A325222 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.

1, 0, 1, 0, 8, 1, 0, 136, 88, 1, 0, 3968, 6240, 816, 1, 0, 176896, 513536, 195216, 7376, 1, 0, 11184128, 51880064, 39572864, 5352544, 66424, 1, 0, 951878656, 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

Offset: 0

Author

Paul D. Hanna, Apr 13 2019

Keywords

• tabl
• nonn

Comments

Equals a row reversal of triangle A322231.

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.

Examples

			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! + ...
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).
This triangle of coefficients T(n,j) of x^(2*n)*k^(2*j)/(2*n)! in e.g.f. D(x,k) begins:
1;
0, 1;
0, 8, 1;
0, 136, 88, 1;
0, 3968, 6240, 816, 1;
0, 176896, 513536, 195216, 7376, 1;
0, 11184128, 51880064, 39572864, 5352544, 66424, 1;
0, 951878656, 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; ...
RELATED SERIES.
The related series S(x,k), where D(x,k)^2 - k^2*S(x,k)^2 = 1, starts
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! + ...
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).
		

Crossrefs

Cf. A325220 (S), A325221(C).

Cf. A322231 (row reversal).

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)!*polcoeff(polcoeff(D, 2*n, x), 2*j, k)}
for(n=0, N, for(j=0, n, print1( T(n,j), ", ")) ; print(""))

Formula

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:

(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)!/(n!*2^n) )^2 = A001818(n), the squares of the odd double factorials.

Column T(n,n+1) = 2^n*A002105(n+1), for n>=0, where A002105 gives the reduced tangent numbers.