A325220 - OEIS

%I #15 Apr 26 2019 20:39:30

%S 1,2,1,16,28,1,272,1032,270,1,7936,52736,36096,2456,1,353792,3646208,

%T 4766048,1035088,22138,1,22368256,330545664,704357760,319830400,

%U 27426960,199284,1,1903757312,38188155904,120536980224,93989648000,18598875760,702812568,1793606,1,209865342976,5488365862912,24060789342208,28745874079744,10324483102720,1002968825344,17753262208,16142512,1,29088885112832,961530104709120,5590122715250688,9498855414644736,5416305638467584,1013356176688128,51882638754240,445736371872,145282674,1

%N E.g.f.: S(x,k) = -i * sn( i * Integral C(x,k) dx, k) such that C(x,k) = cn( i * Integral C(x,k) dx, k), where S(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n+1)*k^(2*j)/(2*n+1)!, as a triangle of coefficients T(n,j) read by rows.

%C Equals a row reversal of triangle A322230.

%C Appears to equal EG1 triangle A162005, which has other formulas.

%C Compare to sn(x,k) = Integral cn(x,k)*dn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions (see triangle A060628).

%F E.g.f. S = S(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n+1)*k^(2*j)/(2*n+1)!, along with related series C = C(x,k) and D = D(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+1)!*(2*n)!/(n!^2*4^n) = A079484(n), the product of two consecutive odd double factorials.

%F Column T(n,0) = A000182(n), where A000182 is the tangent numbers.

%e E.g.f.: 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 such that S(x,k) = cn( i * Integral C(x,k) dx, k) and C(x,k)^2 - S(x,k)^2 = 1.

%e This triangle of coefficients T(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in e.g.f. S(x,k) begins:

%e 1;

%e 2, 1;

%e 16, 28, 1;

%e 272, 1032, 270, 1;

%e 7936, 52736, 36096, 2456, 1;

%e 353792, 3646208, 4766048, 1035088, 22138, 1;

%e 22368256, 330545664, 704357760, 319830400, 27426960, 199284, 1;

%e 1903757312, 38188155904, 120536980224, 93989648000, 18598875760, 702812568, 1793606, 1;

%e 209865342976, 5488365862912, 24060789342208, 28745874079744, 10324483102720, 1002968825344, 17753262208, 16142512, 1;

%e 29088885112832, 961530104709120, 5590122715250688, 9498855414644736, 5416305638467584, 1013356176688128, 51882638754240, 445736371872, 145282674, 1; ...

%e RELATED SERIES.

%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).

%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! + (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! + ...

%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+1)!*polcoeff(polcoeff(S, 2*n+1, x), 2*j, k)}

%o for(n=0, N, for(j=0, n, print1( T(n,j), ", ")) ; print(""))

%Y Cf. A325221 (C), A325222 (D).

%Y Cf. A322230 (row reversal), A162005.

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Apr 13 2019