A060628 -id:A060628 - cp's OEIS frontend

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)!.

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).

A370542 Expansion of the Jacobi elliptic function sn(x,k) at k = 2 (odd powers only).

Original entry on oeis.org

1, -5, 73, -2765, 171409, -16145045, 2168436697, -391723265885, 91633164775201, -26955095234906405, 9737498127795037033, -4237907290209405609965, 2187044171819241257792689, -1320533769141977996485790645, 922274662722967857470247551737, -737730926392606318468533810754685

Offset: 0

Paul D. Hanna, Mar 23 2024

sign

			E.g.f.: S(x) = x - 5*x^3/3! + 73*x^5/5! - 2765*x^7/7! + 171409*x^9/9! - 16145045*x^11/11! + 2168436697*x^13/13! - 391723265885*x^15/15! + ...
where S(x) = sn(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. A000182 (unsigned sn(x,1)), A060628 (sn(x,k)).

Cf. A370543 (cn(x,2)), A370544 (dn(x,2)), A249282.

# a(n) = (2*n+1)! * [x^(2*n+1)] sn(x, 2).
sn_list := proc(k, len) local n; seq((2*n+1)!*coeff(series(JacobiSN(z, k), z,
2*len + 2), z, 2*n + 1), n = 0..len) end:
sn_list(2, 15);  # Peter Luschny, Mar 25 2024

nmax = 20;
DeleteCases[CoefficientList[JacobiSN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax] + 1)! (* Jean-François Alcover, Mar 28 2024 *)

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

a(n) = (-1)^n * Sum_{k=0..n} A060628(n,k)*4^k for n >= 0.
E.g.f. S(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)! satisfies the following formulas, where sn, cn, and dn are Jacobi elliptic functions.
(1) S(x) = sn(x,k) at k = 2.
(2.a) S(x) = sn(2*x,1/2)/2.
(2.b) S(x) = sn(x,1/2) * cn(x,1/2) * dn(x,1/2) / (1 - sn(x,1/2)^4/4).
(3.a) S(x) = Series_Reversion( Integral 1/sqrt( (1-x^2)*(1-4*x^2) ) dx ).
(3.b) S(x) = Integral sqrt(1 - S(x)^2) * sqrt(1 - 4*S(x)^2) dx.
(4.a) S(x) = sin( Integral sqrt(1 - 4*S(x)^2) dx ).
(4.b) S(x) = sin( 2 * Integral sqrt(1 - S(x)^2) dx ) / 2.
(5.a) S(x) = sqrt(1 - cn(x,2)^2).
(5.b) S(x) = sqrt(1 - dn(x,2)^2) / 2.
O.g.f.: x/(1 + 5*x - 4*1*2^2*3*x^2/(1 + 5*3^2*x - 4*3*4^2*5*x^2/(1 + 5*5^2*x - 4*5*6^2*7*x^2/(1 + 5*7^2*x - 4*7*8^2*9*x^2/(1 + 5*9^2*x - ...))))) = x - 5*x^2 + 73*x^3 - 2765*x^4 + 171409*x^5 - 16145045*x^6 + ... (continued fraction, see Wall, 94.17, p. 374).
a(n) ~ (-1)^n * 2^(4*n+4) * agm(1,2)^(2*n+2) * n^(2*n + 3/2) / (Pi^(2*n + 3/2) * 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.

A032348 Coefficients of Jacobi function c(3,m).

Original entry on oeis.org

1, 1228, 165826, 13180268, 834687179, 47152124264, 2504055894564, 128453495887560, 6460701405171285, 321298267540551700, 15875718186751193446, 781562415106660985428, 38396599486084770569951, 1884152729554433297404688

Offset: 0

Simon Plouffe

nonn

A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.

Cf. A060628 (3rd lower diagonal).

j = 3; max = 17; coes = CoefficientList[#, k]& /@ ((CoefficientList[ Series[ JacobiSN[x, k], {x, 0, 2*max}], x] // Select[#, # =!= 0 &] &)*Table[(-1)^n*(2*n+1)!, {n, 0, max-1}] ) ; coes[[j+1 ;; -1]][[All, j+1]] (* Jean-François Alcover, May 14 2013 *)

a(n) = (104*n*9^(n+4) + 3*7^(2*n+7) - (24*n+36)*5^(2*n+7) + (32*n^2+54)*3^(2*n+8) -256*n^3-1248*n^2-1328*n-135) / 12288. - Vaclav Kotesovec after Fransen, Jul 30 2013

Typo in a(7) fixed by Jean-François Alcover, May 14 2013

A032430 Coefficients of Jacobi elliptic function c(7,m).

Original entry on oeis.org

1, 8071256, 74197080276, 128453495887560, 107266611330420090, 59752013018382750024, 25835579116799316507780, 9424979520638053300516632, 3051808875538951440990525939, 906467723949073501017465886864, 252583298644057469403578416269848

Offset: 7

Simon Plouffe

nonn

A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.

Cf. A060628 (7th lower diagonal).

a(n) = T(2*n+1,0,7) where T(1,0,0) = 1; T(n,i,j) = 0 if i+j < 0 or i+j > n/2; T(2*n,i,j) = (2*j+1) * T(2*n-1,i,j) + (2*i+2) * T(2*n-1,i+1,j-1) + (2*n-2*i-2*j+1) * T(2*n-1,i,j-1), and T(2*n+1,i,j) = (2*i+1) * T(2*n-1,i,j) + (2*j+2) * T(2*n,i-1,j+1) + (2*n-2*i-2*j+2) * T(2*n-1,i-1,j). - Sean A. Irvine, Jun 20 2020

Offset corrected and more terms from Sean A. Irvine, Jun 20 2020

A032431 Coefficients of Jacobi elliptic function c(8,m).

Original entry on oeis.org

1, 72641337, 1859539731885, 6460701405171285, 9412382749388124015, 8470841585571575617239, 5632500127524872577252027, 3051808875538951440990525939, 1429953329302734392093044646982, 602297594518030428986818986545686, 234170438234669757816987374536542702

Offset: 8

Simon Plouffe

nonn

A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.

Cf. A060628 (8th lower diagonal).

a(n) = T(2*n+1,0,8) where T(1,0,0) = 1; T(n,i,j) = 0 if i+j < 0 or i+j > n/2; T(2*n,i,j) = (2*j+1) * T(2*n-1,i,j) + (2*i+2) * T(2*n-1,i+1,j-1) + (2*n-2*i-2*j+1) * T(2*n-1,i,j-1), and T(2*n+1,i,j) = (2*i+1) * T(2*n-1,i,j) + (2*j+2) * T(2*n,i-1,j+1) + (2*n-2*i-2*j+2) * T(2*n-1,i-1,j). - Sean A. Irvine, Jun 20 2020

Offset corrected and more terms from Sean A. Irvine, Jun 20 2020

A032432 Coefficients of Jacobi elliptic function c(9,m).

Original entry on oeis.org

1, 653772070, 46535238000235, 321298267540551700, 803475280086029066515, 1146456994425541774291534, 1149330973559307337432235521, 906467723949073501017465886864, 602297594518030428986818986545686, 352513571679334580855533139395470836

Offset: 9

Simon Plouffe

nonn

A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.

Cf. A060628 (9th lower diagonal).

a(n) = T(2*n+1,0,9) where T(1,0,0) = 1; T(n,i,j) = 0 if i+j < 0 or i+j > n/2; T(2*n,i,j) = (2*j+1) * T(2*n-1,i,j) + (2*i+2) * T(2*n-1,i+1,j-1) + (2*n-2*i-2*j+1) * T(2*n-1,i,j-1), and T(2*n+1,i,j) = (2*i+1) * T(2*n-1,i,j) + (2*j+2) * T(2*n,i-1,j+1) + (2*n-2*i-2*j+2) * T(2*n-1,i-1,j). - Sean A. Irvine, Jun 20 2020

Offset corrected and more terms from Sean A. Irvine, Jun 20 2020

A032433 Coefficients of Jacobi elliptic function c(10,m).

Original entry on oeis.org

1, 5883948671, 1163848723925346, 15875718186751193446, 67362921649153881472361, 150221961163114696686151695, 223559382769795167319093086664, 252583298644057469403578416269848, 234170438234669757816987374536542702, 187377221472810770345920109207417275058

Offset: 10

Simon Plouffe

nonn

A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.

Cf. A060628 (10th lower diagonal).

a(n) = T(2*n+1,0,10) where T(1,0,0) = 1; T(n,i,j) = 0 if i+j < 0 or i+j > n/2; T(2*n,i,j) = (2*j+1) * T(2*n-1,i,j) + (2*i+2) * T(2*n-1,i+1,j-1) + (2*n-2*i-2*j+1) * T(2*n-1,i,j-1), and T(2*n+1,i,j) = (2*i+1) * T(2*n-1,i,j) + (2*j+2) * T(2*n,i-1,j+1) + (2*n-2*i-2*j+2) * T(2*n-1,i-1,j). - Sean A. Irvine, Jun 20 2020

Offset corrected and more terms from Sean A. Irvine, Jun 20 2020

cp's OEIS Frontend

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)!.

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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)!.

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A370542 Expansion of the Jacobi elliptic function sn(x,k) at k = 2 (odd powers only).

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

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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.

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A032348 Coefficients of Jacobi function c(3,m).

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A032430 Coefficients of Jacobi elliptic function c(7,m).

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A032431 Coefficients of Jacobi elliptic function c(8,m).

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)!.

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.