|
| |
|
|
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.
|
|
4
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
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.
|
|
|
LINKS
|
|
|
|
FORMULA
|
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.
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.
|
|
|
EXAMPLE
|
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! + ...
|
|
|
PROG
|
(PARI) 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(""))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
