A060627 -id:A060627 - cp's OEIS frontend

A002754 Related to coefficient of m in Jacobi elliptic function cn(z, m).

0, 0, 4, 44, 408, 3688, 33212, 298932, 2690416, 24213776, 217924020, 1961316220, 17651846024, 158866614264, 1429799528428, 12868195755908, 115813761803232, 1042323856229152, 9380914706062436, 84428232354561996, 759854091191058040

Offset: 0

N. J. A. Sloane

A. Cayley, An Elementary Treatise on Elliptic Functions. Bell, London, 1895, p. 56.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Roland Bacher and Philippe Flajolet, Pseudo-factorials, elliptic functions, and continued fractions, arXiv:0901.1379 [math.CA], 2009.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. Cayley, An Elementary Treatise on Elliptic Functions (page images), G. Bell and Sons, London, 1895, p. 56.
G. Viennot, Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi, J. Combin. Theory, A 29 (1980), 121-133.
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.
Index entries for linear recurrences with constant coefficients, signature (11,-19,9).

Cf. A060627, A014832.

[(9^n-8*n-1)/16: n in [0..25]]; // Vincenzo Librandi, Jun 29 2011

a[ n_] := If[ n < 0, 0, (-1)^n (2 n)! Coefficient[ SeriesCoefficient[ JacobiCN[x, m], {x, 0, 2 n}], m, 1]]; (* Michael Somos, Dec 27 2014 *)
LinearRecurrence[{11, -19, 9}, {0, 0, 4}, 21] (* Jean-François Alcover, Sep 21 2017 *)

{a(n) = (9^n - 8*n -1) / 16}; /* Michael Somos, Jun 27 2003 */

From Michael Somos, Jun 27 2003: (Start)
G.f.: 4*x^2/((1-x)^2*(1-9*x)).
a(n) = (9^n-8*n-1)/16. (End)
a(n+2) = 4*A014832(n+1). [Bruno Berselli, Jun 29 2011]

More terms from Paolo Dominici (pl.dm(AT)libero.it) using formulas 16.22.1 and 16.22.2 of Abramowitz and Stegun's Handbook of Mathematical Functions.

A322231 E.g.f.: C(x,k) = 1 + Integral S(x,k)D(x,k)^2 dx, such that C(x,k)^2 - S(x,k)^2 = 1, and D(x,k)^2 - k^2S(x,k)^2 = 1, as a triangle of coefficients read by rows.

Original entry on oeis.org

1, 1, 0, 1, 8, 0, 1, 88, 136, 0, 1, 816, 6240, 3968, 0, 1, 7376, 195216, 513536, 176896, 0, 1, 66424, 5352544, 39572864, 51880064, 11184128, 0, 1, 597864, 139127640, 2458228480, 8258202240, 6453433344, 951878656, 0, 1, 5380832, 3535586112, 137220256000, 994697838080, 1889844670464, 978593947648, 104932671488, 0, 1, 48427552, 88992306208, 7233820923904, 102950036177920, 398800479698944, 485265505927168, 178568645312512, 14544442556416, 0

Offset: 0

Paul D. Hanna, Dec 14 2018

Equals a row reversal of triangle A325222.
Compare to cn(x,k) = 1 - Integral sn(x,k)*dn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions (see triangle A060627).
Compare also to Michael Pawellek's generalized elliptic functions.

			E.g.f.: C(x,k) = 1 + x^2/2! + (8*k^2 + 1)*x^4/4! + (136*k^4 + 88*k^2 + 1)*x^6/6! + (3968*k^6 + 6240*k^4 + 816*k^2 + 1)*x^8/8! + (176896*k^8 + 513536*k^6 + 195216*k^4 + 7376*k^2 + 1)*x^10/10! + (11184128*k^10 + 51880064*k^8 + 39572864*k^6 + 5352544*k^4 + 66424*k^2 + 1)*x^12/12! + (951878656*k^12 + 6453433344*k^10 + 8258202240*k^8 + 2458228480*k^6 + 139127640*k^4 + 597864*k^2 + 1)*x^14/14! + ...
such that C(x,k)^2 - S(x,k)^2 = 1.
This triangle of coefficients T(n,j) of x^(2*n)*k^(2*j)/(2*n)! in e.g.f. C(x,k) begins:
1;
1, 0;
1, 8, 0;
1, 88, 136, 0;
1, 816, 6240, 3968, 0;
1, 7376, 195216, 513536, 176896, 0;
1, 66424, 5352544, 39572864, 51880064, 11184128, 0;
1, 597864, 139127640, 2458228480, 8258202240, 6453433344, 951878656, 0;
1, 5380832, 3535586112, 137220256000, 994697838080, 1889844670464, 978593947648, 104932671488, 0;
1, 48427552, 88992306208, 7233820923904, 102950036177920, 398800479698944, 485265505927168, 178568645312512, 14544442556416, 0; ...
RELATED SERIES.
The related series S(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
S(x,k) = x + (2*k^2 + 1)*x^3/3! + (16*k^4 + 28*k^2 + 1)*x^5/5! + (272*k^6 + 1032*k^4 + 270*k^2 + 1)*x^7/7! + (7936*k^8 + 52736*k^6 + 36096*k^4 + 2456*k^2 + 1)*x^9/9! + (353792*k^10 + 3646208*k^8 + 4766048*k^6 + 1035088*k^4 + 22138*k^2 + 1)*x^11/11! + (22368256*k^12 + 330545664*k^10 + 704357760*k^8 + 319830400*k^6 + 27426960*k^4 + 199284*k^2 + 1)*x^13/13! + ...
The related series D(x,k), where D(x,k)^2 - k^2*S(x,k)^2 = 1, starts
D(x,k) = 1 + k^2*x^2/2! + (5*k^4 + 4*k^2)*x^4/4! + (61*k^6 + 148*k^4 + 16*k^2)*x^6/6! + (1385*k^8 + 6744*k^6 + 2832*k^4 + 64*k^2)*x^8/8! + (50521*k^10 + 410456*k^8 + 383856*k^6 + 47936*k^4 + 256*k^2)*x^10/10! + (2702765*k^12 + 32947964*k^10 + 54480944*k^8 + 17142784*k^6 + 780544*k^4 + 1024*k^2)*x^12/12! + (199360981*k^14 + 3402510924*k^12 + 8760740640*k^10 + 5199585280*k^8 + 686711040*k^6 + 12555264*k^4 + 4096*k^2)*x^14/14! + ...

Paul D. Hanna, Table of n, a(n) for n = 0..860 terms of this triangle read by rows 0..40.

Cf. A322230 (S), A322232 (D), A001818 (row sums), A002105.

Cf. A325222 (row reversal).

N=10;
{S=x;C=1;D=1; for(i=1,2*N, S = intformal(C*D^2 +O(x^(2*N+1))); C = 1 + intformal(S*D^2); D = 1 + k^2*intformal(S*C*D));}
for(n=0,N, for(j=0,n, print1( (2*n)!*polcoeff(polcoeff(C,2*n,x),2*j,k),", ")) ;print(""))

E.g.f. C = C(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n) * k^(2*j) / (2*n)!, along with related series S = S(x,k) and D = D(x,k), satisfies:
(1a) S = Integral C*D^2 dx.
(1b) C = 1 + Integral S*D^2 dx.
(1c) D = 1 + k^2 * Integral S*C*D dx.
(2a) C^2 - S^2 = 1.
(2b) D^2 - k^2*S^2 = 1.
(3a) C + S = exp( Integral D^2 dx ).
(3b) D + k*S = exp( k * Integral C*D dx ).
(4a) S = sinh( Integral D^2 dx ).
(4b) S = sinh( k * Integral C*D dx ) / k.
(4c) C = cosh( Integral D^2 dx ).
(4d) D = cosh( k * Integral C*D dx ).
(5a) d/dx S = C*D^2.
(5b) d/dx C = S*D^2.
(5c) d/dx D = k^2 * S*C*D.
From Paul D. Hanna, Mar 31 2019, Apr 20 2019 (Start):
Given sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions, with i^2 = -1, k' = sqrt(1-k^2), then
(6a) S = -i * sn( i * Integral D dx, k),
(6b) C = cn( i * Integral D dx, k),
(6c) D = dn( i * Integral D dx, k).
(7a) S = sc( Integral D dx, k') = sn(Integral D dx, k')/cn(Integral D dx, k'),
(7b) C = nc( Integral D dx, k') = 1/cn(Integral D dx, k'),
(7c) D = dc( Integral D dx, k') = dn(Integral D dx, k')/cn(Integral D dx, k'). (End)
Row sums equal ( (2*n)!/(n!*2^n) )^2 = A001818(n), the squares of the odd double factorials.
Diagonal T(n+1,n) = 2^n*A002105(n+1), for n>=0, where A002105 gives the reduced tangent numbers.

A370543 Expansion of the Jacobi elliptic function cn(x,k) at k = 2 (even powers only).

Original entry on oeis.org

1, -1, 17, -433, 20321, -1584289, 179967473, -28151779537, 5812048858049, -1529741412486721, 499975227342256337, -198676311845589783793, 94327947921149101192481, -52736138158762405338195169, 34291374178966525773142501553, -25660133983889999165774819970577

Offset: 0

Paul D. Hanna, Mar 25 2024

sign

			E.g.f.: C(x) = 1 - x^2/2! + 17*x^4/4! - 433*x^6/6! + 20321*x^8/8! - 1584289*x^10/10! + 179967473*x^12/12! - 28151779537*x^14/14! + ...
where C(x) = cn(x,2).

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.

Paul D. Hanna, Table of n, a(n) for n = 0..301

Cf. A028296 (cn(x,1)), A060627 (cn(x,k)).

Cf. A370542 (sn(x,2)), A370544 (dn(x,2)), A249282.

# a(n) = (2*n)! * [x^(2*n)] cn(x, 2).
cn_list := proc(k, len) local n; seq((2*n)!*coeff(series(JacobiCN(z, k), z,
2*len + 2), z, 2*n), n = 0..len) end:
cn_list(2, 15);  # Peter Luschny, Mar 25 2024

nmax = 20;
DeleteCases[CoefficientList[JacobiCN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax])! (* Jean-François Alcover, Mar 28 2024 *)

/* C(x) = Jacobi Elliptic Function cn(x,k) at k = 2: */
{a(n) = my(k=2,C=1,S=x,D=1); for(i=1,n,
S = intformal(C*D + x*O(x^(2*n+1)));
C = 1 - intformal(S*D);
D = 1 - k^2*intformal(S*C)); (2*n)!*polcoeff(C,2*n)}
for(n=0,20,print1(a(n),", "))

a(n) = (-1)^n * Sum_{k=0..n-1} A060627(n,k)*4^k for n >= 1, with a(0) = 1.
E.g.f. C(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! satisfies the following formulas, where sn, cn, and dn are Jacobi elliptic functions.
(1) C(x) = cn(x,k) at k = 2.
(2.a) C(x) = dn(2*x, 1/2).
(2.b) C(x) = (4 - 2*sn(x,1/2)^2 + sn(x,1/2)^4) / (4 - sn(x,1/2)^4).
(3) C(x) = 1 - Integral sqrt(1 - C(x)^2) * sqrt(4*C(x)^2 - 3) dx.
(4) C(x) = cos( Integral sqrt(4*C(x)^2 - 3) dx ).
(5.a) C(x) = sqrt(1 - sn(x,2)^2).
(5.b) C(x) = sqrt(3 + dn(x,2)^2) / 2.
O.g.f.: 1/(1 + x/(1 + 4*2^2*x/(1 + 3^2*x/(1 + 4*4^2*x/(1 + 5^2*x/(1 + 4*6^2*x/(1 + 7^2*x/(1 + ...)))))))) = 1 - x + 17*x^2 - 433*x^3 + 20321*x^4 - 1584289*x^5 + ... (continued fraction, see Wall, 94.18, p. 374). - [See formula in A060627 by Peter Bala, Apr 25 2017].
a(n) ~ (-1)^n * 2^(4*n+2) * agm(1,2)^(2*n+1) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)), where agm(1,2) = A068521 is the arithmetic-geometric mean. - Vaclav Kotesovec, Mar 28 2024

A181612 Triangle T(n,m) of the coefficients JacobiDC(x,y) = sum_{n>=0} sum_{m=0..n} (-1)^m* T(n,m) x^(2n) y^(2m)/(2*n)!.

Original entry on oeis.org

1, 1, 1, 5, 6, 1, 61, 107, 47, 1, 1385, 3116, 2142, 412, 1, 50521, 138933, 130250, 45530, 3693, 1, 2702765, 8783986, 10430983, 5353260, 1036715, 33218, 1, 199360981, 747603679, 1074680289, 728130163, 226132303

Offset: 0

R. J. Mathar, Jan 30 2011

			The triangle starts in row n=0 as
1;
1, 1;
5, 6, 1;
61, 107, 47, 1;
1385, 3116, 2142, 412, 1;
50521, 138933, 130250, 45530, 3693, 1;

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover. Section 16.22.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052v2 [math.CA].
NIST Digital Library of Mathematical Functions, NIST Handbook of Mathematical Functions, Chapter 22.
Eric W. Weisstein, Jacobi Elliptic Functions.

Cf. A060627, A060628, A181613, A000364 (apparently the column m=0).

A181612 := proc(n,m) JacobiDC(z,k) ; coeftayl(%,z=0,2*n) ; (-1)^m*coeftayl(%,k=0,2*m)*(2*n)! ; end proc:
seq( seq(A181612(n,m),m=0..n),n=0..10) ;

nmax = 8; se = Series[JacobiDC[x, y], {x, 0, 2*nmax}]; t[n_, m_] := Coefficient[se, x, 2*n]*(2*n)! // Coefficient[#, y, m]& // Abs; Table[t[n, m], {n, 0, nmax}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)

From Peter Bala, Aug 23 2011: (Start)
The elliptic function dc(x,k) (JacobiDC(x,k) in Maple notation) is defined as dn(x,k)/cn(x,k) where dn(x,k) and cn(x,k) are the Jacobian elliptic functions of modulus k. The Taylor expansions begin
dn(x,k) = 1-k^2*x^2/2!+k^2*(4+k^2)*x^4/4!-k^2*(16+44*k^2+k^4)*x^6/6!+...
cn(x,k) = 1-x^2/2!+(1+4*k^2)*x^4/4!-(1+44*k^2+16*k^4)*x^6/6!+... and hence
dc(x,k) = 1+(1-k^2)*x^2/2!+(5-6*k^2+k^4)*x^4/4!+(61-107*k^2+47*k^4-k^6)*x^6/6!+....
The coefficients for cn(x,k) are in A060627. The coefficients of dn(x,k) may be obtained by row reversal of A060627.
The expansion for dc(x,k) can also be obtained directly from that of dn(x,k) since by Jacobi's imaginary transformations we have dc(x,k) = dn(i*x,k'), where the complementary modulus k' is given by k' = sqrt(1-k^2).
By Jacobi's real transformation the reciprocal of dc(x,k) is given by 1/dc(x,k) = dc(x*k,1/k).
The row polynomials of this table can be calculated using nested derivatives as follows (see [Dominici, Theorem 4.1 and Example 4.5]):
Let f(x) = sqrt(1-(1-k^2)*sin^2(x)). Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0.
See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x).
Then the coefficient of x^(2*n)/(2*n)! in the expansion of dc(x,k) is given by (-1)^n*D^(2*n)[f](0).
(End)

A181613 Triangle T(n,m) of the coefficients JacobiNC(x,y) = sum_{n>0} sum_{m=0..n-1} (-1)^m* T(n,m) x^(2n) y^(2m)/(2*n)!.

Original entry on oeis.org

1, 5, 4, 61, 76, 16, 1385, 2424, 1104, 64, 50521, 113672, 79728, 16832, 256, 2702765, 7432604, 7052528, 2586112, 264448, 1024, 199360981, 647923188, 775638816, 408850432, 85975296, 4205568, 4096, 19391512145, 72718170544, 105138354912, 72490884224, 23551644928, 2939602944, 67162112, 16384

Offset: 1

R. J. Mathar, Jan 30 2011

The column m=0 is apparently A000364.

			The triangle starts in row n=1 as:
1;
5, 4;
61, 76, 16;
1385, 2424, 1104, 64;
50521, 113672, 79728, 16832, 256;

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover. Section 16.22.
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
P. Bala, A triangle for calculating A181613
D. Dominici, Nested derivatives: A simple method for Computing series expansions of inverse functions, arXiv:math/0501052v2 [math.CA].
NIST Digital Library of Mathematical Functions, NIST Handbook of Mathematical Functions, Chapter 22.
Eric W. Weisstein, Jacobi Elliptic Functions.

Cf. A060627, A060628, A181612.

A181613 := proc(n,m) JacobiNC(z,k) ; coeftayl(%,z=0,2*n) ; (-1)^m*coeftayl(%,k=0,2*m)*(2*n)! ; end proc:
seq( seq(A181613(n,m),m=0..n-1),n=1..10) ;

nmax = 8; se = Series[JacobiNC[x, y], {x, 0, 2*nmax}]; t[n_, m_] := Coefficient[se, x, 2*n]*(2*n)! // Coefficient[#, y, m]& // Abs; Table[t[n, m], {n, 1, nmax}, {m, 0, n-1}] // Flatten (* Jean-François Alcover, Jan 10 2014 *)

From Peter Bala, Aug 23 2011: (Start)
The Taylor expansion of the Jacobian elliptic function cn(u,k) begins
cn(u,k) = 1-u^2/2!+(1+4*k^2)*u^4/4!-(1+44*k^2+16*k^4)*u^6/6!+... - see A060627.
The Taylor expansion of the reciprocal function 1/cn(u,k) can be obtained directly from this by using Jacobi's imaginary transformation
1/cn(u,k) = cn(i*u,sqrt(1-k^2)) [Abramowitz and Stegun, 16.20] to yield
1/cn(u,k) = 1+u^2/2!+(5-4*k^2)*u^4/4!+(61-76*k^2+16*k^4)*u^6/6!+....
The coefficient polynomials R(2*n,k) of this expansion can be calculated as follows (apply [Dominici, Theorem 4.1]):
Let f(x) = sqrt(k^2-cos^2(x)). Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0. Then R(2*n,k) = D^(2*n)[f](0).
See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x).
(End)
G.f. 1/(1 - x/(1 - 2^2*(1 - k^2)*x/(1 - 3^2*x/(1 - 4^2*(1 - k^2)*x/(1 - 5^2*x/(1 - ...)))))) = 1 + x + (5 - 4*k^2)*x^2 + (61 - 76*k^2 + 16*k^4)*x^3 + ... (see Wall, 94.19, p. 374).

A325221 E.g.f.: C(x,k) = cn( i * Integral C(x,k) dx, k), where C(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2n)k^(2j)/(2n)!, as a triangle of coefficients T(n,j) read by rows.

Original entry on oeis.org

1, 1, 0, 5, 4, 0, 61, 148, 16, 0, 1385, 6744, 2832, 64, 0, 50521, 410456, 383856, 47936, 256, 0, 2702765, 32947964, 54480944, 17142784, 780544, 1024, 0, 199360981, 3402510924, 8760740640, 5199585280, 686711040, 12555264, 4096, 0, 19391512145, 441239943664, 1632067372896, 1569971730560, 419867864320, 26090711040, 201199616, 16384, 0, 2404879675441, 70347660061552, 353538702361888, 502094919789184, 227204970315520, 30892394850304, 965223559168, 3220652032, 65536, 0

Offset: 0

Paul D. Hanna, Apr 13 2019

Equals a row reversal of triangle A322232.

Compare to cn(x,k) = 1 - Integral sn(x,k)*dn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions (see triangle A060627).

			E.g.f.: C(x,k) = 1 + x^2/2! + (5 + 4*k^2)*x^4/4! + (61 + 148*k^2 + 16*k^4)*x^6/6! + (1385 + 6744*k^2 + 2832*k^4 + 64*k^6)*x^8/8! + (50521 + 410456*k^2 + 383856*k^4 + 47936*k^6 + 256*k^8)*x^10/10! + (2702765 + 32947964*k^2 + 54480944*k^4 + 17142784*k^6 + 780544*k^8 + 1024*k^10)*x^12/12! + (199360981 + 3402510924*k^2 + 8760740640*k^4 + 5199585280*k^6 + 686711040*k^8 + 12555264*k^10 + 4096*k^12)*x^14/14! + ...
such that C(x,k) = cn( i * Integral C(x,k) dx, k).
This triangle of coefficients T(n,j) of x^(2*n)*k^(2*j)/(2*n)! in e.g.f. C(x,k) begins:
1;
1, 0;
5, 4, 0;
61, 148, 16, 0;
1385, 6744, 2832, 64, 0;
50521, 410456, 383856, 47936, 256, 0;
2702765, 32947964, 54480944, 17142784, 780544, 1024, 0;
199360981, 3402510924, 8760740640, 5199585280, 686711040, 12555264, 4096, 0;
19391512145, 441239943664, 1632067372896, 1569971730560, 419867864320, 26090711040, 201199616, 16384, 0;
2404879675441, 70347660061552, 353538702361888, 502094919789184, 227204970315520, 30892394850304, 965223559168, 3220652032, 65536, 0; ...
RELATED SERIES.
The related series S(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
S(x,k) = x + (2 + 1*k^2)*x^3/3! + (16 + 28*k^2 + 1*k^4)*x^5/5! + (272 + 1032*k^2 + 270*k^4 + 1*k^6)*x^7/7! + (7936 + 52736*k^2 + 36096*k^4 + 2456*k^6 + 1*k^8)*x^9/9! + (353792 + 3646208*k^2 + 4766048*k^4 + 1035088*k^6 + 22138*k^8 + 1*k^10)*x^11/11! + (22368256 + 330545664*k^2 + 704357760*k^4 + 319830400*k^6 + 27426960*k^8 + 199284*k^10 + 1*k^12)*x^13/13! + (1903757312 + 38188155904*k^2 + 120536980224*k^4 + 93989648000*k^6 + 18598875760*k^8 + 702812568*k^10 + 1793606*k^12 + 1*k^14)*x^15/15! + ...
The related series D(x,k), where D(x,k)^2 - k^2*S(x,k)^2 = 1, starts
D(x,k) = 1 + k^2*x^2/2! + (8*k^2 + 1*k^4)*x^4/4! + (136*k^2 + 88*k^4 + 1*k^6)*x^6/6! + (3968*k^2 + 6240*k^4 + 816*k^6 + 1*k^8)*x^8/8! + (176896*k^2 + 513536*k^4 + 195216*k^6 + 7376*k^8 + 1*k^10)*x^10/10! + (11184128*k^2 + 51880064*k^4 + 39572864*k^6 + 5352544*k^8 + 66424*k^10 + 1*k^12)*x^12/12! + (951878656*k^2 + 6453433344*k^4 + 8258202240*k^6 + 2458228480*k^8 + 139127640*k^10 + 597864*k^12 + 1*k^14)*x^14/14! + ...

Cf. A325220 (S), A325222(D).

Cf. A322232 (row reversal).

N=10;
{S=x; C=1; D=1; for(i=1, 2*N, S = intformal(C^2*D +O(x^(2*N+1))); C = 1 + intformal(S*C*D); D = 1 + k^2*intformal(S*C^2)); }
{T(n,j) = (2*n)!*polcoeff(polcoeff(C, 2*n, x), 2*j, k)}
for(n=0, N, for(j=0, n, print1( T(n,j), ", ")) ; print(""))

E.g.f. C = C(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n)*k^(2*j)/(2*n)!, along with related series S = S(x,k) and D = D(x,k), satisfies:
(1a) S = Integral C^2*D dx.
(1b) C = 1 + Integral S*C*D dx.
(1c) D = 1 + k^2 * Integral S*C^2 dx.
(2a) C^2 - S^2 = 1.
(2b) D^2 - k^2*S^2 = 1.
(3a) C + S = exp( Integral C*D dx ).
(3b) D + k*S = exp( k * Integral C^2 dx ).
(4a) S = sinh( Integral C*D dx ).
(4b) S = sinh( k * Integral C^2 dx ) / k.
(4c) C = cosh( Integral C*D dx ).
(4d) D = cosh( k * Integral C^2 dx ).
(5a) d/dx S = C^2*D.
(5b) d/dx C = S*C*D.
(5c) d/dx D = k^2 * S*C^2.
Given sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions, with i^2 = -1, k' = sqrt(1-k^2), then
(6a) S = -i * sn( i * Integral C dx, k),
(6b) C = cn( i * Integral C dx, k),
(6c) D = dn( i * Integral C dx, k).
(7a) S = sc( Integral C dx, k') = sn(Integral C dx, k')/cn(Integral C dx, k'),
(7b) C = nc( Integral C dx, k') = 1/cn(Integral C dx, k'),
(7c) D = dc( Integral C dx, k') = dn(Integral C dx, k')/cn(Integral C dx, k').
Row sums equal ( (2*n)!/(n!*2^n) )^2 = A001818(n), the squares of the odd double factorials.
Column T(n,0) = A000364(n), for n>=0, where A000364 is the secant numbers.

A370544 Expansion of the Jacobi elliptic function dn(x,k) at k = 2 (even powers only).

Original entry on oeis.org

1, -4, 32, -832, 41216, -3168256, 359518208, -56319950848, 11624409595904, -3059387770077184, 999955757611876352, -397353151288859164672, 188655750511199441125376, -105472284295853235792510976, 68582751548430569936978444288, -51320267059211655419226235076608

Offset: 0

Paul D. Hanna, Mar 25 2024

sign

			E.g.f.: D(x) = 1 - 4*x^2/2! + 32*x^4/4! - 832*x^6/6! + 41216*x^8/8! - 3168256*x^10/10! + 359518208*x^12/12! - 56319950848*x^14/14! + ...
where D(x) = dn(x,2).

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.

Paul D. Hanna, Table of n, a(n) for n = 0..301

Cf. A028296 (dn(x,1)), A060627 (cn(x,k)).

Cf. A370542 (sn(x,2)), A370543 (cn(x,2)), A249282.

# a(n) = (2*n)! * [x^(2*n)] dn(x, 2).
dn_list := proc(k, len) local n; seq((2*n)!*coeff(series(JacobiDN(z, k), z,
2*len + 2), z, 2*n), n = 0..len) end:
dn_list(2, 15);  # Peter Luschny, Mar 25 2024

nmax = 20;
DeleteCases[CoefficientList[JacobiDN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax])! (* Jean-François Alcover, Mar 28 2024 *)

/* D(x) = Jacobi Elliptic Function dn(x,k) at k = 2: */
{a(n) = my(k=2, C=1,S=x,D=1); for(i=1,n,
S = intformal(C*D + x*O(x^(2*n+1)));
C = 1 - intformal(S*D);
D = 1 - k^2*intformal(S*C)); (2*n)!*polcoeff(D,2*n)}
for(n=0,20,print1(a(n),", "))

a(n) = (-1)^n * Sum_{k=0..n-1} A060627(n,k)*4^(n-k) for n >= 1, with a(0) = 1.
E.g.f. D(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! satisfies the following formulas, where sn, cn, and dn are Jacobi elliptic functions.
(1) D(x) = dn(x,k) at k = 2.
(2.a) D(x) = cn(2*x, 1/2).
(2.b) D(x) = (4 - 8*sn(x,1/2)^2 + sn(x,1/2)^4) / (4 - sn(x,1/2)^4).
(3) D(x) = 1 - Integral sqrt(1 - D(x)^2) * sqrt(3 + D(x)^2) dx.
(4) D(x) = cos( Integral sqrt(3 + D(x)^2) dx ).
(5.a) D(x) = sqrt(1 - 4*sn(x,2)^2).
(5.b) D(x) = sqrt(4*cn(x,2)^2 - 3).
O.g.f. 1/(1 + 4*x/(1 + 2^2*x/(1 + 4*3^2*x/(1 + 4^2*x/(1 + 4*5^2*x/(1 + 6^2*x/(1 + 4*7^2*x/(1 + ...)))))))) = 1 - 4*x + 32*x^2 - 832*x^3 + 41216*x^4 - 3168256*x^5 + ... (continued fraction, see Wall, 94.19, p. 374).
a(n) ~ (-1)^n * 2^(4*n+3) * agm(1,2)^(2*n+1) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)), where agm(1,2) = A068521 is the arithmetic-geometric mean. - Vaclav Kotesovec, Mar 28 2024

A190904 a(n) = Sum_{k=0..n-1} cos(Pik/2)binomial(n-1,k)a(n-1-k)a(k) for n > 0, a(0) = 1.

Original entry on oeis.org

1, 1, 1, 0, -3, -12, -27, 0, 441, 3024, 11529, 0, -442827, -4390848, -23444883, 0, 1636819569, 21224560896, 145703137041, 0, -16106380394643, -257991277243392, -2164638920874507, 0, 347592265948756521

Offset: 0

Peter Luschny, Jul 26 2011

sign

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
Eric W. Weisstein, Jacobi Elliptic Functions.

Cf. A060627, A060628, A104203, A159600, A193541, A193544.

A190904 := proc(n) option remember; `if`(n=0,1,add(((1-irem(k,2))*(-1)^ iquo(k,2))*binomial(n-1,k)*A190904(n-1-k)*A190904(k),k=0..n-1)) end:

a[0] = 1;
a[n_] := a[n] =
  Sum[Mod[(k+1)^3, 4, -1] Binomial[n-1, k] a[n-k-1] a[k], {k, 0, n-1}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jun 24 2019 *)

Let F(n,x) = Sum_{k=0..n-1} cos(Pi*k*x)*binomial(n-1,k)*F(n-1-k,x)* F(k,x), then
F(n, 0)   = n! = A000142(n),
F(n, 1/2) = a(n),
F(n, 1)   = 2^n*Euler_{n}(1) = A_{n}(-1) = A155585(n).
a(2*n) = A159601(n)*(-1)^floor((n-1)/2).
a(2*n+1) = A104203(2*n+1).
From Peter Bala, Aug 25 2011: (Start)
The sequence entries may be calculated as follows: Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0. The coefficients in the expansion of D^n[f](x) in powers of f(x) can be found in A145271. Then we have
a(2*n) = D^(2*n)[sqrt(1+sin^2(x))](0)
a(2*n+1) = D^(2*n)[sqrt(1-x^4)](0).
The generating function involves the Jacobian elliptic functions. Define E(u,k) := cn(i*u,k)-i*sn(i*u,k) = 1+u+u^2/2!+(1+k^2)*u^3/3!+(1+4*k^2)*u^4/4!+..., where cn(u,k) and sn(u,k) are Jacobian elliptic functions of modulus k (see A060627 and A060628). Then the e.g.f. A(u) for this sequence is
A(u) := E(u,i) = 1+u+u^2/2!-3*u^4/4!-12*u^5/5!-27*u^6/6!+....
Proof: Using well-known properties of the Jacobian elliptic functions (see for example Abramowitz and Stegun, Chapter 16) we find the generating function A(u) satisfies the differential equation
(d/du)A(u) = dn(i*u,i)*A(u) = 1/2*(A(i*u)+A(-i*u))*A(u), which leads to a recurrence for the coefficients of A(u):
a(n+1) = sum{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*a(2*k)*a(n-2*k) with a(0) = 1. This recurrence is equivalent to the defining recurrence for this sequence given above.
End proof.
The generating function A(u) satisfies 1/A(u) = A(-u).
Compare entries of this sequence with those of A104203, A159600, A193541 and A193544.
(End)

A291527 E.g.f. A(x,k) satisfies: sn(A(x,k), k) = k * sn(x,k), where sn(,) and cn(,) are Jacobi Elliptic functions.

Original entry on oeis.org

1, -1, 0, 1, 1, 4, -10, -4, 9, -1, -44, 75, 224, -299, -180, 225, 1, 408, 92, -7400, 4758, 19592, -15876, -12600, 11025, -1, -3688, -23387, 194160, 155702, -1313312, 264586, 2445840, -1289925, -1323000, 893025, 1, 33212, 804210, -3980044, -20402105, 64915224, 74573980, -279362392, -18229761, 414859500, -144802350, -196465500, 108056025, -1, -298932, -22347185, 33998224, 1349961795, -1942776004, -12484642765, 21458573952, 32679754381, -72263858940, -19224079875, 92046754800, -20560114575, -39332393100, 18261468225, 1, 2690416, 581249144, 2783246128, -71371497796, -59230867280, 1313526021896, -606679979408, -7350770598874, 7512502827344, 15289334428104, -22529210886000, -9997446759300, 25906255174800, -3292683193800, -10226422206000, 4108830350625

Offset: 1

Paul D. Hanna, Aug 25 2017

Compare to the law of sines of a spherical triangle: sin(A)/sin(a) = k.

The series reversion of e.g.f. A(x,k) wrt x equals A(k*x, 1/k) / k.

			This irregular triangle of coefficients T(n,r) in A(x,k) begins:
[1],
[-1, 0, 1],
[1, 4, -10, -4, 9],
[-1, -44, 75, 224, -299, -180, 225],
[1, 408, 92, -7400, 4758, 19592, -15876, -12600, 11025],
[-1, -3688, -23387, 194160, 155702, -1313312, 264586, 2445840, -1289925, -1323000, 893025],
[1, 33212, 804210, -3980044, -20402105, 64915224, 74573980, -279362392, -18229761, 414859500, -144802350, -196465500, 108056025],
[-1, -298932, -22347185, 33998224, 1349961795, -1942776004, -12484642765, 21458573952, 32679754381, -72263858940, -19224079875, 92046754800, -20560114575, -39332393100, 18261468225],
[1, 2690416, 581249144, 2783246128, -71371497796, -59230867280, 1313526021896, -606679979408, -7350770598874, 7512502827344, 15289334428104, -22529210886000, -9997446759300, 25906255174800, -3292683193800, -10226422206000, 4108830350625], ...
where e.g.f. A(x,k) = Sum_{n>=1, r=1..2*n-1} T(n,r) * x^(2*n-1) * k^(2*r-1) / (2*n-1)!.
E.g.f.: A(x,k) = k*x + (k^5 - k)*x^3/3! +
(9*k^9 - 4*k^7 - 10*k^5 + 4*k^3 + k)*x^5/5! +
(225*k^13 - 180*k^11 - 299*k^9 + 224*k^7 + 75*k^5 - 44*k^3 - k)*x^7/7! +
(11025*k^17 - 12600*k^15 - 15876*k^13 + 19592*k^11 + 4758*k^9 - 7400*k^7 + 92*k^5 + 408*k^3 + k)*x^9/9! +
(893025*k^21 - 1323000*k^19 - 1289925*k^17 + 2445840*k^15 + 264586*k^13 - 1313312*k^11 + 155702*k^9 + 194160*k^7 - 23387*k^5 - 3688*k^3 - k)*x^11/11! +
(108056025*k^25 - 196465500*k^23 - 144802350*k^21 + 414859500*k^19 - 18229761*k^17 - 279362392*k^15 + 74573980*k^13 + 64915224*k^11 - 20402105*k^9 - 3980044*k^7 + 804210*k^5 + 33212*k^3 + k)*x^13/13! +
(18261468225*k^29 - 39332393100*k^27 - 20560114575*k^25 + 92046754800*k^23 - 19224079875*k^21 - 72263858940*k^19 + 32679754381*k^17 + 21458573952*k^15 - 12484642765*k^13 - 1942776004*k^11 + 1349961795*k^9 + 33998224*k^7 - 22347185*k^5 - 298932*k^3 - k)*x^15/15! +...
such that
(1) sn(A(x,k), k) = k * sn(x,k),
(2) cn(A(x,k), k) = dn(x,k),
(3) dn(A(k*x,1/k)/k, k) = cn(x,k),
(4) A(k * A(x,k), 1/k) = k * x,
(5) A(A(x,1/k) / k, k) = x / k.
RELATED SERIES.
Let A^r(x,k) denote the r-th iteration of A(x,k) wrt x, then
sn( A^r(x,k), k) = k^r * sn(x,k).
For example, sn( A(A(x,k), k), k) = k^2 * sn(x,k), where
A(A(x,k), k) = k^2*x + (k^8 + k^6 - k^4 - k^2)*x^3/3! + (9*k^14 + 6*k^12 - k^10 - 20*k^8 - 9*k^6 + 14*k^4 + k^2)*x^5/5! + (225*k^20 + 135*k^18 - 180*k^16 - 300*k^14- 434*k^12 + 210*k^10 + 524*k^8 - 44*k^6 - 135*k^4 - k^2)*x^7/7! + (11025*k^26 + 6300*k^24 - 13230*k^22 - 23940*k^20 - 2961*k^18 + 6552*k^16 + 18332*k^14 + 22712*k^12 - 17825*k^10 - 12852*k^8 + 4658*k^6 + 1228*k^4 + k^2)*x^9/9! + (893025*k^32 + 496125*k^30 - 1393875*k^28 - 2433375*k^26 - 335475*k^24 + 3138345*k^22 + 866745*k^20 - 82995*k^18 + 562771*k^16 - 2154361*k^14 - 783465*k^12 + 1194707*k^10 + 201343*k^8 - 158445*k^6 - 11069*k^4 - k^2)*x^11/11! +...
Related Jacobi elliptic functions sn(,), cn(,), and dn(,) begin:
sn(x,k) = x + (-k^2 - 1)*x^3/3! + (k^4 + 14*k^2 + 1)*x^5/5! + (-k^6 - 135*k^4 - 135*k^2 - 1)*x^7/7! + (k^8 + 1228*k^6 + 5478*k^4 + 1228*k^2 + 1)*x^9/9! + (-k^10 - 11069*k^8 - 165826*k^6 - 165826*k^4 - 11069*k^2 - 1)*x^11/11! + (k^12 + 99642*k^10 + 4494351*k^8 + 13180268*k^6 + 4494351*k^4 + 99642*k^2 + 1)*x^13/13! + (-k^14 - 896803*k^12 - 116294673*k^10 - 834687179*k^8 - 834687179*k^6 - 116294673*k^4 - 896803*k^2 - 1)*x^15/15! +...
where sn(x,k) = sn(A(x,k), k)/k.
cn(x,k) = 1 - x^2/2! + (4*k^2 + 1)*x^4/4! + (-16*k^4 - 44*k^2 - 1)*x^6/6! + (64*k^6 + 912*k^4 + 408*k^2 + 1)*x^8/8! + (-256*k^8 - 15808*k^6 - 30768*k^4 - 3688*k^2 - 1)*x^10/10! + (1024*k^10 + 259328*k^8 + 1538560*k^6 + 870640*k^4 + 33212*k^2 + 1)*x^12/12! + (-4096*k^12 - 4180992*k^10 - 65008896*k^8 - 106923008*k^6 - 22945056*k^4 - 298932*k^2 - 1)*x^14/14! +...
where cn(x,k) = dn(A(k*x,1/k)/k, k),
and cn(2*A(x,k), k) = -1 + 2*dn(x,k)^2 / (1 - k^6*sn(x,k)^4).
dn(x,k) = 1 - k^2*x^2/2! + (k^4 + 4*k^2)*x^4/4! + (-k^6 - 44*k^4 - 16*k^2)*x^6/6! + (k^8 + 408*k^6 + 912*k^4 + 64*k^2)*x^8/8! + (-k^10 - 3688*k^8 -30768*k^6 - 15808*k^4 - 256*k^2)*x^10/10! + (k^12 + 33212*k^10 + 870640*k^8 + 1538560*k^6 + 259328*k^4 + 1024*k^2)*x^12/12! + (-k^14 - 298932*k^12 - 22945056*k^10 - 106923008*k^8 - 65008896*k^6 - 4180992*k^4 - 4096*k^2)*x^14/14! +...
where dn(x,k) = cn(A(x,k),k).

Paul D. Hanna, Table of n, a(n) for n = 1..900 of rows 1..30 read as a flattened table.

Cf. A060627, A002754, A060628, A001818.

Cf. A291560.

/* Find A such that sn(A,k) = k * sn(x,k) */
{T(n,r) = my(A=x,V=[k],S=x,C=1-x^2/2);
for(m=0,n, V=concat(V,[0,0]); A = x*Ser(V);
S = intformal(C*subst(C,x,A));
C = 1 - intformal(S*subst(C,x,A));
V[#V] = -polcoeff(subst(S,x,A)/S,#V-1,x););
(2*n-1)!*polcoeff(V[2*n-1],2*r-1,k)}
for(n=1,10, for(r=1,2*n-1, print1(T(n,r),", "));print(""))

{T(n, k) = my(A, m); if( n<0 || k>=(m=2*n+1), 0, A = intformal(1 / sqrt((1 - x^2) * (1 - y^2*x^2) + x*O(x^m))); A = subst(A, x, y * serreverse(A)); m! * polcoeff( polcoeff(A, m), 2*k+1))}; /* Michael Somos, Aug 27 2017 */

E.g.f. A(x,k) = Sum_{n>=1, r=1..2*n-1} T(n,r) * x^(2*n-1) * k^(2*r-1)/(2*n-1)!, satisfies:
(1) sn(A(x,k), k) = k * sn(x,k),
(2) cn(A(x,k), k) = dn(x,k),
(3) dn(A(k*x,1/k)/k, k) = cn(x,k),
(4) A(k*A(x,k), 1/k) = k*x,
(5) A(A(x,1/k)/k, k) = x/k,
(6) sn( A^r(x,k), k) = k^r * sn(x,k) where A^r(x,k) = A( A^{r-1}(x,k), k) is the r-th iteration of A(x,k) wrt x, with A^0(x,k) = x.
Row sums of n-th row equals zero for n>1.
T(n+1,1) = (-1)^n for n>=0.
T(n+1, 2*n+1) = ( (2*n)! / (n!*2^n) )^2 = A001818(n) for n>=0.

A171660 Triangle T(n,m) of the expansion coefficients of JacobiCN(x,y) + JacobiDN(x,y) = Sum_{n>=0} Sum_{k=0..n} (-1)^nT(n,m)x^(2n)y^(2m)/(2n)!.

Original entry on oeis.org

2, 1, 1, 1, 8, 1, 1, 60, 60, 1, 1, 472, 1824, 472, 1, 1, 3944, 46576, 46576, 3944, 1, 1, 34236, 1129968, 3077120, 1129968, 34236, 1, 1, 303028, 27126048, 171931904, 171931904, 27126048, 303028, 1, 1, 2706800, 653677408, 8874639488, 19720976896

Offset: 0

Roger L. Bagula, Dec 14 2009

Row sums are 2*A000364(n).

Since the coefficients of JacobiCN are in A060627 and the coefficients of JacobiDN are obtained by row-reversal of A060627, this triangle here is a symmetrized variant, adding A060627 and its mirrored version.

			2;
1, 1;
1, 8, 1;
1, 60, 60, 1;
1, 472, 1824, 472, 1;
1, 3944, 46576, 46576, 3944, 1;
1, 34236, 1129968, 3077120, 1129968, 34236, 1;
1, 303028, 27126048, 171931904, 171931904, 27126048, 303028, 1;
1, 2706800, 653677408, 8874639488, 19720976896, 8874639488, 653677408, 2706800, 1;
1, 24279312, 15877769376, 440712200064, 1948265426688, 1948265426688, 440712200064, 15877769376, 24279312, 1;
1, 218186164, 388726995744, 21489645169920, 176743676925696, 343497841920000, 176743676925696, 21489645169920, 388726995744, 218186164, 1;

Cf. A060627.

A171660 := proc(n,m) JacobiCN(z,k) +JacobiDN(z,k) ; coeftayl(%,z=0,2*n) ; (-1)^n*coeftayl(%,k=0,2*m)*(2*n)! ; end proc: # R. J. Mathar, Jan 30 2011

p[t_] = JacobiCN[t, x] + JacobiDN[t, x]
a = Table[ CoefficientList[FullSimplify[ExpandAll[(-1)^Floor[n/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 20, 2}]
Flatten[a]

cp's OEIS Frontend

A002754 Related to coefficient of m in Jacobi elliptic function cn(z, m).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A322231 E.g.f.: C(x,k) = 1 + Integral S(x,k)*D(x,k)^2 dx, such that C(x,k)^2 - S(x,k)^2 = 1, and D(x,k)^2 - k^2*S(x,k)^2 = 1, as a triangle of coefficients read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A370543 Expansion of the Jacobi elliptic function cn(x,k) at k = 2 (even powers only).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A181612 Triangle T(n,m) of the coefficients JacobiDC(x,y) = sum_{n>=0} sum_{m=0..n} (-1)^m* T(n,m) *x^(2*n) *y^(2*m)/(2*n)!.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A181613 Triangle T(n,m) of the coefficients JacobiNC(x,y) = sum_{n>0} sum_{m=0..n-1} (-1)^m* T(n,m) *x^(2*n) *y^(2*m)/(2*n)!.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A325221 E.g.f.: C(x,k) = cn( i * Integral C(x,k) dx, k), where C(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n)*k^(2*j)/(2*n)!, as a triangle of coefficients T(n,j) read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A370544 Expansion of the Jacobi elliptic function dn(x,k) at k = 2 (even powers only).

Views

Author

Keywords

Examples

References

Links

A322231 E.g.f.: C(x,k) = 1 + Integral S(x,k)D(x,k)^2 dx, such that C(x,k)^2 - S(x,k)^2 = 1, and D(x,k)^2 - k^2S(x,k)^2 = 1, as a triangle of coefficients read by rows.

A181612 Triangle T(n,m) of the coefficients JacobiDC(x,y) = sum_{n>=0} sum_{m=0..n} (-1)^m* T(n,m) x^(2n) y^(2m)/(2*n)!.

A181613 Triangle T(n,m) of the coefficients JacobiNC(x,y) = sum_{n>0} sum_{m=0..n-1} (-1)^m* T(n,m) x^(2n) y^(2m)/(2*n)!.

A325221 E.g.f.: C(x,k) = cn( i * Integral C(x,k) dx, k), where C(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2n)k^(2j)/(2n)!, as a triangle of coefficients T(n,j) read by rows.

A190904 a(n) = Sum_{k=0..n-1} cos(Pik/2)binomial(n-1,k)a(n-1-k)a(k) for n > 0, a(0) = 1.

A171660 Triangle T(n,m) of the expansion coefficients of JacobiCN(x,y) + JacobiDN(x,y) = Sum_{n>=0} Sum_{k=0..n} (-1)^nT(n,m)x^(2n)y^(2m)/(2n)!.