A002754 -id:A002754 - cp's OEIS frontend

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

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

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.

A172259 Let CK(m) denote the complete elliptic integral of the first kind. a(n) is the n-th smallest integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

Original entry on oeis.org

1, 2, 5, 14, 38, 101, 275, 746, 2026, 5507, 14969, 40689, 110604, 300652, 817255, 2221528, 6038739, 16414993, 44620576, 121291299, 329703934, 896228212, 2436200862, 6622280533, 18001224835, 48932402358, 133012060152, 361564266077, 982833574297, 2671618645410

Offset: 1

Michel Lagneau, Jan 30 2010

nonn

F(z,k) = Integral_{t=0..z} 1/(sqrt(1-t^2)*sqrt(1-k^2*t^2)) dt and the complete elliptic integral CK is defined by CK(k) = F(1,sqrt(1-k^2)). We calculate the values of CK(k) with k = 1/p, p = 1,2,3, ... and we propose a very interesting property: a(n+1)/a(n) tends toward e = 2.7182818... when n tends to infinity. For example, a(8) / a(7) = 2.718281581; a(9) / a(8) = 2.7182817562.

			a(3) = 38 because floor(CK(1/37)) = 4 and floor(CK(1/38)) = 5.

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.
M. Abramowitz and I. Stegun, "Elliptic Integrals", Chapter 17 of Handbook of Mathematical Functions. Dover Publications Inc., New York, 1046 p., (1965).
A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

A. Cayley, An Elementary Treatise on Elliptic Functions, G. Bell and Sons, London, 1895, p. 56.
F. Clarke, The Taylor Series Coefficients of the Jacobi Elliptic Functions, slides. [broken link]

Cf. elliptic functions: A001936, A002318, A001937, A001934, A001938, A002754, A001939, A001940, A001941, A002753, A006089, A004005.

a0:=1:for p from 1 to 1000 do:a:= evalf(EllipticCK(1/p)):if floor(a)=a0+1 then print(p):a0:=floor(a):else fi:od:

F(z,k) = Integral_{t=0..z} 1/(sqrt(1-t^2)*sqrt(1-k^2*t^2)) dt. CK is defined by CK(k) = F(1,sqrt(1-k^2)). a(n) is the n-th integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

cp's OEIS Frontend

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

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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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A172259 Let CK(m) denote the complete elliptic integral of the first kind. a(n) is the n-th smallest integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

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cp's OEIS Frontend

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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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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A172259 Let CK(m) denote the complete elliptic integral of the first kind. a(n) is the n-th smallest integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

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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).