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A322622
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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.
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4
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1, 1, 1, 6, 6, 1, 1, 40, 140, 140, 40, 1, 1, 224, 2562, 7000, 7000, 2562, 224, 1, 1, 1152, 39384, 260736, 605808, 605808, 260736, 39384, 1152, 1, 1, 5632, 541310, 8131200, 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
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OFFSET
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1,4
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COMMENTS
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See A322194 for another description of the e.g.f. of this sequence.
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LINKS
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FORMULA
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E.g.f.: S(x,y) and related series C(x,y) satisfy the following identities.
(1) C(x,y)^2 - S(x,y)^2 = 1.
(2a) C(x,y) = cosh(x) * cosh(y) / (1 - sinh(x)*sinh(y)).
(2b) S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
(3a) cosh(x) = C(x,y) * cosh(y) / (1 + sinh(y)*S(x,y)).
(3b) sinh(x) = (S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)).
(3c) cosh(y) = C(x,y) * cosh(x) / (1 + sinh(x)*S(x,y)).
(3d) sinh(y) = (S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)).
(4a) exp(x) = (C(x,y)*cosh(y) + S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)).
(4b) exp(y) = (C(x,y)*cosh(x) + S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)).
(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).
(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).
(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).
(6a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) / (cosh(y) + sinh(y)*C(x,y)).
(6b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) / (cosh(x) + sinh(x)*C(x,y)).
(6c) C(x,y) + S(x,y) = (cosh(x) + sinh(x)*cosh(y)) / (cosh(y) - sinh(y)*cosh(x)).
(6d) C(x,y) + S(x,y) = (cosh(y) + sinh(y)*cosh(x)) / (cosh(x) - sinh(x)*cosh(y)).
SPECIAL ARGUMENTS.
S(x, y=0) = sinh(x).
S(x, y=x) = 2*sinh(x) / (1 - sinh(x)^2).
S(x, y=-x) = 0.
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EXAMPLE
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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! + ...
where S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
This irregular triangle of coefficients of x^(2*n+1-k)*y^k/(2*n+1)! in C(x,y) begins
1, 1;
1, 6, 6, 1;
1, 40, 140, 140, 40, 1;
1, 224, 2562, 7000, 7000, 2562, 224, 1;
1, 1152, 39384, 260736, 605808, 605808, 260736, 39384, 1152, 1;
1, 5632, 541310, 8131200, 39032400, 80735424, 80735424, 39032400, 8131200, 541310, 5632, 1;
1, 26624, 6908772, 225065984, 2101762520, 8105175936, 15355426944, 15355426944, 8105175936, 2101762520, 225065984, 6908772, 26624, 1; ...
RELATED SERIES.
The series C(x,y), such that C(x,y)^2 - S(x,y)^2 = 1, begins
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! + ...
The e.g.f. may be written with coefficients of x^(2*n+1-k)*y^k/((2*n+1-k)!*k!), as follows:
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!) + ...
these coefficients are described by triangle A322194.
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PROG
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(PARI) {T(n, k) = my(X=x+x*O(x^(2*n+1-k)), Y=y+y*O(y^k));
S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y));
(2*n+1)!*polcoeff(polcoeff(S, 2*n+1-k, x), k, y)}
/* Print as a triangle */
for(n=0, 10, for(k=0, 2*n+1, print1( T(n, k), ", ")); print(""))
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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