 2018

G.f. cos(x)/(1 - 3*sin(x)^2) = 1 + 5*x^2/2! + 157*x^4/4! + 12425*x^6/6! + ... - Peter Bala, Feb 06 2017

A098432 Coefficients of polynomials S(n,x) related to Springer numbers.

Original entry on oeis.org

1, 8, 7, 128, 304, 177, 3072, 13952, 21080, 10199, 98304, 724992, 2016000, 2441056, 1051745, 3932160, 42762240, 187643904, 407505664, 428605352, 169913511, 188743680, 2839019520, 17974591488, 60428242944, 111985428352

Offset: 0

Ralf Stephan, Sep 07 2004

			S(0,x) = 1,
S(1,x) = 8*x + 7,
S(2,x) = 128*x^2 + 304*x + 177,
S(3,x) = 3072*x^3 + 13952*x^2 + 21080*x + 10199.

A. Randrianarivony and J. Zeng, Une famille des polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Cf. A001586. S(n, 1/2) = A000464(n+1), S(n, -1/2) = A000281(n).
Leading coefficients are A051189. Constant terms are in A098433.
Cf. A001586. S(n, 1/2) = A000464(n), S(n, -1/2) = A000281(n).

S(n,x)=if(n<1,1,(2*x+2)*(2*x+4)*S(n-1,x+2)-(2*x+1)^2*S(n-1,x))

Recurrence: S(0, x)=1, S(n, x)=(2x+2)(2x+4)S(n-1, x+2)-(2x+1)^2S(n-1, x).

G.f.: Sum[n>=0, S(n, x)t^n] = 1/(1+t-4*2(x+1)t/(1-4*2(x+2)t/(1+t-4*4(x+3)t/(1-4+4(x+4)t/...)))).

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.

A151775 Triangle read by rows: T(n,k) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.

Original entry on oeis.org

1, 1, 2, 5, 28, 24, 61, 662, 1320, 720, 1385, 24568, 83664, 100800, 40320, 50521, 1326122, 6749040, 13335840, 11491200, 3628800, 2702765, 98329108, 692699304, 1979524800, 2739623040, 1836172800, 479001600, 199360981, 9596075582

Offset: 0

N. J. A. Sloane, Jun 24 2009, at the suggestion of Alexander R. Povolotsky

From Emeric Deutsch, Jun 27 2009: (Start)
T(n,0) = A000364(n), the Euler (or secant) numbers.
Sum of entries in row n = A000281(n).
(End)

			Triangle begins:
      1;
      1,       2;
      5,      28,      24;
     61,     662,    1320,      720;
   1385,   24568,   83664,   100800,    40320;
  50521, 1326122, 6749040, 13335840, 11491200, 3628800;

Alois P. Heinz, Rows n = 0..100, flattened
C. Radoux, The Hankel Determinant of Exponential Polynomials: A Very Short Proof and a New Result Concerning Euler Numbers, Amer. Math. Monthly, 109 (2002), 277-278.

A subtriangle of A008294.

Cf. A000364, A000281. [Emeric Deutsch, Jun 27 2009]

A151775 := proc(n,k) if n= 0 then 1 ; else taylor( (tan(x))^(2*k)/cos(x),x=0,2*n+1) ; diff(%,x$(2*n)) ; coeftayl(%,x=0,0) ; fi; end: for n from 0 to 10 do for k from 0 to n do printf("%d ", A151775(n,k)) ; od: printf("\n") ; od: # R. J. Mathar, Jun 24 2009
T := proc (n, k) if n = 0 and k = 0 then 1 elif n = 0 then 0 else simplify(subs(x = 0, diff(tan(x)^(2*k)/cos(x), `$`(x, 2*n)))) end if end proc: for n from 0 to 7 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form; Emeric Deutsch, Jun 27 2009
# alternative Maple program:
T:= (n, k)-> (2*n)!*coeff(series(tan(x)^(2*k)/cos(x), x, 2*n+1), x, 2*n):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Aug 06 2017

T[n_, k_] := (2n)! SeriesCoefficient[Tan[x]^(2k)/Cos[x], {x, 0, 2n}];
T[0, 0] = 1;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 10 2019, after Alois P. Heinz *)

More values from R. J. Mathar and Emeric Deutsch, Jun 24 2009

A262143 Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} c(n,i)*x^i/i ) for n >= 1, where c(n,k) is Shanks' array of generalized Euler and class numbers.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 8, 33, 23, 1, 16, 208, 1011, 371, 1, 30, 768, 14336, 65985, 10515, 1, 46, 2211, 94208, 2091520, 7536099, 461869, 1, 64, 5043, 412860, 24313856, 535261184, 1329205857, 28969177, 1, 96, 9984, 1361948, 164276421, 11025776640, 211966861312, 334169853267, 2454072147

Offset: 1

Peter Bala, Sep 13 2015

Shanks' array c(n,k) n >= 1, k >= 0, is A235605.
We conjecture that the entries of the present array are all integers. More generally, we conjecture that for r = 0,1,2,... and for each n >= 1, the expansion of exp( Sum_{i >= 1} c(n,i + r)*x^i/i ) has integer coefficients. The case n = 1 was conjectured by Hanna in A255895.
For the similarly defined array associated with Shanks' d(n,k) array see A262144.

			The square array begins (row indexing n starts at 1)
1  1    3      23        371         10515           461869 ..
1  3   33    1011      65985       7536099       1329205857 ..
1  8  208   14336    2091520     535261184     211966861312 ..
1 16  768   94208   24313856   11025776640    7748875976704 ..
1 30 2211  412860  164276421  115699670490  126686112278631 ..
1 46 5043 1361948  778121381  787337024970 1239870854518999 ..
1 64 9984 3716096 2891509760 3978693525504 8522989918683136 ..
...
Array as a triangle
1
1  1
1  3    3
1  8   33      23
1 16  208    1011      371
1 30  768   14336    65985        10515
1 46 2211   94208  2091520      7536099       461869
1 64 5043  412860  24313856   535261184   1329205857 28969177
1 96 9984 1361948 164276421 11025776640 211966861312 ...
...

P. Bala, Notes on logarithmic differentiation, the binomial transform and series reversion
William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia, The generating function for the Dirichlet series Lm(s) Mathematics of Computation, Vol. 81, No. 278, April 2012.
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Cf. A000233 (column 1), A000364 (c(1,n)), A000281 (c(2,n)), A000436 (c(3,n)), A000490 (c(4,n)), A000187 (c(5,n)), A000192 (c(6,n)), A064068 (c(7,n)), A235605, A235606, A255881, A255895, A262144, A262145.

cp's OEIS Frontend

A000825 Expansion of cos x (1 + sin x ) /cos 2x.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A156134 Q_2n(sqrt(2)) (see A104035).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A098432 Coefficients of polynomials S(n,x) related to Springer numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A151775 Triangle read by rows: T(n,k) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A262143 Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} c(n,i)*x^i/i ) for n >= 1, where c(n,k) is Shanks' array of generalized Euler and class numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.