A322230 E.g.f.: S(x,k) = Integral C(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.

1, 1, 2, 1, 28, 16, 1, 270, 1032, 272, 1, 2456, 36096, 52736, 7936, 1, 22138, 1035088, 4766048, 3646208, 353792, 1, 199284, 27426960, 319830400, 704357760, 330545664, 22368256, 1, 1793606, 702812568, 18598875760, 93989648000, 120536980224, 38188155904, 1903757312, 1, 16142512, 17753262208, 1002968825344, 10324483102720, 28745874079744, 24060789342208, 5488365862912, 209865342976, 1, 145282674, 445736371872, 51882638754240, 1013356176688128, 5416305638467584, 9498855414644736, 5590122715250688, 961530104709120, 29088885112832

Offset

0,3

Comments

Equals a row reversal of triangle A325220.

Appears to be a row reversal of EG1 triangle A162005, which has other formulas.

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).

Compare also to Michael Pawellek's generalized elliptic functions.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..860 terms of this triangle read by rows 0..40.

Formula

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:

(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.

From Paul D. Hanna, Mar 31 2019, Apr 20 2019 (Start):

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+1)!*(2*n)!/(n!^2*4^n) = A079484(n), the product of two consecutive odd double factorials.

Main diagonal equals A000182, the tangent numbers.

Example

E.g.f.: 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! + ...
such that C(x,k)^2 - S(x,k)^2 = 1.
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:
1;
1, 2;
1, 28, 16;
1, 270, 1032, 272;
1, 2456, 36096, 52736, 7936;
1, 22138, 1035088, 4766048, 3646208, 353792;
1, 199284, 27426960, 319830400, 704357760, 330545664, 22368256;
1, 1793606, 702812568, 18598875760, 93989648000, 120536980224, 38188155904, 1903757312;
1, 16142512, 17753262208, 1002968825344, 10324483102720, 28745874079744, 24060789342208, 5488365862912, 209865342976; ...
RELATED SERIES.
The related series C(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
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! + ...
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+1)!*polcoeff(polcoeff(S, 2*n+1, x), 2*j, k), ", ")) ; print(""))

Crossrefs

Cf. A000182 (diagonal), A162006, A162007, A162008, A162009, A162010.

Cf. A322231 (C), A322232 (D).

Cf. A325220 (row reversal), A162005.

Sequence in context: A089963 A362013 A336911 * A353910 A353913 A012411

Adjacent sequences: A322227 A322228 A322229 * A322231 A322232 A322233

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 14 2018

Status

approved