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A322231 E.g.f.: C(x,k) = 1 + Integral S(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. 4
1, 1, 0, 1, 8, 0, 1, 88, 136, 0, 1, 816, 6240, 3968, 0, 1, 7376, 195216, 513536, 176896, 0, 1, 66424, 5352544, 39572864, 51880064, 11184128, 0, 1, 597864, 139127640, 2458228480, 8258202240, 6453433344, 951878656, 0, 1, 5380832, 3535586112, 137220256000, 994697838080, 1889844670464, 978593947648, 104932671488, 0, 1, 48427552, 88992306208, 7233820923904, 102950036177920, 398800479698944, 485265505927168, 178568645312512, 14544442556416, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Equals a row reversal of triangle A325222.

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).

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

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

EXAMPLE

E.g.f.: 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! + ...

such that C(x,k)^2 - S(x,k)^2 = 1.

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;

1, 8, 0;

1, 88, 136, 0;

1, 816, 6240, 3968, 0;

1, 7376, 195216, 513536, 176896, 0;

1, 66424, 5352544, 39572864, 51880064, 11184128, 0;

1, 597864, 139127640, 2458228480, 8258202240, 6453433344, 951878656, 0;

1, 5380832, 3535586112, 137220256000, 994697838080, 1889844670464, 978593947648, 104932671488, 0;

1, 48427552, 88992306208, 7233820923904, 102950036177920, 398800479698944, 485265505927168, 178568645312512, 14544442556416, 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*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! + ...

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)!*polcoeff(polcoeff(C, 2*n, x), 2*j, k), ", ")) ; print(""))

CROSSREFS

Cf. A322230 (S), A322232 (D), A001818 (row sums), A002105.

Cf. A325222 (row reversal).

Sequence in context: A160389 A011104 A232227 * A261168 A329086 A265294

Adjacent sequences:  A322228 A322229 A322230 * A322232 A322233 A322234

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 14 2018

STATUS

approved

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Last modified November 19 19:09 EST 2019. Contains 329323 sequences. (Running on oeis4.)