A322218 - OEIS

%I #47 Mar 24 2019 19:42:33

%S 1,1,1,4,1,20,16,24,1,56,336,288,384,128,192,1,120,2352,6448,12736,

%T 5888,10176,5760,3840,1280,1920,1,220,10032,93280,214016,472704,

%U 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

%N 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.

%C 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.

%C Right border equals A002866.

%C Row sums equal the secant numbers (A000364).

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

%H Paul D. Hanna, <a href="/A322218/b322218.txt">Table of n, a(n) for n = 0..4525, as a flattened irregular triangle read by rows 0..30.</a>

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

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

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

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

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

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

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

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

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

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

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

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

%F PARTICULAR ARGUMENTS.

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

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

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

%F FORMULAS FOR TERMS.

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

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

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

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

%e 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

%e 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! + ...

%e such that C(x,q) = cosh( Integral C(q*x,q) dx ).

%e This irregular triangle of coefficients T(n,k) of x^(2*n)*q^(2*k)/(2*n)! in C(x,q) begins:

%e 1;

%e 1;

%e 1, 4;

%e 1, 20, 16, 24;

%e 1, 56, 336, 288, 384, 128, 192;

%e 1, 120, 2352, 6448, 12736, 5888, 10176, 5760, 3840, 1280, 1920;

%e 1, 220, 10032, 93280, 214016, 472704, 385472, 431616, 294912, 341504, 141056, 164352, 69120, 46080, 15360, 23040;

%e 1, 364, 32032, 740168, 4072640, 11702912, 18676672, 30112640, 23848704, 27599616, 17884032, 20958208, 13595136, 11074560, 5992448, 5945856, 2673664, 2300928, 967680, 645120, 215040, 322560;

%e 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; ...

%e RELATED SERIES.

%e 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! +  ...

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

%t 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}];

%t CoefficientList[#, q^2]& /@ (CoefficientList[c[x, q], x] Range[0, m]!) // DeleteCases[#, {}]& // Flatten (* _Jean-François Alcover_, Dec 17 2018 *)

%o (PARI) {T(n,k) = my(S=x,C=1); for(i=1,2*n,

%o S = intformal(C*subst(C,x,q*x) +O(x^(2*n+1)));

%o C = 1 + intformal(S*subst(C,x,q*x)));

%o (2*n)!*polcoeff( polcoeff(C,2*n,x),2*k,q)}

%o for(n=0,10, for(k=0,n*(n-1)/2, print1( T(n,k),", "));print(""))

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

%K nonn,tabf

%O 0,4

%A _Paul D. Hanna_, Dec 16 2018