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A322622 E.g.f.: S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), where S(x,y) = Sum_{n>=0} Sum_{k=0..2*n+1} T(n,k) * x^(2*n+1-k)*y^k/(2*n+1)!, as a triangle of coefficients T(n,k) read by rows.

%I A322622 #16 Jan 01 2019 10:02:07
%S A322622 1,1,1,6,6,1,1,40,140,140,40,1,1,224,2562,7000,7000,2562,224,1,1,1152,
%T A322622 39384,260736,605808,605808,260736,39384,1152,1,1,5632,541310,8131200,
%U A322622 39032400,80735424,80735424,39032400,8131200,541310,5632,1,1,26624,6908772,225065984,2101762520,8105175936,15355426944,15355426944,8105175936,2101762520,225065984,6908772,26624,1,1,122880,83702010,5726156800,100096487400,680914730496,2234443571360,3951806601600,3951806601600,2234443571360,680914730496,100096487400,5726156800,83702010,122880,1
%N A322622 E.g.f.: S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), where S(x,y) = Sum_{n>=0} Sum_{k=0..2*n+1} T(n,k) * x^(2*n+1-k)*y^k/(2*n+1)!, as a triangle of coefficients T(n,k) read by rows.
%C A322622 See A322194 for another description of the e.g.f. of this sequence.
%H A322622 Paul D. Hanna, <a href="/A322622/b322622.txt">Table of n, a(n) for n = 1..2652 terms of this triangle read by rows 0..50.</a>
%F A322622 E.g.f.: S(x,y) and related series C(x,y) satisfy the following identities.
%F A322622 (1) C(x,y)^2 - S(x,y)^2 = 1.
%F A322622 (2a) C(x,y) = cosh(x) * cosh(y) / (1 - sinh(x)*sinh(y)).
%F A322622 (2b) S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
%F A322622 (3a) cosh(x) = C(x,y) * cosh(y) / (1 + sinh(y)*S(x,y)).
%F A322622 (3b) sinh(x) = (S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)).
%F A322622 (3c) cosh(y) = C(x,y) * cosh(x) / (1 + sinh(x)*S(x,y)).
%F A322622 (3d) sinh(y) = (S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)).
%F A322622 (4a) exp(x) = (C(x,y)*cosh(y) + S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)).
%F A322622 (4b) exp(y) = (C(x,y)*cosh(x) + S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)).
%F A322622 (5a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) * (cosh(y) - sinh(y)*C(x,y)) / (1 - sinh(y)^2*S(x,y)^2).
%F A322622 (5b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) * (cosh(x) - sinh(x)*C(x,y)) / (1 - sinh(x)^2*S(x,y)^2).
%F A322622 (5c) C(x,y) + S(x,y) = (cosh(x) + sinh(x)*cosh(y)) * (cosh(y) + sinh(y)*cosh(x)) / (1 - sinh(x)^2*sinh(y)^2).
%F A322622 (6a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) / (cosh(y) + sinh(y)*C(x,y)).
%F A322622 (6b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) / (cosh(x) + sinh(x)*C(x,y)).
%F A322622 (6c) C(x,y) + S(x,y) = (cosh(x) + sinh(x)*cosh(y)) / (cosh(y) - sinh(y)*cosh(x)).
%F A322622 (6d) C(x,y) + S(x,y) = (cosh(y) + sinh(y)*cosh(x)) / (cosh(x) - sinh(x)*cosh(y)).
%F A322622 SPECIAL ARGUMENTS.
%F A322622 S(x, y=0) = sinh(x).
%F A322622 S(x, y=x) = 2*sinh(x) / (1 - sinh(x)^2).
%F A322622 S(x, y=-x) = 0.
%e A322622 E.g.f.: S(x,y) = (1*x + 1*y) + (1*x^3 + 6*x^2*y + 6*x*y^2 + 1*y^3)/3! + (1*x^5 + 40*x^4*y + 140*x^3*y^2 + 140*x^2*y^3 + 40*x*y^4 + 1*y^5)/5! + (1*x^7 + 224*x^6*y + 2562*x^5*y^2 + 7000*x^4*y^3 + 7000*x^3*y^4 + 2562*x^2*y^5 + 224*x*y^6 + 1*y^7)/7! + ...
%e A322622 where S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
%e A322622 This irregular triangle of coefficients of x^(2*n+1-k)*y^k/(2*n+1)! in C(x,y) begins
%e A322622 1, 1;
%e A322622 1, 6, 6, 1;
%e A322622 1, 40, 140, 140, 40, 1;
%e A322622 1, 224, 2562, 7000, 7000, 2562, 224, 1;
%e A322622 1, 1152, 39384, 260736, 605808, 605808, 260736, 39384, 1152, 1;
%e A322622 1, 5632, 541310, 8131200, 39032400, 80735424, 80735424, 39032400, 8131200, 541310, 5632, 1;
%e A322622 1, 26624, 6908772, 225065984, 2101762520, 8105175936, 15355426944, 15355426944, 8105175936, 2101762520, 225065984, 6908772, 26624, 1; ...
%e A322622 RELATED SERIES.
%e A322622 The series C(x,y), such that C(x,y)^2 - S(x,y)^2 = 1, begins
%e A322622 C(x,y) = 1 + (1*x^2 + 2*x*y + 1*y^2)/2! + (1*x^4 + 16*x^3*y + 30*x^2*y^2 + 16*x*y^3 + 1*y^4)/4! + (1*x^6 + 96*x^5*y + 615*x^4*y^2 + 1040*x^3*y^3 + 615*x^2*y^4 + 96*x*y^5 + 1*y^6)/6! + (1*x^8 + 512*x^7*y + 10220*x^6*y^2 + 43904*x^5*y^3 + 68390*x^4*y^4 + 43904*x^3*y^5 + 10220*x^2*y^6 + 512*x*y^7 + 1*y^8)/8! + ...
%e A322622 The e.g.f. may be written with coefficients of x^(2*n+1-k)*y^k/((2*n+1-k)!*k!), as follows:
%e A322622 S(x,y) = (1*x + 1*y) + (1*x^3/3! + 2*x^2*y/2! + 2*x*y^2/2! + 1*y^3/3!) + (1*x^5/5! + 8*x^4*y/4! + 14*x^3*y^2/(3!*2!) + 14*x^2*y^3/(2!*3!) + 8*x*y^4/4! + 1*y^5/5!) + (1*x^7/7! + 32*x^6*y/6! + 122*x^5*y^2/(5!*2!) + 200*x^4*y^3/(4!*3!) + 200*x^3*y^4/(3!*4!) + 122*x^2*y^5/(2!*5!) + 32*x*y^6/6! + 1*y^7/7!) + ...
%e A322622 these coefficients are described by triangle A322194.
%o A322622 (PARI) {T(n, k) = my(X=x+x*O(x^(2*n+1-k)), Y=y+y*O(y^k));
%o A322622 S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y));
%o A322622 (2*n+1)!*polcoeff(polcoeff(S, 2*n+1-k, x), k, y)}
%o A322622 /* Print as a triangle */
%o A322622 for(n=0, 10, for(k=0, 2*n+1, print1( T(n, k), ", ")); print(""))
%Y A322622 Cf. A322620 (C + S), A322621 (C), A322625 (main diagonal), A322194.
%K A322622 nonn,tabf
%O A322622 1,4
%A A322622 _Paul D. Hanna_, Dec 20 2018