A325220 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.

1, 2, 1, 16, 28, 1, 272, 1032, 270, 1, 7936, 52736, 36096, 2456, 1, 353792, 3646208, 4766048, 1035088, 22138, 1, 22368256, 330545664, 704357760, 319830400, 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

Offset: 0

Author

Paul D. Hanna, Apr 13 2019

Keywords

• nonn
• tabl

Comments

Equals a row reversal of triangle A322230.

Appears to equal 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).

Examples

			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! + ...
such that S(x,k) = cn( i * Integral C(x,k) dx, k) and 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;
2, 1;
16, 28, 1;
272, 1032, 270, 1;
7936, 52736, 36096, 2456, 1;
353792, 3646208, 4766048, 1035088, 22138, 1;
22368256, 330545664, 704357760, 319830400, 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; ...
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! + (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! + ...
which also satisfies C(x,k) = cn( i * Integral C(x,k) dx, k).
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! + (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! + ...
		

Crossrefs

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

Cf. A322230 (row reversal), A162005.

Programs

PARI

N=10;
{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)); }
{T(n,j) = (2*n+1)!*polcoeff(polcoeff(S, 2*n+1, x), 2*j, k)}
for(n=0, N, for(j=0, n, print1( T(n,j), ", ")) ; print(""))

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^2*D dx.

(1b) C = 1 + Integral S*C*D dx.

(1c) D = 1 + k^2 * Integral S*C^2 dx.

(2a) C^2 - S^2 = 1.

(2b) D^2 - k^2*S^2 = 1.

(3a) C + S = exp( Integral C*D dx ).

(3b) D + k*S = exp( k * Integral C^2 dx ).

(4a) S = sinh( Integral C*D dx ).

(4b) S = sinh( k * Integral C^2 dx ) / k.

(4c) C = cosh( Integral C*D dx ).

(4d) D = cosh( k * Integral C^2 dx ).

(5a) d/dx S = C^2*D.

(5b) d/dx C = S*C*D.

(5c) d/dx D = k^2 * S*C^2.

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 C dx, k),

(6b) C = cn( i * Integral C dx, k),

(6c) D = dn( i * Integral C dx, k).

(7a) S = sc( Integral C dx, k') = sn(Integral C dx, k')/cn(Integral C dx, k'),

(7b) C = nc( Integral C dx, k') = 1/cn(Integral C dx, k'),

(7c) D = dc( Integral C dx, k') = dn(Integral C dx, k')/cn(Integral C dx, k').

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

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