A322190 -id:A322190 - cp's OEIS frontend

A322620 E.g.f.: A(x,y) = (cosh(x)cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)sinh(y)), where A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n+k)!, as a square table of coefficients T(n,k) read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 16, 30, 16, 1, 1, 40, 140, 140, 40, 1, 1, 96, 615, 1040, 615, 96, 1, 1, 224, 2562, 7000, 7000, 2562, 224, 1, 1, 512, 10220, 43904, 68390, 43904, 10220, 512, 1, 1, 1152, 39384, 260736, 605808, 605808, 260736, 39384, 1152, 1, 1, 2560, 147645, 1482240, 4998210, 7322112, 4998210, 1482240, 147645, 2560, 1, 1, 5632, 541310, 8131200, 39032400, 80735424, 80735424, 39032400, 8131200, 541310, 5632, 1, 1, 12288, 1948650, 43310080, 291662415, 831080448, 1161583500, 831080448, 291662415, 43310080, 1948650, 12288, 1

Offset: 0

Paul D. Hanna, Dec 20 2018

Compare to the addition theorem of Jacobi's elliptic functions: cn(x+y) + i*sn(x+y) = (cn(x) + i*sn(x)*dn(y)) * (cn(y) + i*sn(y)*dn(x)) / (1 - k^2*sn(x)^2*sn(y)^2), where the modulus k is implicit.

See A322190 for another description of the e.g.f. of this sequence.

			E.g.f.: A(x,y) = 1 + (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2)/2! + (1*x^3 + 6*x^2*y + 6*x*y^2 + 1*y^3)/3! + (1*x^4 + 16*x^3*y + 30*x^2*y^2 + 16*x*y^3 + 1*y^4)/4! + (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^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^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! + (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! + ...
where A(x,y) = (cosh(x)*cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
This square table of coefficients of x^n*y^k/(n+k)! in A(x,y) begins
1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 2, 6, 16, 40, 96, 224, 512, 1152, ...;
1, 6, 30, 140, 615, 2562, 10220, 39384, 147645, ...;
1, 16, 140, 1040, 7000, 43904, 260736, 1482240, 8131200, ...;
1, 40, 615, 7000, 68390, 605808, 4998210, 39032400, 291662415, ...;
1, 96, 2562, 43904, 605808, 7322112, 80735424, 831080448, 8105175936, ...;
1, 224, 10220, 260736, 4998210, 80735424, 1161583500, 15355426944, ...;
1, 512, 39384, 1482240, 39032400, 831080448, 15355426944, 256124504064, ...; ...
This sequence may be written as a triangle, starting as
1,
1, 1,
1, 2, 1,
1, 6, 6, 1;
1, 16, 30, 16, 1;
1, 40, 140, 140, 40, 1;
1, 96, 615, 1040, 615, 96, 1;
1, 224, 2562, 7000, 7000, 2562, 224, 1;
1, 512, 10220, 43904, 68390, 43904, 10220, 512, 1;
1, 1152, 39384, 260736, 605808, 605808, 260736, 39384, 1152, 1;
1, 2560, 147645, 1482240, 4998210, 7322112, 4998210, 1482240, 147645, 2560, 1; ...
RELATED SERIES.
The series expansions for C(x,y) and S(x,y) are given by
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! + ...
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 A(x,y) = C(x,y) + S(x,y) such that C(x,y)^2 - S(x,y)^2 = 1.
The e.g.f. may be written with coefficients of x^n*y^k/(n!*k!), as follows:
A(x,y) = 1 + (1*x + 1*y) + (1*x^2/2! + 1*x*y + 1*y^2/2!) + (1*x^3/3! + 2*x^2*y/2! + 2*x*y^2/2! + 1*y^3/3!) + (1*x^4/4! + 4*x^3*y/3! + 5*x^2*y^2/(2!*2!) + 4*x*y^3/3! + 1*y^4/4!) + (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^6/6! + 16*x^5*y/5! + 41*x^4*y^2/(4!*2!) + 52*x^3*y^3/(3!*3!) + 41*x^2*y^4/(2!*4!) + 16*x*y^5/5! + 1*y^6/6!) + (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!) + (1*x^8/8! + 64*x^7*y/7! + 365*x^6*y^2/(6!*2!) + 784*x^5*y^3/(5!*3!) + 977*x^4*y^4/(4!*4!) + 784*x^3*y^5/(3!*5!) + 365*x^2*y^6/(2!*6!) + 64*x*y^7/7! + 1*y^8/8!) + ...
these coefficients are described by table A322190.

Paul D. Hanna, Table of n, a(n) for n = 0..1325 terms of this square table read by antidiagonals across rows 0..50.

Cf. A322621 (C(x,y)), A322622 (S(x,y)), A322623 (antidiagonal sums), A322624 (main diagonal), A322625, A057711 (column 1).

Cf. A322190, A245140, A245155, A245166.

nmax = 12;
t[n_, k_] := SeriesCoefficient[(Cosh[x] Cosh[y] + Sinh[x] + Sinh[y])/(1 - Sinh[x] Sinh[y]), {x, 0, n}, {y, 0, k}] (n + k)!;
tt = Table[t[n, k], {n, 0, nmax}, {k, 0, nmax}];
T[n_, k_] := tt[[n+1, k+1]];
Table[T[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 26 2018 *)

{T(n,k) = my(X=x+x*O(x^n),Y=y+y*O(y^k));
C = cosh(X)*cosh(Y)/(1 - sinh(X)*sinh(Y));
S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y));
(n+k)!*polcoeff(polcoeff(C + S,n,x),k,y)}
/* Print as a square table */
for(n=0,10,for(k=0,10,print1( T(n,k),", "));print(""))
/* Print as a triangle */
for(n=0,15,for(k=0,n,print1( T(n-k,k),", "));print(""))

E.g.f.: A(x,y) = (cosh(x) + sinh(x)*cosh(y)) * (cosh(y) + sinh(y)*cosh(x)) / (1 - sinh(x)^2*sinh(y)^2).
E.g.f.: A(x,y) = (cosh(x) + sinh(x)*cosh(y)) / (cosh(y) - sinh(y)*cosh(x)).
E.g.f.: A(x,y) = (cosh(y) + sinh(y)*cosh(x)) / (cosh(x) - sinh(x)*cosh(y)).
E.g.f.: A(x,y) = C(x,y) + S(x,y) such that the following identities hold.
(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).
(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)).
SPECIAL ARGUMENTS.
A(x, y=0) = exp(x).
A(x, y=x) = (1 + sinh(x)) / (1 - sinh(x)).
A(x, y=-x) = 1.
FORMULAS FOR TERMS.
a(n) = binomial(n,k) * A322190(n,k).
Sum_{k=0..n} 2^k * T(n,k) = A245140(n).
Sum_{k=0..n} 3^k * T(n,k) = A245155(n).
Sum_{k=0..n} 2^(n-k) * 3^k * T(n,k) = A245166(n).

A245140 E.g.f.: (cosh(2x) + sinh(2x)cosh(x)) / (cosh(x) - sinh(x)cosh(2*x)).

Original entry on oeis.org

1, 3, 9, 45, 297, 2433, 23949, 275145, 3612177, 53348193, 875453589, 15802999545, 311196040857, 6638817262353, 152521855979229, 3754366520240745, 98575724288354337, 2749997026637342913, 81230299711952152869, 2532707187355266614745, 83124358113443446120617, 2864579803637260793877873

Offset: 0

Paul D. Hanna, Jul 12 2014

nonn

Limit (a(n)/n!)^(-1/n) = log(t) = 0.609377863436... where t is the tribonacci constant and satisfies 1 + t + t^2 = t^3.

In general, the radius of convergence r of the e.g.f. (cosh(p*x) + sinh(p*x)*cosh(q*x)) / (cosh(q*x) - sinh(q*x)*cosh(p*x)), where p and q are positive integers, equals r = log(t) such that t is the positive real root that satisfies: 1 + t^p + t^q = t^(p+q).

			E.g.f.: A(x) = 1 + 3*x + 9*x^2/2! + 45*x^3/3! + 297*x^4/4! + 2433*x^5/5! +...
such that A(x) = B(x)*C(x), where
B(x) = 1 + x + x^2/2! + 13*x^3/3! + 49*x^4/4! + 361*x^5/5! + 3121*x^6/6! +...
C(x) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 64*x^4/4! + 602*x^5/5! + 5344*x^6/6! +...
are the e.g.f.s of A245138 and A245139, respectively.
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 9*x^2/2! + 297*x^4/4! + 23949*x^6/6! + 3612177*x^8/8! +...
A1(x) = 3*x + 45*x^3/3! + 2433*x^5/5! + 275145*x^7/7! + 53348193*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm is an odd function:
log(A(x)) = 3*x + 18*x^3/3! + 570*x^5/5! + 46158*x^7/7! + 6959250*x^9/9! + 1686709398*x^11/11! + 599570355930*x^13/13! + 3754366520240745*x^15/15! +...
thus A(x)*A(-x) = 1.

Cf. A245138, A245139, A245155, A245166.

Cf. A322620, A322190.

{a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff( (cosh(2*X) + sinh(2*X)*cosh(X)) / (cosh(X) - sinh(X)*cosh(2*X)), n)}
for(n=0,30,print1(a(n),", "))

{a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(X) + sinh(X)*cosh(2*X)) * (cosh(2*X) + sinh(2*X)*cosh(X)) / (1 - sinh(X)^2*sinh(2*X)^2),n)}
for(n=0,30,print1(a(n),", "))

E.g.f.: (cosh(x) + sinh(x)*cosh(2*x)) * (cosh(2*x) + sinh(2*x)*cosh(x)) / (1 - sinh(x)^2*sinh(2*x)^2).
E.g.f.: (cosh(x)*cosh(2*x) + sinh(x) + sinh(2*x)) / (1 - sinh(x)*sinh(2*x)). - Paul D. Hanna, Dec 22 2018
E.g.f.: A(x) = B(x)*C(x), where B(x) and C(x) are the e.g.f.s of A245138 and A245139, respectively.
Let e.g.f. A(x) = A0(x) + A1(x) where A0(x) = (A(x)+A(-x))/2 and A1(x) = (A(x)-A(-x))/2, then:
(1) A0(x)^2 - A1(x)^2 = 1.
(2) exp(x) = (A0(x) + A1(x)*cosh(2*x)) * (cosh(2*x) - sinh(2*x)*A0(x)) / (1 - sinh(2*x)^2*A1(x)^2).
(3) exp(2*x) = (A0(x) + A1(x)*cosh(x)) * (cosh(x) - sinh(x)*A0(x)) / (1 - sinh(x)^2*A1(x)^2).
From Paul D. Hanna, Dec 22 2018: (Start)
(4) exp(x) = (A0(x)*cosh(2*x) + A1(x) - sinh(2*x)) / (1 + sinh(2*x)*A1(x)).
(5) exp(2*x) = (A0(x)*cosh(x) + A1(x) - sinh(x)) / (1 + sinh(x)*A1(x)). (End)
FORMULAS FOR TERMS.
From Paul D. Hanna, Dec 22 2018: (Start)
a(n) = Sum_{k=0..n} 2^k * A322620(n,k).
a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * A322190(n,k). (End)

A245155 E.g.f.: (cosh(3x) + sinh(3x)cosh(x)) / (cosh(x) - sinh(x)cosh(3*x)).

Original entry on oeis.org

1, 4, 16, 100, 832, 8644, 107776, 1567780, 26063872, 487466884, 10129985536, 231560895460, 5774444019712, 155997355725124, 4538464905527296, 141469868440031140, 4703786933664612352, 166172927821116399364, 6215792183431115309056, 245422172388559255422820

Offset: 0

Paul D. Hanna, Jul 12 2014

nonn

Limit (a(n)/n!)^(-1/n) = log(t) = 0.4812118250596... where t = (1+sqrt(5))/2 satisfies 1 + t + t^3 = t^4.

			E.g.f.: A(x) = 1 + 4*x + 16*x^2/2! + 100*x^3/3! + 832*x^4/4! + 8644*x^5/5! +...
such that A(x) = B(x)*C(x), where
B(x) = 1 + x + x^2/2! + 28*x^3/3! + 109*x^4/4! + 1036*x^5/5! + 12421*x^6/6! +...
C(x) = 1 + 3*x + 9*x^2/2! + 36*x^3/3! + 189*x^4/4! + 2148*x^5/5! + 26109*x^6/6! +...
are the e.g.f.s of A245153 and A245154, respectively.
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 16*x^2/2! + 832*x^4/4! + 107776*x^6/6! + 26063872*x^8/8! +...
A1(x) = 4*x + 100*x^3/3! + 8644*x^5/5! + 1567780*x^7/7! + 487466884*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm is an odd function:
log(A(x)) = 4*x + 36*x^3/3! + 1860*x^5/5! + 240996*x^7/7! + 58280580*x^9/9! + 22651336356*x^11/11! + 12912049359300*x^13/13! + 10148316042271716*x^15/15! +...
thus A(x)*A(-x) = 1.

Cf. A245153, A245154, A245140, A245166.

Cf. A322620, A322190.

{a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff( (cosh(3*X) + sinh(3*X)*cosh(X)) / (cosh(X) - sinh(X)*cosh(3*X)), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(X) + sinh(X)*cosh(3*X)) * (cosh(3*X) + sinh(3*X)*cosh(X)) / (1 - sinh(X)^2*sinh(3*X)^2), n)}
for(n=0, 30, print1(a(n), ", "))

E.g.f.: (cosh(x) + sinh(x)*cosh(3*x)) * (cosh(3*x) + sinh(3*x)*cosh(x)) / (1 - sinh(x)^2*sinh(3*x)^2).
E.g.f.: (cosh(x)*cosh(3*x) + sinh(x) + sinh(3*x)) / (1 - sinh(x)*sinh(3*x)). - Paul D. Hanna, Dec 22 2018
E.g.f.: A(x) = B(x)*C(x), where B(x) and C(x) are the e.g.f.s of A245153 and A245154, respectively.
Let e.g.f. A(x) = A0(x) + A1(x) where A0(x) = (A(x)+A(-x))/2 and A1(x) = (A(x)-A(-x))/2, then:
(1) A0(x)^2 - A1(x)^2 = 1.
(2) exp(x) = (A0(x) + A1(x)*cosh(3*x)) * (cosh(3*x) - sinh(3*x)*A0(x)) / (1 - sinh(3*x)^2*A1(x)^2).
(3) exp(3*x) = (A0(x) + A1(x)*cosh(x)) * (cosh(x) - sinh(x)*A0(x)) / (1 - sinh(x)^2*A1(x)^2).
From Paul D. Hanna, Dec 22 2018: (Start)
(4) exp(x) = (A0(x)*cosh(3*x) + A1(x) - sinh(3*x)) / (1 + sinh(3*x)*A1(x)).
(5) exp(3*x) = (A0(x)*cosh(x) + A1(x) - sinh(x)) / (1 + sinh(x)*A1(x)). (End)
FORMULAS FOR TERMS.
From Paul D. Hanna, Dec 22 2018: (Start)
a(n) = Sum_{k=0..n} 3^k * A322620(n,k).
a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * A322190(n,k). (End)
a(n) ~ 2*sqrt(2*Pi/5) * n^(n+1/2) / (exp(n) * (log((1+sqrt(5))/2))^(n+1)). - Vaclav Kotesovec, Nov 04 2014

A245166 E.g.f.: (cosh(3x) + sinh(3x)cosh(2x)) / (cosh(2x) - sinh(2x)cosh(3x)).

Original entry on oeis.org

1, 5, 25, 215, 2425, 33875, 569125, 11160035, 250047025, 6302723075, 176522216125, 5438291613155, 182773714292425, 6654680279353475, 260930805319957525, 10961922511422743075, 491220886240696086625, 23388149451193115459075, 1179066988050425638569325

Offset: 0

Paul D. Hanna, Jul 12 2014

nonn

Limit (a(n)/n!)^(-1/n) = log(t) = 0.3570506972213... where t satisfies 1 + t^2 + t^3 = t^5.

			E.g.f.: A(x) = 1 + 5*x + 25*x^2/2! + 215*x^3/3! + 2425*x^4/4! + 33875*x^5/5! +...
such that A(x) = B(x)*C(x), where
B(x) = 1 + 2*x + 4*x^2/2! + 62*x^3/3! + 448*x^4/4! + 5882*x^5/5! +...
C(x) = 1 + 3*x + 9*x^2/2! + 63*x^3/3! + 513*x^4/4! + 8043*x^5/5! +...
are the e.g.f.s of A245164 and A245165, respectively.
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 25*x^2/2! + 2425*x^4/4! + 569125*x^6/6! + 250047025*x^8/8! +...
A1(x) = 5*x + 215*x^3/3! + 33875*x^5/5! + 11160035*x^7/7! + 6302723075*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm is an odd function:
log(A(x)) = 5*x + 90*x^3/3! + 8250*x^5/5! + 1946910*x^7/7! + 855018450*x^9/9! + 603621345030*x^11/11! + 624997732481850*x^13/13! +...
thus A(x)*A(-x) = 1.

Cf. A245164, A245165, A245140, A245155.

Cf. A322620, A322190.

With[{nn=20},CoefficientList[Series[(Cosh[3x]+Sinh[3x]Cosh[2x])/(Cosh[2x]-Sinh[2x]Cosh[3x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 21 2024 *)

{a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff( (cosh(3*X) + sinh(3*X)*cosh(2*X)) / (cosh(2*X) - sinh(2*X)*cosh(3*X)), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(2*X) + sinh(2*X)*cosh(3*X)) * (cosh(3*X) + sinh(3*X)*cosh(2*X)) / (1 - sinh(2*X)^2*sinh(3*X)^2), n)}
for(n=0, 30, print1(a(n), ", "))

E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(3*x)) * (cosh(3*x) + sinh(3*x)*cosh(2*x)) / (1 - sinh(2*x)^2*sinh(3*x)^2).
E.g.f.: (cosh(2*x)*cosh(3*x) + sinh(2*x) + sinh(3*x)) / (1 - sinh(2*x)*sinh(3*x)). - Paul D. Hanna, Dec 22 2018
E.g.f.: A(x) = B(x)*C(x), where B(x) and C(x) are the e.g.f.s of A245164 and A245165, respectively.
Let e.g.f. A(x) = A0(x) + A1(x) where A0(x) = (A(x)+A(-x))/2 and A1(x) = (A(x)-A(-x))/2, then:
(1) A0(x)^2 - A1(x)^2 = 1.
(2) exp(2*x) = (A0(x) + A1(x)*cosh(3*x)) * (cosh(3*x) - sinh(3*x)*A0(x)) / (1 - sinh(3*x)^2*A1(x)^2).
(3) exp(3*x) = (A0(x) + A1(x)*cosh(2*x)) * (cosh(2*x) - sinh(2*x)*A0(x)) / (1 - sinh(2*x)^2*A1(x)^2).
From Paul D. Hanna, Dec 22 2018: (Start)
(4) exp(2*x) = (A0(x)*cosh(3*x) + A1(x) - sinh(3*x)) / (1 + sinh(3*x)*A1(x)).
(5) exp(3*x) = (A0(x)*cosh(2*x) + A1(x) - sinh(2*x)) / (1 + sinh(2*x)*A1(x)). (End)
FORMULAS FOR TERMS.
a(n) == 0 (mod 5) for n>0.
From Paul D. Hanna, Dec 22 2018: (Start)
a(n) = Sum_{k=0..n} 2^(n-k) * 3^k * A322620(n,k).
a(n) = Sum_{k=0..n} 2^(n-k) * 3^k * binomial(n,k) * A322190(n,k). (End)

A322195 a(n) = [x^ny^n/n!^2] (cosh(x)cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), for n >= 0.

Original entry on oeis.org

1, 1, 5, 52, 977, 29056, 1257125, 74628352, 5823720257, 578189787136, 71175865436645, 10640402473418752, 1898906773603283537, 398777106584112726016, 97349216334148411738565, 27336588856134172778954752, 8749733524560004884480322817, 3166642491794673645738520477696, 1286690426658870521915733780962885, 583275073512315366760585608772452352, 293315345943354969193393028030640310097

Offset: 0

Paul D. Hanna, Dec 19 2018

nonn

a(n) = A322190(n,n) for n >= 0.

a(n) = A322624(n) / binomial(2*n,n) for n >= 0.

Vaclav Kotesovec, Table of n, a(n) for n = 0..246 (terms 0..100 from Paul D. Hanna)

Cf. A322190, A322624.

{A322190(n, k) = my(X=x+x*O(x^n), Y=y+y*O(y^k));
C = cosh(X)*cosh(Y)/(1 - sinh(X)*sinh(Y));
S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y));
n!*k!*polcoeff(polcoeff( C + S, n, x), k, y)}
for(n=0,20, print1( A322190(n,n),", "))

a(n) ~ c * n^(2*n + 1/2) / (exp(2*n) * (log(1+sqrt(2)))^(2*n)), where c = 9.2665697179347261408645686858011097... - Vaclav Kotesovec, Dec 31 2018

A322193 E.g.f.: C(x,y) = cosh(x)cosh(y) / (1 - sinh(x)sinh(y)), where C(x,y) = Sum_{n>=0} Sum_{k=0..2n} T(n,k) x^(2n-k)y^k/((2n-k)!k!), as a triangle of coefficients T(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 5, 4, 1, 1, 16, 41, 52, 41, 16, 1, 1, 64, 365, 784, 977, 784, 365, 64, 1, 1, 256, 3281, 12352, 23801, 29056, 23801, 12352, 3281, 256, 1, 1, 1024, 29525, 196864, 589217, 1049344, 1257125, 1049344, 589217, 196864, 29525, 1024, 1, 1, 4096, 265721, 3146752, 14677961, 37789696, 63318641, 74628352, 63318641, 37789696, 14677961, 3146752, 265721, 4096, 1, 1, 16384, 2391485, 50335744, 366476657, 1360482304, 3140590685, 5010663424, 5823720257, 5010663424, 3140590685, 1360482304, 366476657, 50335744, 2391485, 16384, 1

Offset: 0

Paul D. Hanna, Dec 20 2018

See A322621 for another description of the e.g.f. of this sequence.

			E.g.f.: C(x,y) = 1 + (1*x^2/2! + 1*x*y + 1*y^2/2!) + (1*x^4/4! + 4*x^3*y/3! + 5*x^2*y^2/(2!*2!) + 4*x*y^3/3! + 1*y^4/4!) + (1*x^6/6! + 16*x^5*y/5! + 41*x^4*y^2/(4!*2!) + 52*x^3*y^3/(3!*3!) + 41*x^2*y^4/(2!*4!) + 16*x*y^5/5! + 1*y^6/6!) + (1*x^8/8! + 64*x^7*y/7! + 365*x^6*y^2/(6!*2!) + 784*x^5*y^3/(5!*3!) + 977*x^4*y^4/(4!*4!) + 784*x^3*y^5/(3!*5!) + 365*x^2*y^6/(2!*6!) + 64*x*y^7/7! + 1*y^8/8!) + ...
where C(x,y) = cosh(x)*cosh(y) / (1 - sinh(x)*sinh(y)).
This irregular triangle of coefficients of x^(2*n-k)*y^k/((2*n-k)!*k!) in C(x,y) begins
1;
1, 1, 1;
1, 4, 5, 4, 1;
1, 16, 41, 52, 41, 16, 1;
1, 64, 365, 784, 977, 784, 365, 64, 1;
1, 256, 3281, 12352, 23801, 29056, 23801, 12352, 3281, 256, 1;
1, 1024, 29525, 196864, 589217, 1049344, 1257125, 1049344, 589217, 196864, 29525, 1024, 1;
1, 4096, 265721, 3146752, 14677961, 37789696, 63318641, 74628352, 63318641; 37789696, 14677961, 3146752, 265721, 4096, 1; ...
RELATED SERIES.
The series S(x,y), such that C(x,y)^2 - S(x,y)^2 = 1, begins
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!) + ...
The e.g.f. may be written with coefficients of x^(2*n-k)*y^k/(2*n)!, as follows:
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! + ...
these coefficients are described by triangle A322621.

Paul D. Hanna, Table of n, a(n) for n = 0..2600 terms of this triangle as read by rows 0..50.

Cf. A322190 (C + S), A322194 (S), A322195 (main diagonal).

T[n_, k_] := (2n-k)! k! SeriesCoefficient[Cosh[x] Cosh[y]/(1-Sinh[x] Sinh[y]), {x, 0, 2n-k}, {y, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, 2n}] // Flatten (* Jean-François Alcover, Dec 29 2018 *)

{T(n, k) = my(X=x+x*O(x^(2*n-k)), Y=y+y*O(y^k));
C = cosh(X)*cosh(Y)/(1 - sinh(X)*sinh(Y));
(2*n-k)!*k!*polcoeff(polcoeff(C, 2*n-k, x), k, y)}
/* Print as a triangle */
for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print(""))

E.g.f.: C(x,y) and related series S(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.
C(x, y=0) = cosh(x).
C(x, y=x) = cosh(x)^2 / (1 - sinh(x)^2).
C(x, y=-x) = 1.

A322194 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..2n+1} T(n,k) * x^(2n+1-k)y^k/((2n+1-k)!k!), as a triangle of coefficients T(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 8, 14, 14, 8, 1, 1, 32, 122, 200, 200, 122, 32, 1, 1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1, 1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1, 1, 2048, 88574, 786944, 2939528, 6297728, 8948384, 8948384, 6297728, 2939528, 786944, 88574, 2048, 1, 1, 8192, 797162, 12584960, 73330760, 226744832, 446442272, 614111360, 614111360, 446442272, 226744832, 73330760, 12584960, 797162, 8192, 1

Offset: 1

Paul D. Hanna, Dec 20 2018

See A322622 for another description of the e.g.f. of this sequence.

			E.g.f.: 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!) + ...
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-k)!*k!) in C(x,y) begins
1, 1;
1, 2, 2, 1;
1, 8, 14, 14, 8, 1;
1, 32, 122, 200, 200, 122, 32, 1;
1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1;
1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1;
1, 2048, 88574, 786944, 2939528, 6297728, 8948384, 8948384, 6297728, 2939528, 786944, 88574, 2048, 1;
1, 8192, 797162, 12584960, 73330760, 226744832, 446442272, 614111360, 614111360, 446442272, 226744832, 73330760, 12584960, 797162, 8192, 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! + 1*x*y + 1*y^2/2!) + (1*x^4/4! + 4*x^3*y/3! + 5*x^2*y^2/(2!*2!) + 4*x*y^3/3! + 1*y^4/4!) + (1*x^6/6! + 16*x^5*y/5! + 41*x^4*y^2/(4!*2!) + 52*x^3*y^3/(3!*3!) + 41*x^2*y^4/(2!*4!) + 16*x*y^5/5! + 1*y^6/6!) + (1*x^8/8! + 64*x^7*y/7! + 365*x^6*y^2/(6!*2!) + 784*x^5*y^3/(5!*3!) + 977*x^4*y^4/(4!*4!) + 784*x^3*y^5/(3!*5!) + 365*x^2*y^6/(2!*6!) + 64*x*y^7/7! + 1*y^8/8!) + ...
The e.g.f. may be written with coefficients of x^(2*n+1-k)*y^k/(2*n+1)!, as follows:
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! + ...
these coefficients are described by triangle A322622.

Paul D. Hanna, Table of n, a(n) for n = 1..2652

Cf. A322190 (C + S), A322193 (C), A322196 (main diagonal).

Cf. A322622.

T[n_, k_] := (2n-k+1)! k! SeriesCoefficient[(Sinh[x] + Sinh[y])/(1 - Sinh[x] Sinh[y]), {x, 0, 2n-k+1}, {y, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, 2n+1}] // Flatten (* Jean-François Alcover, Dec 29 2018 *)

{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-k)!*k!*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(""))

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.

cp's OEIS Frontend

A322197 Antidiagonal sums of square table A322190.

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A322620 E.g.f.: A(x,y) = (cosh(x)*cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), where A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n+k)!, as a square table of coefficients T(n,k) read by antidiagonals.

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A245140 E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(x)) / (cosh(x) - sinh(x)*cosh(2*x)).

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A245155 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(x)) / (cosh(x) - sinh(x)*cosh(3*x)).

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A245166 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(2*x)) / (cosh(2*x) - sinh(2*x)*cosh(3*x)).

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A322195 a(n) = [x^n*y^n/n!^2] (cosh(x)*cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), for n >= 0.

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A322193 E.g.f.: C(x,y) = cosh(x)*cosh(y) / (1 - sinh(x)*sinh(y)), where C(x,y) = Sum_{n>=0} Sum_{k=0..2*n} T(n,k) * x^(2*n-k)*y^k/((2*n-k)!*k!), as a triangle of coefficients T(n,k) read by rows.

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A322194 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-k)!*k!), as a triangle of coefficients T(n,k) read by rows.

A322620 E.g.f.: A(x,y) = (cosh(x)cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)sinh(y)), where A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n+k)!, as a square table of coefficients T(n,k) read by antidiagonals.

A245140 E.g.f.: (cosh(2x) + sinh(2x)cosh(x)) / (cosh(x) - sinh(x)cosh(2*x)).

A245155 E.g.f.: (cosh(3x) + sinh(3x)cosh(x)) / (cosh(x) - sinh(x)cosh(3*x)).

A245166 E.g.f.: (cosh(3x) + sinh(3x)cosh(2x)) / (cosh(2x) - sinh(2x)cosh(3x)).

A322195 a(n) = [x^ny^n/n!^2] (cosh(x)cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), for n >= 0.

A322193 E.g.f.: C(x,y) = cosh(x)cosh(y) / (1 - sinh(x)sinh(y)), where C(x,y) = Sum_{n>=0} Sum_{k=0..2n} T(n,k) x^(2n-k)y^k/((2n-k)!k!), as a triangle of coefficients T(n,k) read by rows.

A322194 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..2n+1} T(n,k) * x^(2n+1-k)y^k/((2n+1-k)!k!), as a triangle of coefficients T(n,k) read by rows.

A322196 a(n) = [x^(n+1)y^n/((n+1)!n!)] (cosh(x)cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)sinh(y)), for n >= 0.