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%I #46 Apr 26 2019 20:40:14
%S 1,1,0,1,8,0,1,88,136,0,1,816,6240,3968,0,1,7376,195216,513536,176896,
%T 0,1,66424,5352544,39572864,51880064,11184128,0,1,597864,139127640,
%U 2458228480,8258202240,6453433344,951878656,0,1,5380832,3535586112,137220256000,994697838080,1889844670464,978593947648,104932671488,0,1,48427552,88992306208,7233820923904,102950036177920,398800479698944,485265505927168,178568645312512,14544442556416,0
%N 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.
%C Equals a row reversal of triangle A325222.
%C 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).
%C Compare also to Michael Pawellek's generalized elliptic functions.
%H Paul D. Hanna, <a href="/A322231/b322231.txt">Table of n, a(n) for n = 0..860 terms of this triangle read by rows 0..40.</a>
%F 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:
%F (1a) S = Integral C*D^2 dx.
%F (1b) C = 1 + Integral S*D^2 dx.
%F (1c) D = 1 + k^2 * Integral S*C*D dx.
%F (2a) C^2 - S^2 = 1.
%F (2b) D^2 - k^2*S^2 = 1.
%F (3a) C + S = exp( Integral D^2 dx ).
%F (3b) D + k*S = exp( k * Integral C*D dx ).
%F (4a) S = sinh( Integral D^2 dx ).
%F (4b) S = sinh( k * Integral C*D dx ) / k.
%F (4c) C = cosh( Integral D^2 dx ).
%F (4d) D = cosh( k * Integral C*D dx ).
%F (5a) d/dx S = C*D^2.
%F (5b) d/dx C = S*D^2.
%F (5c) d/dx D = k^2 * S*C*D.
%F From _Paul D. Hanna_, Mar 31 2019, Apr 20 2019 (Start):
%F 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
%F (6a) S = -i * sn( i * Integral D dx, k),
%F (6b) C = cn( i * Integral D dx, k),
%F (6c) D = dn( i * Integral D dx, k).
%F (7a) S = sc( Integral D dx, k') = sn(Integral D dx, k')/cn(Integral D dx, k'),
%F (7b) C = nc( Integral D dx, k') = 1/cn(Integral D dx, k'),
%F (7c) D = dc( Integral D dx, k') = dn(Integral D dx, k')/cn(Integral D dx, k'). (End)
%F Row sums equal ( (2*n)!/(n!*2^n) )^2 = A001818(n), the squares of the odd double factorials.
%F Diagonal T(n+1,n) = 2^n*A002105(n+1), for n>=0, where A002105 gives the reduced tangent numbers.
%e 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! + ...
%e such that C(x,k)^2 - S(x,k)^2 = 1.
%e This triangle of coefficients T(n,j) of x^(2*n)*k^(2*j)/(2*n)! in e.g.f. C(x,k) begins:
%e 1;
%e 1, 0;
%e 1, 8, 0;
%e 1, 88, 136, 0;
%e 1, 816, 6240, 3968, 0;
%e 1, 7376, 195216, 513536, 176896, 0;
%e 1, 66424, 5352544, 39572864, 51880064, 11184128, 0;
%e 1, 597864, 139127640, 2458228480, 8258202240, 6453433344, 951878656, 0;
%e 1, 5380832, 3535586112, 137220256000, 994697838080, 1889844670464, 978593947648, 104932671488, 0;
%e 1, 48427552, 88992306208, 7233820923904, 102950036177920, 398800479698944, 485265505927168, 178568645312512, 14544442556416, 0; ...
%e RELATED SERIES.
%e The related series S(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
%e 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! + ...
%e The related series D(x,k), where D(x,k)^2 - k^2*S(x,k)^2 = 1, starts
%e 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! + ...
%o (PARI) N=10;
%o {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));}
%o for(n=0,N, for(j=0,n, print1( (2*n)!*polcoeff(polcoeff(C,2*n,x),2*j,k),", ")) ;print(""))
%Y Cf. A322230 (S), A322232 (D), A001818 (row sums), A002105.
%Y Cf. A325222 (row reversal).
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Dec 14 2018
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