OFFSET
1,6
COMMENTS
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.
LINKS
FORMULA
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.
EXAMPLE
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).
PROG
(PARI) /* 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(""))
(PARI) {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 */
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Paul D. Hanna, Aug 25 2017
STATUS
approved