cp's OEIS Frontend

A322219 E.g.f.: S(x,q) = Integral C(x,q) * C(q*x,q) dx, such that C(x,q)^2 - S(x,q)^2 = 1, where S(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.

%I A322219 #49 Mar 24 2019 19:42:51
%S A322219 1,1,1,1,10,1,4,1,35,91,85,20,16,24,1,84,966,1324,2737,632,1200,288,
%T A322219 384,128,192,1,165,5082,26818,50941,64329,69816,49872,27568,19776,
%U A322219 16448,10176,5760,3840,1280,1920,1,286,18447,279136,954239,2550054,2455233,4013788,2929104,3264864,1176640,1815552,834752,731136,394752,491264,141056,164352,69120,46080,15360,23040,1,455,53053,1780207,15627183,51699869,128611679,187372653,213804652,257006976,245800968,195109120,161177792,123750592,83792000,69316224,52893696,35140992,28215808,18433536,12687360,8411648,5945856,2673664,2300928,967680,645120,215040,322560
%N A322219 E.g.f.: S(x,q) = Integral C(x,q) * C(q*x,q) dx, such that C(x,q)^2 - S(x,q)^2 = 1, where S(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 A322219 Compare to Jacobi's elliptic function sn(x,k) = Integral cn(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 A322219 Right border equals A002866.
%C A322219 Row sums equal the tangent numbers (A000182).
%C A322219 Last n terms in row n of this triangle and of triangle A322218 are equal for n>0.
%H A322219 Paul D. Hanna, <a href="/A322219/b322219.txt">Table of n, a(n) for n = 0..4990, as a flattened irregular triangle read by rows 0..30.</a>
%F A322219 E.g.f. S(x,q) and related series C(x,q) satisfy:
%F A322219 (1) C(x,q)^2 - S(x,q)^2 = 1.
%F A322219 (2) C(x,q) = 1 + Integral S(x,q) * C(q*x,q) dx.
%F A322219 (3) S(x,q) = Integral C(x,q) * C(q*x,q) dx.
%F A322219 (4a) C(x,q) + S(x,q) = exp( Integral C(q*x,q) dx ).
%F A322219 (4b) C(x,q) = cosh( Integral C(q*x,q) dx ).
%F A322219 (4c) S(x,q) = sinh( Integral C(q*x,q) dx ).
%F A322219 (5) C(q*x,q) = 1 + q * Integral S(q*x,q) * C(q^2*x,q) dx.
%F A322219 (6) S(q*x,q) = q * Integral C(q*x,q) * C(q^2*x,q) dx.
%F A322219 (7a) C(q*x,q) + S(q*x,q) = exp( q * Integral C(q^2*x,q) dx ).
%F A322219 (7b) C(q*x,q) = cosh( q * Integral C(q^2*x,q) dx ).
%F A322219 (7c) S(q*x,q) = sinh( q * Integral C(q^2*x,q) dx ).
%F A322219 PARTICULAR ARGUMENTS.
%F A322219 S(x,q=0) = sinh(x).
%F A322219 S(x,q=1) = tan(x).
%F A322219 S(x,q=i) = -i * sl(i*x), where sl(x) is the sine lemniscate function (A104203).
%F A322219 FORMULAS FOR TERMS.
%F A322219 T(n, n*(n+1)/2) = 2^(n-1)*n! for n >= 1.
%F A322219 T(n, n*(n+1)/2 - k) = A322218(n, n*(n-1)/2 - k) for k = 0..n-1, n > 0.
%F A322219 Sum_{k=0..n*(n+1)/2} T(n,k) = A000182(n+1) for n >= 0.
%F A322219 Sum_{k=0..n*(n+1)/2} T(n,k)*(-1)^k = A104203(2*n+1) for n >= 0.
%e A322219 E.g.f. S(x,q) = Sum_{n>=0} sum_{k=0..n*(n+1)/2} T(n,k)*x^(2*n+1)*q^(2*k)/(2*n+1)! starts
%e A322219 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 A322219 such that S(x,q) = sinh( Integral C(q*x,q) dx ) and C(x,q)^2 = 1 + S(x,q)^2.
%e A322219 This irregular triangle of coefficients T(n,k) of x^(2*n+1)*q^(2*k)/(2*n+1)! in S(x,q) begins:
%e A322219 1;
%e A322219 1, 1;
%e A322219 1, 10, 1, 4;
%e A322219 1, 35, 91, 85, 20, 16, 24;
%e A322219 1, 84, 966, 1324, 2737, 632, 1200, 288, 384, 128, 192;
%e A322219 1, 165, 5082, 26818, 50941, 64329, 69816, 49872, 27568, 19776, 16448, 10176, 5760, 3840, 1280, 1920;
%e A322219 1, 286, 18447, 279136, 954239, 2550054, 2455233, 4013788, 2929104, 3264864, 1176640, 1815552, 834752, 731136, 394752, 491264, 141056, 164352, 69120, 46080, 15360, 23040;
%e A322219 1, 455, 53053, 1780207, 15627183, 51699869, 128611679, 187372653, 213804652, 257006976, 245800968, 195109120, 161177792, 123750592, 83792000, 69316224, 52893696, 35140992, 28215808, 18433536, 12687360, 8411648, 5945856, 2673664, 2300928, 967680, 645120, 215040, 322560;
%e A322219 1, 680, 129948, 8212360, 163115238, 1001312104, 3705217660, 7815443320, 15434182497, 17298854576, 23429393056, 21144463040, 25624143104, 18454639872, 18756800128, 12036914176, 12076688384, 7122865152, 7609525248, 4420732928, 4042876928, 2553473024, 2465701888, 1353586688, 1234018304, 619528192, 587358208, 311279616, 255467520, 117383168, 108036096, 42778624, 36814848, 15482880, 10321920, 3440640, 5160960; ...
%e A322219 RELATED SERIES.
%e A322219 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 A322219 such that C(x,q) = cosh( Integral C(q*x,q) dx ).
%t A322219 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 A322219 CoefficientList[#, q^2]& /@ (CoefficientList[s[x, q], x] Range[0, m-1]!) // DeleteCases[#, {}]& // Flatten (* _Jean-François Alcover_, Dec 17 2018 *)
%o A322219 (PARI) {T(n,k) = my(S=x,C=1); for(i=1,2*n,
%o A322219 S = intformal(C*subst(C,x,q*x) +O(x^(2*n+1)));
%o A322219 C = 1 + intformal(S*subst(C,x,q*x)));
%o A322219 (2*n+1)!*polcoeff( polcoeff(S,2*n+1,x),2*k,q)}
%o A322219 for(n=0,10, for(k=0,n*(n+1)/2, print1( T(n,k),", "));print(""))
%Y A322219 Cf. A322218 (C(x,q)), A000182 (row sums), A104203, A002866.
%K A322219 nonn,tabf
%O A322219 0,5
%A A322219 _Paul D. Hanna_, Dec 16 2018