A060628 -id:A060628 - cp's OEIS frontend

A032349 Number of paths from (0,0) to (3n,0) that stay in first quadrant (but may touch horizontal axis), where each step is (2,1),(1,2) or (1,-1) and start with (2,1).

1, 4, 24, 172, 1360, 11444, 100520, 911068, 8457504, 80006116, 768464312, 7474561164, 73473471344, 728745517972, 7284188537672, 73301177482172, 742009157612608, 7550599410874820, 77193497566719320, 792498588659426924

Offset: 1

Emeric Deutsch

a(n) is the maximum number of distinct sets that can be obtained as complete parenthesizations of “S_1 union S_2 intersect S_3 union S_4 intersect S_5 union ... union S_{2*n}”, where n union and n-1 intersection operations alternate, starting with a union, and S_1, S_2, ... , S_{2*n} are sets. - Alexander Burstein, Nov 22 2023

			From _Alexander Burstein_, Feb 14 2025: (Start)
a(2) = 4 as the maximum number of distinct sets obtained as complete parenthesizations of S_1 u(nion) S_2 (i)n(tersect) S_3 u(nion) S_4:
S_1 u (S_2 n (S_3 u S_4)),
S_1 u ((S_2 n S_3) u S_4) = (S_1 u (S_2 n S_3)) u S_4,
(S_1 u S_2) n (S_3 u S_4),
((S_1 u S_2) n S_3) u S_4. (End)

G. C. Greubel, Table of n, a(n) for n = 1..950
Emeric Deutsch, Problem 10658, American Math. Monthly, 107, 2000, 368-370.
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See pp. 7, 11.

Convolution of A027307 with itself.

Cf. A060628 diagonal(-6).

RecurrenceTable[{n*(2*n-1)*a[n] == (28*n^2-65*n+36)*a[n-1] - (64*n^2-323*n+408)*a[n-2] - 3*(n-4)*(2*n-5)*a[n-3],a[1]==1,a[2]==4,a[3]==24},a,{n,20}] (* Vaclav Kotesovec, Oct 08 2012 *)

a(n):=2*sum((2*n+i-1)!/(i!*(n-i-1)!*(n+i+1)!),i,0,n-1); /* Vladimir Kruchinin, Oct 18 2011 */

vector(30, n, 2*sum(k=0, n-1, (2*n+k-1)!/(k!*(n-k-1)!*(n+k+1)!))) \\ Altug Alkan, Oct 06 2015

{a(n) = my(A=1); for(i=1,n, A = 1 + x*(A + sqrt(A +x*O(x^n)))^2); polcoeff(A,n)}
for(n=0,30, print1(a(n),", ")) \\ Paul D. Hanna, Jun 11 2016

G.f.: z*A^2, where A is the g.f. of A027307.
a(n) = 2*Sum_{i=0..n-1} (2*n+i-1)!/(i!*(n-i-1)!*(n+i+1)!). - Vladimir Kruchinin, Oct 18 2011
D-finite with recurrence: n*(2*n-1)*a(n) = (28*n^2-65*n+36)*a(n-1) - (64*n^2-323*n+408)*a(n-2) - 3*(n-4)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ sqrt(45*sqrt(5)-100)*((11+5*sqrt(5))/2)^n/(5*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012
G.f. A(x) satisfies: A(x) = 1 + x * ( A(x) + sqrt(A(x)) )^2. - Paul D. Hanna, Jun 11 2016
From Peter Bala, May 07 2023: (Start)
n*(2*n-1)*(5*n-9)*a(n) = 2*(55*n^3-209*n^2+255*n-99)*a(n-1) + (n-3)*(2*n-3)*(5*n-4)*a(n-2) with a(1) = 1 and a(2) = 4.
G.f.: A(x) = series reversion of x*(1 - x)^2/(1 + x)^2. (End)

A060627 1 + Sum_{n >= 1} Sum_{k = 0..n-1} (-1)^nT(n,k)y^(2k)x^(2n)/(2n)! = JacobiCN(x,y).

Original entry on oeis.org

1, 1, 4, 1, 44, 16, 1, 408, 912, 64, 1, 3688, 30768, 15808, 256, 1, 33212, 870640, 1538560, 259328, 1024, 1, 298932, 22945056, 106923008, 65008896, 4180992, 4096, 1, 2690416, 586629984, 6337665152, 9860488448, 2536974336, 67047424, 16384

Offset: 1

Vladeta Jovovic, Apr 13 2001

Essentially same triangle as triangle given by [1, 0, 9, 0, 25, 0, 49, 0, 81, 0, 121, ...] DELTA [0, 4, 0, 16, 0, 36, 0, 64, 0, 100, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jun 13 2004

For the recurrence of the row polynomials b_n(y^2) for cn(x|y^2) = Sum_{n >=0} b_n(y^2)*x^(2*n)/(2*n)! see the Fricke reference, where y=k. - Wolfdieter Lang, Jul 05 2016

			The first rows of triangle T(n, k), n >= 1, k = 0..n-1, are:
[1], [1, 4], [1, 44, 16], [1, 408, 912, 64], [1, 3688, 30768, 15808, 256], [1, 33212, 870640, 1538560, 259328, 1024], [1, 298932, 22945056, 106923008, 65008896, 4180992, 4096], [1, 2690416, 586629984, 6337665152, 9860488448, 2536974336, 67047424, 16384], ...

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 526.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(5.2.20).
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. 575, 16.22.2.
P. Bala, A triangle for calculating A060627
F. Clarke, The Taylor Series Coefficients of the Jacobi Elliptic Functions, slides. [broken link]
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052 [math.CA], 2005.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Erster Teil, p. 399 with p. 397, Springer, Berlin, Heidelberg, 2012.
Eric W. Weisstein, Jacobi Elliptic Functions

Row sums: A000364.

Cf. A002754, A060628, A145271, A181612, A181613.

A060627 := proc(n,m) JacobiCN(z,k) ; coeftayl(%,z=0,2*n) ; (-1)^n*coeftayl(%,k=0,2*m)*(2*n)! ; end proc: # R. J. Mathar, Jan 30 2011

nmax = 8; se = Series[JacobiCN[x, y], {x, 0, 2*nmax} ]; t[n_, m_] := (-1)^n*Coefficient[se, x, 2*n] *(2*n)! // Coefficient[#, y, m]&; Table[t[n, m], {n, 1, nmax}, {m, 0, n-1}] // Flatten (* Jean-François Alcover, Mar 26 2013 *)

JacobiCN(x, y) = 1 - 1/2*x^2 + (1/24 + 1/6*y^2)*x^4 + ( - 1/720 - 11/180*y^2 - 1/45*y^4)*x^6 + (1/40320 + 17/1680*y^2 + 19/840*y^4 + 1/630*y^6)*x^8 + ( - 1/3628800 - 247/56700*y^6 - 461/453600*y^2 - 641/75600*y^4 - 1/14175*y^8)*x^10 + O(x^12).
From Peter Bala, Aug 23 2011: (Start)
The Taylor expansion of the Jacobian elliptic function cn(x,k) begins
  cn(x,k) = 1 - x^2/2! + (1+4*k^2)*x^4/4! - (1+44*k^2+16*k^4)*x^6/6! + ....
The coefficient polynomials in this expansion can be calculated using nested derivatives as follows (see [Dominici, Theorem 4.1 and Example 4.5]):
Let f(x) = sqrt(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.
Then the coefficient polynomial R(2*n,k) of x^(2*n)/(2*n)! in the expansion of cn(x,k) is given by 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). See A181613 for the expansion of the reciprocal function 1/cn(x,k).
(End)
G.f. 1/(1 - x/(1 - (2*k)^2*x/(1 - 3^2*x/(1 - (4*k)^2*x/(1 - 5^2*x/(1 - ...)))))) = 1 + x + (1 + 4*k^2)*x^2 + (1 + 44*k^2 + 16*k^4)*x^3 + ... (see Wall, 94.19, p. 374). - Peter Bala, Apr 25 2017

A322230 E.g.f.: S(x,k) = Integral C(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, 2, 1, 28, 16, 1, 270, 1032, 272, 1, 2456, 36096, 52736, 7936, 1, 22138, 1035088, 4766048, 3646208, 353792, 1, 199284, 27426960, 319830400, 704357760, 330545664, 22368256, 1, 1793606, 702812568, 18598875760, 93989648000, 120536980224, 38188155904, 1903757312, 1, 16142512, 17753262208, 1002968825344, 10324483102720, 28745874079744, 24060789342208, 5488365862912, 209865342976, 1, 145282674, 445736371872, 51882638754240, 1013356176688128, 5416305638467584, 9498855414644736, 5590122715250688, 961530104709120, 29088885112832

Offset: 0

Paul D. Hanna, Dec 14 2018

Equals a row reversal of triangle A325220.
Appears to be a row reversal of EG1 triangle A162005, which has other formulas.
Compare to sn(x,k) = Integral cn(x,k)*dn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions (see triangle A060628).
Compare also to Michael Pawellek's generalized elliptic functions.

			E.g.f.: 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! + ...
such that C(x,k)^2 - S(x,k)^2 = 1.
This triangle of coefficients T(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in e.g.f. S(x,k) begins:
1;
1, 2;
1, 28, 16;
1, 270, 1032, 272;
1, 2456, 36096, 52736, 7936;
1, 22138, 1035088, 4766048, 3646208, 353792;
1, 199284, 27426960, 319830400, 704357760, 330545664, 22368256;
1, 1793606, 702812568, 18598875760, 93989648000, 120536980224, 38188155904, 1903757312;
1, 16142512, 17753262208, 1002968825344, 10324483102720, 28745874079744, 24060789342208, 5488365862912, 209865342976; ...
RELATED SERIES.
The related series C(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
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! + ...
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. A000182 (diagonal), A162006, A162007, A162008, A162009, A162010.
Cf. A322231 (C), A322232 (D).
Cf. A325220 (row reversal), A162005.

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+1)!*polcoeff(polcoeff(S,2*n+1,x),2*j,k),", ")) ;print(""))

E.g.f. S = S(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n+1) * k^(2*j) / (2*n+1)!, along with related series C = C(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+1)!*(2*n)!/(n!^2*4^n) = A079484(n), the product of two consecutive odd double factorials.
Main diagonal equals A000182, the tangent numbers.

A002753 Coefficients of elliptic function sn.

Original entry on oeis.org

1, 1, 14, 135, 5478, 165826, 13180268, 834687179, 109645021894, 11966116940238, 2347836365864484, 393938089395885894, 107947764316226205276, 25835579116799316507780

Offset: 0

N. J. A. Sloane

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, Eq. 16.22.1 and 16.22.2.
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..100
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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, Eq. 16.22.1 and 16.22.2.
A. Cayley, An Elementary Treatise on Elliptic Functions (page images), G. Bell and Sons, London, 1895, p. 56.
C. L. Mallows, Letter to N. J. A. Sloane, May 16 1973.
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.

Cf. A006089, A060628.

lim = 14; se = Series[ JacobiSN[u, m], {u, 0, 2 lim -1}]; a[n_] := (-1)^n*Coefficient[ SeriesCoefficient[se, 2n + 1]*(2n + 1)!, m^Floor[n/2]]; a[0] = a[1] = 1; Table[a[n], {n, 0, lim-1}] (* Jean-François Alcover, Sep 21 2011 *)

a(n) ~ 2^(4*n + 13/2) * n^(2*n + 1) / (exp(2*n) * Pi^(2*n + 2)). - Vaclav Kotesovec, Apr 10 2018

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

A279832 Numerators of the coefficients in g.f. A(x) such that: sn(x,i*A(x)) = x, where i^2 = -1, and sn(x,k) is a Jacobi elliptic function.

Original entry on oeis.org

1, 3, 27, 1129, 6177, 228496227, 507769159, 3411091723167, 226108446954939, 2799063804718849119, 56928279095622876861, 175898907783132547767087, 2387767743416733035533529, 617528637834242429324813087883, 26943941094191660800993918030539, 4813884370789026772162811298692933153, 41249694296981783922826921997571040581, 69502372123801730691426662081268221528029689, 19796290340432197210800092215751765052273983, 16957540878135184586375745347497078257426299617863, 1168637136489375278109169401471800538288143908488069

Offset: 0

Paul D. Hanna, Dec 21 2016

Denominators are given by A279833.

			This sequence gives the numerators of the coefficients in g.f. A(x), such that
A(x) = 1 + 3/10*x^2 + 27/200*x^4 + 1129/14000*x^6 + 6177/112000*x^8 + 228496227/5605600000*x^10 + 507769159/16016000000*x^12 + 3411091723167/133413280000000*x^14 + 226108446954939/10673062400000000*x^16 + 2799063804718849119/156146902912000000000*x^18 + 56928279095622876861/3690744977920000000000*x^20 + 175898907783132547767087/13072618711792640000000000*x^22 + 2387767743416733035533529/201117210950656000000000000*x^24 + 617528637834242429324813087883/58382315166865930240000000000000*x^26 + 26943941094191660800993918030539/2835712450962059468800000000000000*x^28 + 4813884370789026772162811298692933153/559968137691477883303936000000000000000*x^30 +...
satisfies: sn(x,i*A(x)) = x.
RELATED SERIES.
The Jacobi elliptic function sn(x,k) begins:
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! +...
which equals x when k = i*A(x).
A real transformation of the imaginary modulus i*A(x) yields the series:
A(x)/sqrt(1 + A(x)^2) = sqrt(1/2)*(1 + 3/20*x^2 + 27/800*x^4 + 1681/112000*x^6 + 11667/1280000*x^8 + 45274443/7175168000*x^10 + 613581239/130457600000*x^12 + 62857335822759/17076899840000000*x^14 + 8148919947718779/2732303974400000000*x^16 + 198293692034112113343/79947214290944000000000*x^18 + 4605729854262557732997/2188029022699520000000000*x^20 + 243052910628213000290505027/133863615608756633600000000000*x^22 + 38893821159628323146146353/24505925054234624000000000000*x^24 +...).
sn(2*x, i*A(x)) = 2*x*sqrt(1-x^2)*sqrt(1 + x^2*A(x)^2)/(1 + x^4*A(x)^2) where
sn(2*x, i*A(x)) = 2*x - 12/5*x^5 - 36/25*x^7 + 1332/875*x^9 + 9984/4375*x^11 - 5136624/21896875*x^13 - 266818932/109484375*x^15 - 77131141044/65143203125*x^17 + 33379542432/19159765625*x^19 + 304830773316936/140153685546875*x^21 - 77528188053360024/154869822529296875*x^23 - 145014068636962776/58668332769921875*x^25 +...
The series y = sn(x/2, i*A(x)) satisfies:
x^2*(1 + A(x)^2*y^4)^2  =  4*y^2*(1-y^2)*(1 + A(x)^2*y^2)
where the series y begins:
sn(x/2, i*A(x)) = 1/2*x + 3/320*x^5 + 9/1600*x^7 + 14013/3584000*x^9 + 3729/1280000*x^11 + 6533718813/2870067200000*x^13 + 2402215119/1304576000000*x^15 + 1670885671753959/1092921589760000000*x^17 + 252839176306947/195164569600000000*x^19 + 1427498770243103841051/1279155428655104000000000*x^21 + 4263718777800583142667/4376058045399040000000000*x^23 + 147404533631490298403307261/171345427979208491008000000000*x^25 +...

Paul D. Hanna, Table of n, a(n) for n = 0..50

Cf. A279833, A279834, A279835, A060628.

/* Code to list first N nonzero terms of vector A: */ {N=20;
/* Generate 2*N terms of Jacobi Elliptic Function SN: */
SN = serreverse(intformal(1/sqrt((1-x^2)*(1-k^2*x^2) +x*O(x^(2*N+2))) ));
/* Print N terms of this sequence: */
A=[1]; print1(A[1],", ");
for(i=1,N, A = concat(A,[0,0]);
A[#A] = 3*polcoeff(x - subst(SN,k,I*Ser(A)),#A+2);
print1( numerator(A[#A]),", ") );}

G.f. A = A(x) satisfies:
(1) sd( x*sqrt(1+A^2), A/sqrt(1+A^2) ) = x*sqrt(1+A^2), where sd(x,k) = sn(x,k)/dn(x,k) is a Jacobi elliptic function.
(2) sn(2*x, i*A(x)) = 2*x*sqrt(1-x^2)*sqrt(1 + x^2*A^2)/(1 + x^4*A^2).
(3) y = sn(x/2, i*A(x)) is a solution to the equation:
x^2*(1 + A^2*y^4)^2 = 4*y^2*(1-y^2)*(1 + A^2*y^2).

A279833 Denominators of the coefficients in g.f. A(x) such that: sn(x,i*A(x)) = x, where i^2 = -1, and sn(x,k) is a Jacobi elliptic function.

Original entry on oeis.org

1, 10, 200, 14000, 112000, 5605600000, 16016000000, 133413280000000, 10673062400000000, 156146902912000000000, 3690744977920000000000, 13072618711792640000000000, 201117210950656000000000000, 58382315166865930240000000000000, 2835712450962059468800000000000000

Offset: 0

Paul D. Hanna, Dec 21 2016

Numerators are given by A279832.

			This sequence gives the denominators of the coefficients in g.f. A(x), such that
A(x) = 1 + 3/10*x^2 + 27/200*x^4 + 1129/14000*x^6 + 6177/112000*x^8 + 228496227/5605600000*x^10 + 507769159/16016000000*x^12 + 3411091723167/133413280000000*x^14 + 226108446954939/10673062400000000*x^16 + 2799063804718849119/156146902912000000000*x^18 + 56928279095622876861/3690744977920000000000*x^20 + 175898907783132547767087/13072618711792640000000000*x^22 + 2387767743416733035533529/201117210950656000000000000*x^24 + 617528637834242429324813087883/58382315166865930240000000000000*x^26 + 26943941094191660800993918030539/2835712450962059468800000000000000*x^28 + 4813884370789026772162811298692933153/559968137691477883303936000000000000000*x^30 +...
satisfies: sn(x,i*A(x)) = x.
RELATED SERIES.
The Jacobi elliptic function sn(x,k) begins:
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! +...
which equals x when k = i*A(x).
A real transformation of the imaginary modulus i*A(x) yields the series:
A(x)/sqrt(1 + A(x)^2) = sqrt(1/2)*(1 + 3/20*x^2 + 27/800*x^4 + 1681/112000*x^6 + 11667/1280000*x^8 + 45274443/7175168000*x^10 + 613581239/130457600000*x^12 + 62857335822759/17076899840000000*x^14 + 8148919947718779/2732303974400000000*x^16 + 198293692034112113343/79947214290944000000000*x^18 + 4605729854262557732997/2188029022699520000000000*x^20 + 243052910628213000290505027/133863615608756633600000000000*x^22 + 38893821159628323146146353/24505925054234624000000000000*x^24 +...).
sn(2*x, i*A(x)) = 2*x*sqrt(1-x^2)*sqrt(1 + x^2*A(x)^2)/(1 + x^4*A(x)^2) where
sn(2*x, i*A(x)) = 2*x - 12/5*x^5 - 36/25*x^7 + 1332/875*x^9 + 9984/4375*x^11 - 5136624/21896875*x^13 - 266818932/109484375*x^15 - 77131141044/65143203125*x^17 + 33379542432/19159765625*x^19 + 304830773316936/140153685546875*x^21 - 77528188053360024/154869822529296875*x^23 - 145014068636962776/58668332769921875*x^25 +...
The series y = sn(x/2, i*A(x)) satisfies:
x^2*(1 + A(x)^2*y^4)^2  =  4*y^2*(1-y^2)*(1 + A(x)^2*y^2)
where the series y begins:
sn(x/2, i*A(x)) = 1/2*x + 3/320*x^5 + 9/1600*x^7 + 14013/3584000*x^9 + 3729/1280000*x^11 + 6533718813/2870067200000*x^13 + 2402215119/1304576000000*x^15 + 1670885671753959/1092921589760000000*x^17 + 252839176306947/195164569600000000*x^19 + 1427498770243103841051/1279155428655104000000000*x^21 + 4263718777800583142667/4376058045399040000000000*x^23 + 147404533631490298403307261/171345427979208491008000000000*x^25 +...

Paul D. Hanna, Table of n, a(n) for n = 0..50

Cf. A279832, A279834, A279835, A060628.

/* Code to list first N nonzero terms of vector A: */ {N=20;
/* Generate 2*N terms of Jacobi Elliptic Function SN: */
SN = serreverse(intformal(1/sqrt((1-x^2)*(1-k^2*x^2) +x*O(x^(2*N+2))) ));
/* Print N terms of this sequence: */
A=[1]; print1(A[1],", ");
for(i=1,N, A = concat(A,[0,0]);
A[#A] = 3*polcoeff(x - subst(SN,k,I*Ser(A)),#A+2);
print1( denominator(A[#A]),", ") );}

G.f. A = A(x) satisfies:
(1) sd( x*sqrt(1+A^2), A/sqrt(1+A^2) ) = x*sqrt(1+A^2), where sd(x,k) = sn(x,k)/dn(x,k) is a Jacobi elliptic function.
(2) sn(2*x, i*A(x)) = 2*x*sqrt(1-x^2)*sqrt(1 + x^2*A^2)/(1 + x^4*A^2).
(3) y = sn(x/2, i*A(x)) is a solution to the equation:
x^2*(1 + A^2*y^4)^2 = 4*y^2*(1-y^2)*(1 + A^2*y^2).

A279834 Numerators of the coefficients in g.f. A(x) such that: sn(x,-A(x)) = x, where sn(x,m) is a Jacobi elliptic function.

Original entry on oeis.org

1, 3, 9, 212, 774, 2986491, 11962183, 5866732236, 24717532254, 155049859325162, 8766713183100126, 1242400321151564076, 157798597956508868, 141417442289739551841, 3032690837599386922473477, 272243517649610491264579553148, 1244664961615535298800024043306

Offset: 0

Paul D. Hanna, Dec 26 2016

nonn

Denominators are given by A279835.

The g.f. A(x) of this sequence equals the square of the g.f. of A279832.

			This sequence gives the numerators of the coefficients in g.f. A(x), such that
A(x) = 1 + 3/5*x^2 + 9/25*x^4 + 212/875*x^6 + 774/4375*x^8 + 2986491/21896875*x^10 + 11962183/109484375*x^12 + 5866732236/65143203125*x^14 + 24717532254/325716015625*x^16 + 155049859325162/2382612654296875*x^18 + 8766713183100126/154869822529296875*x^20 + 1242400321151564076/24934041427216796875*x^22 + 157798597956508868/3562005918173828125*x^24 + 141417442289739551841/3559956170522705078125*x^26 + 3032690837599386922473477/84510816662372930908203125*x^28 + 272243517649610491264579553148/8344175483159391333221435546875*x^30 + 1244664961615535298800024043306/41720877415796956666107177734375*x^32 + 309586737719752481090144972054844018/11291964076972525306465238189697265625*x^34 + 1428965605601484765267196303905398982/56459820384862626532326190948486328125*x^36 + 1900644020251253780726568413610042774696/81019842252277869073888084011077880859375*x^38 + 10448090522732112432951611797351884498204/478753613308914680891156860065460205078125*x^40 +...
satisfies: sn(x,-A(x)) = x.
RELATED SERIES.
The Jacobi elliptic function sn(x,m) begins:
sn(x,m) = x - (m + 1)*x^3/3! + (m^2 + 14*m + 1)*x^5/5! - (m^3 + 135*m^2 + 135*m + 1)*x^7/7! + (m^4 + 1228*m^3 + 5478*m^2 + 1228*m + 1)*x^9/9! - (m^5 + 11069*m^4 + 165826*m^3 + 165826*m^2 + 11069*m + 1)*x^11/11! + (m^6 + 99642*m^5 + 4494351*m^4 + 13180268*m^3 + 4494351*m^2 + 99642*m + 1)*x^13/13! - (m^7 + 896803*m^6 + 116294673*m^5 + 834687179*m^4 + 834687179*m^3 + 116294673*m^2 + 896803*m + 1)*x^15/15! +...
which equals x when m = -A(x).

Paul D. Hanna, Table of n, a(n) for n = 0..50

Cf. A279835, A279832, A279833, A060628.

/* Code to list first N nonzero terms of vector A: */ {N=20;
/* Generate 2*N terms of Jacobi Elliptic Function SN: */
SN = serreverse(intformal(1/sqrt((1-x^2)*(1-m*x^2) +x*O(x^(2*N+2))) ));
/* Print N terms of this sequence: */
A=[1]; print1(A[1], ", ");
for(i=1, N, A = concat(A, [0, 0]);
A[#A] = 6*polcoeff(x - subst(SN, m, -Ser(A)), #A+2);
print1( numerator(A[#A]), ", ") ); }

G.f. A = A(x) satisfies:
(1) sd( x*sqrt(1+A), A/(1+A) ) = x*sqrt(1+A), where sd(x,m) = sn(x,m)/dn(x,m) is a Jacobi elliptic function.
(2) sn(2*x, -A(x)) = 2*x*sqrt(1-x^2)*sqrt(1 + x^2*A)/(1 + x^4*A).
(3) y = sn(x/2, -A(x)) is a solution to the equation:
x^2*(1 + A*y^4)^2 = 4*y^2*(1-y^2)*(1 + A*y^2).

A279835 Denominators of the coefficients in g.f. A(x) such that: sn(x,-A(x)) = x, where sn(x,m) is a Jacobi elliptic function.

Original entry on oeis.org

1, 5, 25, 875, 4375, 21896875, 109484375, 65143203125, 325716015625, 2382612654296875, 154869822529296875, 24934041427216796875, 3562005918173828125, 3559956170522705078125, 84510816662372930908203125, 8344175483159391333221435546875, 41720877415796956666107177734375, 11291964076972525306465238189697265625

Offset: 0

Paul D. Hanna, Dec 26 2016

nonn

Numerators are given by A279834.

The g.f. A(x) of this sequence equals the square of the g.f. of A279832.

			This sequence gives the denominators of the coefficients in g.f. A(x), such that
A(x) = 1 + 3/5*x^2 + 9/25*x^4 + 212/875*x^6 + 774/4375*x^8 + 2986491/21896875*x^10 + 11962183/109484375*x^12 + 5866732236/65143203125*x^14 + 24717532254/325716015625*x^16 + 155049859325162/2382612654296875*x^18 + 8766713183100126/154869822529296875*x^20 + 1242400321151564076/24934041427216796875*x^22 + 157798597956508868/3562005918173828125*x^24 + 141417442289739551841/3559956170522705078125*x^26 + 3032690837599386922473477/84510816662372930908203125*x^28 + 272243517649610491264579553148/8344175483159391333221435546875*x^30 + 1244664961615535298800024043306/41720877415796956666107177734375*x^32 + 309586737719752481090144972054844018/11291964076972525306465238189697265625*x^34 + 1428965605601484765267196303905398982/56459820384862626532326190948486328125*x^36 + 1900644020251253780726568413610042774696/81019842252277869073888084011077880859375*x^38 + 10448090522732112432951611797351884498204/478753613308914680891156860065460205078125*x^40 +...
satisfies: sn(x,-A(x)) = x.
RELATED SERIES.
The Jacobi elliptic function sn(x,m) begins:
sn(x,m) = x - (m + 1)*x^3/3! + (m^2 + 14*m + 1)*x^5/5! - (m^3 + 135*m^2 + 135*m + 1)*x^7/7! + (m^4 + 1228*m^3 + 5478*m^2 + 1228*m + 1)*x^9/9! - (m^5 + 11069*m^4 + 165826*m^3 + 165826*m^2 + 11069*m + 1)*x^11/11! + (m^6 + 99642*m^5 + 4494351*m^4 + 13180268*m^3 + 4494351*m^2 + 99642*m + 1)*x^13/13! - (m^7 + 896803*m^6 + 116294673*m^5 + 834687179*m^4 + 834687179*m^3 + 116294673*m^2 + 896803*m + 1)*x^15/15! +...
which equals x when m = -A(x).

Paul D. Hanna, Table of n, a(n) for n = 0..50

Cf. A279834, A279832, A279833, A060628.

/* Code to list first N nonzero terms of vector A: */ {N=20;
/* Generate 2*N terms of Jacobi Elliptic Function SN: */
SN = serreverse(intformal(1/sqrt((1-x^2)*(1-m*x^2) +x*O(x^(2*N+2))) ));
/* Print N terms of this sequence: */
A=[1]; print1(A[1], ", ");
for(i=1, N, A = concat(A, [0, 0]);
A[#A] = 6*polcoeff(x - subst(SN, m, -Ser(A)), #A+2);
print1( denominator(A[#A]), ", ") ); }

G.f. A = A(x) satisfies:
(1) sd( x*sqrt(1+A), A/(1+A) ) = x*sqrt(1+A), where sd(x,m) = sn(x,m)/dn(x,m) is a Jacobi elliptic function.
(2) sn(2*x, -A(x)) = 2*x*sqrt(1-x^2)*sqrt(1 + x^2*A)/(1 + x^4*A).
(3) y = sn(x/2, -A(x)) is a solution to the equation:
x^2*(1 + A*y^4)^2 = 4*y^2*(1-y^2)*(1 + A*y^2).

A325220 E.g.f.: S(x,k) = -i * sn( i * Integral C(x,k) dx, k) such that C(x,k) = cn( i * Integral C(x,k) dx, k), where S(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2n+1)k^(2j)/(2n+1)!, as a triangle of coefficients T(n,j) read by rows.

Original entry on oeis.org

1, 2, 1, 16, 28, 1, 272, 1032, 270, 1, 7936, 52736, 36096, 2456, 1, 353792, 3646208, 4766048, 1035088, 22138, 1, 22368256, 330545664, 704357760, 319830400, 27426960, 199284, 1, 1903757312, 38188155904, 120536980224, 93989648000, 18598875760, 702812568, 1793606, 1, 209865342976, 5488365862912, 24060789342208, 28745874079744, 10324483102720, 1002968825344, 17753262208, 16142512, 1, 29088885112832, 961530104709120, 5590122715250688, 9498855414644736, 5416305638467584, 1013356176688128, 51882638754240, 445736371872, 145282674, 1

Offset: 0

Paul D. Hanna, Apr 13 2019

Equals a row reversal of triangle A322230.
Appears to equal EG1 triangle A162005, which has other formulas.
Compare to sn(x,k) = Integral cn(x,k)*dn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions (see triangle A060628).

			E.g.f.: 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! + ...
such that S(x,k) = cn( i * Integral C(x,k) dx, k) and C(x,k)^2 - S(x,k)^2 = 1.
This triangle of coefficients T(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in e.g.f. S(x,k) begins:
1;
2, 1;
16, 28, 1;
272, 1032, 270, 1;
7936, 52736, 36096, 2456, 1;
353792, 3646208, 4766048, 1035088, 22138, 1;
22368256, 330545664, 704357760, 319830400, 27426960, 199284, 1;
1903757312, 38188155904, 120536980224, 93989648000, 18598875760, 702812568, 1793606, 1;
209865342976, 5488365862912, 24060789342208, 28745874079744, 10324483102720, 1002968825344, 17753262208, 16142512, 1;
29088885112832, 961530104709120, 5590122715250688, 9498855414644736, 5416305638467584, 1013356176688128, 51882638754240, 445736371872, 145282674, 1; ...
RELATED SERIES.
The related series C(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
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! + ...
which also satisfies C(x,k) = cn( i * Integral C(x,k) dx, k).
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. A325221 (C), A325222 (D).

Cf. A322230 (row reversal), A162005.

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+1)!*polcoeff(polcoeff(S, 2*n+1, x), 2*j, k)}
for(n=0, N, for(j=0, n, print1( T(n,j), ", ")) ; print(""))

E.g.f. S = S(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n+1)*k^(2*j)/(2*n+1)!, along with related series C = C(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+1)!*(2*n)!/(n!^2*4^n) = A079484(n), the product of two consecutive odd double factorials.
Column T(n,0) = A000182(n), where A000182 is the tangent numbers.

A004005 Coefficients of elliptic function sn.

Original entry on oeis.org

1, 135, 5478, 165826, 4494351, 116294673, 2949965020, 74197080276, 1859539731885, 46535238000235, 1163848723925346, 29100851707716150, 727566807977891803, 18189614152200873621, 454744658216502193656

Offset: 2

N. J. A. Sloane, Simon Plouffe

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(5.2.24).
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 = 2..100
A. Cayley, An Elementary Treatise on Elliptic Functions (page images), G. Bell and Sons, London, 1895, p. 56.
A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.
C. L. Mallows, Letter to N. J. A. Sloane, May 16 1973
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.
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.

Leading terms in rows of triangle in A060628.

A004005:=-(-1-89*z+69*z**2+405*z**3)/(-1+25*z)/(9*z-1)**2/(z-1)**3; # Conjectured by Simon Plouffe in his 1992 dissertation.
A004005 := proc(n) A060628(n,2) ; end proc: seq(A004005(n),n=2..40) ; # R. J. Mathar, Jan 30 2011

maxn = 16; se = Series[JacobiSN[u, m], {u, 0, 2*maxn+1}]; cc = Partition[CoefficientList[se, u], 2][[All, 2]]; cc2 = (CoefficientList[#, m] & /@ cc)*Table[(-1)^n*(2*n+1)!, {n, 0, maxn}]; Table[cc2[[n+1, n-1]], {n, 2, maxn}](* Jean-François Alcover, Feb 17 2012 *)

a(n) = (5^(2*n+1) - (8*n-4)*3^(2*n+1) + 32*n^2 - 32*n -17)/256. - Vaclav Kotesovec after Fransen, Jul 30 2013

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

cp's OEIS Frontend

A032349 Number of paths from (0,0) to (3n,0) that stay in first quadrant (but may touch horizontal axis), where each step is (2,1),(1,2) or (1,-1) and start with (2,1).

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A060627 1 + Sum_{n >= 1} Sum_{k = 0..n-1} (-1)^n*T(n,k)*y^(2*k)*x^(2*n)/(2*n)! = JacobiCN(x,y).

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Maple

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A322230 E.g.f.: S(x,k) = Integral C(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.

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A002753 Coefficients of elliptic function sn.

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A279832 Numerators of the coefficients in g.f. A(x) such that: sn(x,i*A(x)) = x, where i^2 = -1, and sn(x,k) is a Jacobi elliptic function.

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A279833 Denominators of the coefficients in g.f. A(x) such that: sn(x,i*A(x)) = x, where i^2 = -1, and sn(x,k) is a Jacobi elliptic function.

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A279834 Numerators of the coefficients in g.f. A(x) such that: sn(x,-A(x)) = x, where sn(x,m) is a Jacobi elliptic function.

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A060627 1 + Sum_{n >= 1} Sum_{k = 0..n-1} (-1)^nT(n,k)y^(2k)x^(2n)/(2n)! = JacobiCN(x,y).

A322230 E.g.f.: S(x,k) = Integral C(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.

A325220 E.g.f.: S(x,k) = -i * sn( i * Integral C(x,k) dx, k) such that C(x,k) = cn( i * Integral C(x,k) dx, k), where S(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2n+1)k^(2j)/(2n+1)!, as a triangle of coefficients T(n,j) read by rows.