A325221 E.g.f.: C(x,k) = cn( i * Integral C(x,k) dx, k), where C(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, 1, 0, 5, 4, 0, 61, 148, 16, 0, 1385, 6744, 2832, 64, 0, 50521, 410456, 383856, 47936, 256, 0, 2702765, 32947964, 54480944, 17142784, 780544, 1024, 0, 199360981, 3402510924, 8760740640, 5199585280, 686711040, 12555264, 4096, 0, 19391512145, 441239943664, 1632067372896, 1569971730560, 419867864320, 26090711040, 201199616, 16384, 0, 2404879675441, 70347660061552, 353538702361888, 502094919789184, 227204970315520, 30892394850304, 965223559168, 3220652032, 65536, 0

Offset: 0

Author

Paul D. Hanna, Apr 13 2019

Keywords

• nonn
• tabl

Comments

Equals a row reversal of triangle A322232.

Compare to cn(x,k) = 1 - Integral sn(x,k)*dn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions (see triangle A060627).

Examples

			E.g.f.: 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! + ...
such that 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. C(x,k) begins:
1;
1, 0;
5, 4, 0;
61, 148, 16, 0;
1385, 6744, 2832, 64, 0;
50521, 410456, 383856, 47936, 256, 0;
2702765, 32947964, 54480944, 17142784, 780544, 1024, 0;
199360981, 3402510924, 8760740640, 5199585280, 686711040, 12555264, 4096, 0;
19391512145, 441239943664, 1632067372896, 1569971730560, 419867864320, 26090711040, 201199616, 16384, 0;
2404879675441, 70347660061552, 353538702361888, 502094919789184, 227204970315520, 30892394850304, 965223559168, 3220652032, 65536, 0; ...
RELATED SERIES.
The related series S(x,k), where C(x,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 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. A325220 (S), A325222(D).

Cf. A322232 (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(C, 2*n, x), 2*j, k)}
for(n=0, N, for(j=0, n, print1( T(n,j), ", ")) ; print(""))

Formula

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

Column T(n,0) = A000364(n), for n>=0, where A000364 is the secant numbers.