A322218 E.g.f.: C(x,q) = 1 + Integral S(x,q) * C(q*x,q) dx, such that C(x,q)^2 - S(x,q)^2 = 1, where C(x,q) = Sum_{n>=0} sum_{k=0..n*(n-1)/2} T(n,k)*x^n*y^k/n!, as an irregular triangle of coefficients T(n,k) read by rows.

1, 1, 1, 4, 1, 20, 16, 24, 1, 56, 336, 288, 384, 128, 192, 1, 120, 2352, 6448, 12736, 5888, 10176, 5760, 3840, 1280, 1920, 1, 220, 10032, 93280, 214016, 472704, 385472, 431616, 294912, 341504, 141056, 164352, 69120, 46080, 15360, 23040, 1, 364, 32032, 740168, 4072640, 11702912, 18676672, 30112640, 23848704, 27599616, 17884032, 20958208, 13595136, 11074560, 5992448, 5945856, 2673664, 2300928, 967680, 645120, 215040, 322560, 1, 560, 84448, 3952832, 53301248, 230161152, 738249344, 1166436352, 1970874368, 2196244480, 2459786240, 1804101632, 2061498368, 1537437696, 1437724672, 989968384, 921092096, 487923712, 499621888, 282034176, 211599360, 117383168, 108036096, 42778624, 36814848, 15482880, 10321920, 3440640, 5160960

Offset: 0

Author

Paul D. Hanna, Dec 16 2018

Keywords

• nonn
• tabf

Comments

Compare to Jacobi's elliptic function cn(x,k) = 1 - Integral sn(x,k)*dn(x,k) dx such that cn(x,k)^2 + sn(x,k)^2 = 1 and dn(x,k)^2 + k^2*sn(x,k)^2 = 1.

Right border equals A002866.

Row sums equal the secant numbers (A000364).

Last n terms in row n of this triangle and of triangle A322219 are equal for n>0.

Examples

			E.g.f. C(x,q) = Sum_{n>=0} sum_{k=0..n*(n-1)/2} T(n,k) * x^(2*n)*q^(2*k)/(2*n)! starts
C(x,q) = 1 + x^2/2! + (4*q^2 + 1)*x^4/4! + (24*q^6 + 16*q^4 + 20*q^2 + 1)*x^6/6! + (192*q^12 + 128*q^10 + 384*q^8 + 288*q^6 + 336*q^4 + 56*q^2 + 1)*x^8/8! + (1920*q^20 + 1280*q^18 + 3840*q^16 + 5760*q^14 + 10176*q^12 + 5888*q^10 + 12736*q^8 + 6448*q^6 + 2352*q^4 + 120*q^2 + 1)*x^10/10! + (23040*q^30 + 15360*q^28 + 46080*q^26 + 69120*q^24 + 164352*q^22 + 141056*q^20 + 341504*q^18 + 294912*q^16 + 431616*q^14 + 385472*q^12 + 472704*q^10 + 214016*q^8 + 93280*q^6 + 10032*q^4 + 220*q^2 + 1)*x^12/12! + ...
such that C(x,q) = cosh( Integral C(q*x,q) dx ).
This irregular triangle of coefficients T(n,k) of x^(2*n)*q^(2*k)/(2*n)! in C(x,q) begins:
1;
1;
1, 4;
1, 20, 16, 24;
1, 56, 336, 288, 384, 128, 192;
1, 120, 2352, 6448, 12736, 5888, 10176, 5760, 3840, 1280, 1920;
1, 220, 10032, 93280, 214016, 472704, 385472, 431616, 294912, 341504, 141056, 164352, 69120, 46080, 15360, 23040;
1, 364, 32032, 740168, 4072640, 11702912, 18676672, 30112640, 23848704, 27599616, 17884032, 20958208, 13595136, 11074560, 5992448, 5945856, 2673664, 2300928, 967680, 645120, 215040, 322560;
1, 560, 84448, 3952832, 53301248, 230161152, 738249344, 1166436352, 1970874368, 2196244480, 2459786240, 1804101632, 2061498368, 1537437696, 1437724672, 989968384, 921092096, 487923712, 499621888, 282034176, 211599360, 117383168, 108036096, 42778624, 36814848, 15482880, 10321920, 3440640, 5160960; ...
RELATED SERIES.
S(x,q) = x + (q^2 + 1)*x^3/3! + (4*q^6 + q^4 + 10*q^2 + 1)*x^5/5! + (24*q^12 + 16*q^10 + 20*q^8 + 85*q^6 + 91*q^4 + 35*q^2 + 1)*x^7/7! + (192*q^20 + 128*q^18 + 384*q^16 + 288*q^14 + 1200*q^12 + 632*q^10 + 2737*q^8 + 1324*q^6 + 966*q^4 + 84*q^2 + 1)*x^9/9! + (1920*q^30 + 1280*q^28 + 3840*q^26 + 5760*q^24 + 10176*q^22 + 16448*q^20 + 19776*q^18 + 27568*q^16 + 49872*q^14 + 69816*q^12 + 64329*q^10 + 50941*q^8 + 26818*q^6 + 5082*q^4 + 165*q^2 + 1)*x^11/11! +  ...
where C(x,q)^2 - S(x,q)^2 = 1.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..4525, as a flattened irregular triangle read by rows 0..30.

Crossrefs

Cf. A322219 (S(x,q)), A000364 (row sums), A193544.

Programs

Mathematica

rows = 8; m = 2 rows; s[x_, ] = x; c[, ] = 1; Do[s[x, q_] = Integrate[c[x, q] c[q x, q] + O[x]^m // Normal, x]; c[x_, q_] = 1 + Integrate[s[x, q] c[q x, q] + O[x]^m // Normal, x], {m}];
CoefficientList[#, q^2]& /@ (CoefficientList[c[x, q], x] Range[0, m]!) // DeleteCases[#, {}]& // Flatten (* Jean-François Alcover, Dec 17 2018 *)

PARI

{T(n,k) = my(S=x,C=1); for(i=1,2*n,
S = intformal(C*subst(C,x,q*x) +O(x^(2*n+1)));
C = 1 + intformal(S*subst(C,x,q*x)));
(2*n)!*polcoeff( polcoeff(C,2*n,x),2*k,q)}
for(n=0,10, for(k=0,n*(n-1)/2, print1( T(n,k),", "));print(""))

Formula

E.g.f. C(x,q) and related series S(x,q) satisfy:

(1) C(x,q)^2 - S(x,q)^2 = 1.

(2) C(x,q) = 1 + Integral S(x,q) * C(q*x,q) dx.

(3) S(x,q) = Integral C(x,q) * C(q*x,q) dx.

(4a) C(x,q) + S(x,q) = exp( Integral C(q*x,q) dx ).

(4b) C(x,q) = cosh( Integral C(q*x,q) dx ).

(4c) S(x,q) = sinh( Integral C(q*x,q) dx ).

(5) C(q*x,q) = 1 + q * Integral S(q*x,q) * C(q^2*x,q) dx.

(6) S(q*x,q) = q * Integral C(q*x,q) * C(q^2*x,q) dx.

(7a) C(q*x,q) + S(q*x,q) = exp( q * Integral C(q^2*x,q) dx ).

(7b) C(q*x,q) = cosh( q * Integral C(q^2*x,q) dx ).

(7c) S(q*x,q) = sinh( q * Integral C(q^2*x,q) dx ).

PARTICULAR ARGUMENTS.

C(x,q=0) = cosh(x).

C(x,q=1) = 1/cos(x).

C(x,q=i) = cl(i*x), where cl(x) is the cosine lemniscate function (A159600).

FORMULAS FOR TERMS.

T(n, n*(n-1)/2) = 2^(n-1)*n! for n >= 1.

T(n, n*(n-1)/2 - k) = A322219(n, n*(n+1)/2 - k) for k = 0..n-1, n > 0.

Sum_{k=0..n*(n-1)/2} T(n,k) = A000364(n) for n >= 0.

Sum_{k=0..n*(n-1)/2} T(n,k)*(-1)^k = A193544(2*n+1) for n >= 0.