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%I #8 Apr 20 2019 08:32:38
%S 1,0,1,0,8,1,0,136,88,1,0,3968,6240,816,1,0,176896,513536,195216,7376,
%T 1,0,11184128,51880064,39572864,5352544,66424,1,0,951878656,
%U 6453433344,8258202240,2458228480,139127640,597864,1,0,104932671488,978593947648,1889844670464,994697838080,137220256000,3535586112,5380832,1,0,14544442556416,178568645312512,485265505927168,398800479698944,102950036177920,7233820923904,88992306208,48427552,1
%N E.g.f.: D(x,k) = dn( i * Integral C(x,k) dx, k) such that C(x,k) = cn( i * Integral C(x,k) dx, k), where D(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n)*k^(2*j)/(2*n)!, as a triangle of coefficients T(n,j) read by rows.
%C Equals a row reversal of triangle A322231.
%C Compare to dn(x,k) = 1 - k^2 * Integral sn(x,k)*cn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions.
%F E.g.f. D = D(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 C = C(x,k), satisfies:
%F (1a) S = Integral C^2*D dx.
%F (1b) C = 1 + Integral S*C*D dx.
%F (1c) D = 1 + k^2 * Integral S*C^2 dx.
%F (2a) C^2 - S^2 = 1.
%F (2b) D^2 - k^2*S^2 = 1.
%F (3a) C + S = exp( Integral C*D dx ).
%F (3b) D + k*S = exp( k * Integral C^2 dx ).
%F (4a) S = sinh( Integral C*D dx ).
%F (4b) S = sinh( k * Integral C^2 dx ) / k.
%F (4c) C = cosh( Integral C*D dx ).
%F (4d) D = cosh( k * Integral C^2 dx ).
%F (5a) d/dx S = C^2*D.
%F (5b) d/dx C = S*C*D.
%F (5c) d/dx D = k^2 * S*C^2.
%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 C dx, k),
%F (6b) C = cn( i * Integral C dx, k),
%F (6c) D = dn( i * Integral C dx, k).
%F (7a) S = sc( Integral C dx, k') = sn(Integral C dx, k')/cn(Integral C dx, k'),
%F (7b) C = nc( Integral C dx, k') = 1/cn(Integral C dx, k'),
%F (7c) D = dc( Integral C dx, k') = dn(Integral C dx, k')/cn(Integral C dx, k').
%F Row sums equal ( (2*n)!/(n!*2^n) )^2 = A001818(n), the squares of the odd double factorials.
%F Column T(n,n+1) = 2^n*A002105(n+1), for n>=0, where A002105 gives the reduced tangent numbers.
%e E.g.f.: 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! + ...
%e such that D(x,k) = dn( i * Integral C(x,k) dx, k) where C(x,k) = cn( i * Integral C(x,k) dx, k).
%e This triangle of coefficients T(n,j) of x^(2*n)*k^(2*j)/(2*n)! in e.g.f. D(x,k) begins:
%e 1;
%e 0, 1;
%e 0, 8, 1;
%e 0, 136, 88, 1;
%e 0, 3968, 6240, 816, 1;
%e 0, 176896, 513536, 195216, 7376, 1;
%e 0, 11184128, 51880064, 39572864, 5352544, 66424, 1;
%e 0, 951878656, 6453433344, 8258202240, 2458228480, 139127640, 597864, 1;
%e 0, 104932671488, 978593947648, 1889844670464, 994697838080, 137220256000, 3535586112, 5380832, 1;
%e 0, 14544442556416, 178568645312512, 485265505927168, 398800479698944, 102950036177920, 7233820923904, 88992306208, 48427552, 1; ...
%e RELATED SERIES.
%e The related series S(x,k), where D(x,k)^2 - k^2*S(x,k)^2 = 1, starts
%e 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! + ...
%e The related series C(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
%e 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! + ...
%e which also satisfies C(x,k) = cn( i * Integral C(x,k) dx, k).
%o (PARI) N=10;
%o {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)); }
%o {T(n,j) = (2*n)!*polcoeff(polcoeff(D, 2*n, x), 2*j, k)}
%o for(n=0, N, for(j=0, n, print1( T(n,j), ", ")) ; print(""))
%Y Cf. A325220 (S), A325221(C).
%Y Cf. A322231 (row reversal).
%K tabl,nonn
%O 0,5
%A _Paul D. Hanna_, Apr 13 2019
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