cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A322190 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.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 4, 1, 1, 8, 14, 14, 8, 1, 1, 16, 41, 52, 41, 16, 1, 1, 32, 122, 200, 200, 122, 32, 1, 1, 64, 365, 784, 977, 784, 365, 64, 1, 1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1, 1, 256, 3281, 12352, 23801, 29056, 23801, 12352, 3281, 256, 1, 1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1, 1, 1024, 29525, 196864, 589217, 1049344, 1257125, 1049344, 589217, 196864, 29525, 1024, 1, 1, 2048, 88574, 786944, 2939528, 6297728, 8948384, 8948384, 6297728, 2939528, 786944, 88574, 2048, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 19 2018

Keywords

Comments

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 A322620 for another description of the e.g.f. of this sequence.

Examples

			E.g.f.: 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!) + ...
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, ...;
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ...;
1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, ...;
1, 4, 14, 52, 200, 784, 3104, 12352, 49280, 196864, ...;
1, 8, 41, 200, 977, 4808, 23801, 118280, 589217, 2939528, ...;
1, 16, 122, 784, 4808, 29056, 174752, 1049344, 6297728, 37789696, ...;
1, 32, 365, 3104, 23801, 174752, 1257125, 8948384, 63318641, 446442272, ...;
1, 64, 1094, 12352, 118280, 1049344, 8948384, 74628352, 614111360, 5010663424, ...;
1, 128, 3281, 49280, 589217, 6297728, 63318641, 614111360, 5823720257, 54420050048, ...; ...
This sequence may be written as a triangle, starting as
1;
1, 1;
1, 1, 1;
1, 2, 2, 1;
1, 4, 5, 4, 1;
1, 8, 14, 14, 8, 1;
1, 16, 41, 52, 41, 16, 1;
1, 32, 122, 200, 200, 122, 32, 1;
1, 64, 365, 784, 977, 784, 365, 64, 1;
1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1;
1, 256, 3281, 12352, 23801, 29056, 23801, 12352, 3281, 256, 1;
1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 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! + 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!) + ...
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 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*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! + ...
these coefficients are described by table A322620.

Links

Crossrefs

Cf. A322193 (C(x,y)), A322194 (S(x,y)), A322195 (main diagonal), A322196, A322197.
Cf. A322620, A245140, A245155, A245166.

Programs

  • Mathematica
    nmax = 13;
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!;
Table[t[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 26 2018 *)
  • PARI
    {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(""))

Formula

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) = A322620(n,k) / binomial(n,k).
Sum_{k=0..n} 2^k * binomial(n,k) * T(n,k) = A245140(n).
Sum_{k=0..n} 3^k * binomial(n,k) * T(n,k) = A245155(n).
Sum_{k=0..n} 2^(n-k) * 3^k * binomial(n,k) * T(n,k) = A245166(n).

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.

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Dec 20 2018

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A322621 (C(x,y)), A322622 (S(x,y)), A322623 (antidiagonal sums), A322624 (main diagonal), A322625, A057711 (column 1).
Cf. A322190, A245140, A245155, A245166.

Programs

  • Mathematica
    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 *)
  • PARI
    {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(""))

Formula

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(2*x) + sinh(2*x)*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

Views

Author

Paul D. Hanna, Jul 12 2014

Keywords

Comments

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).

Examples

			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.

Crossrefs

Cf. A245138, A245139, A245155, A245166.
Cf. A322620, A322190.

Programs

  • PARI
    {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),", "))
  • PARI
    {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),", "))

Formula

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)

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

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

Views

Author

Paul D. Hanna, Jul 12 2014

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A245164, A245165, A245140, A245155.
Cf. A322620, A322190.

Programs

  • Mathematica
    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 *)
  • PARI
    {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), ", "))
  • PARI
    {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), ", "))

Formula

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)

A245153 E.g.f.: (cosh(x) + sinh(x)*cosh(3*x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2).

Original entry on oeis.org

1, 1, 1, 28, 109, 1036, 12421, 189568, 2377369, 50888656, 889772041, 21056972608, 463426778629, 13171920918976, 338302052475661, 11024635871323648, 331174000888419889, 12111179923298826496, 413871819030803915281, 16886967133601994738688
Offset: 0

Views

Author

Paul D. Hanna, Jul 12 2014

Keywords

Comments

Limit (a(n)/n!)^(-1/n) = log( (1+sqrt(5))/2 ) = 0.4812118250596...

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 28*x^3/3! + 109*x^4/4! + 1036*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + x^2/2! + 109*x^4/4! + 12421*x^6/6! + 2377369*x^8/8! +...
A1(x) = x + 28*x^3/3! + 1036*x^5/5! + 189568*x^7/7! + 50888656*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm of the e.g.f. is an odd function:
Log(A(x)) = x + 27*x^3/3! + 765*x^5/5! + 121527*x^7/7! + 29881305*x^9/9! + 11156851827*x^11/11! + 6479306260245*x^13/13! +...
thus A(x)*A(-x) = 1.

Links

Crossrefs

Cf. A245154, A245155, A245138, A245164.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[(Cosh[x]+Sinh[x]Cosh[3x])/Sqrt[1-Sinh[ x]^2 Sinh[3x]^2 ],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 03 2020 *)
  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(X) + sinh(X)*cosh(3*X)) / sqrt(1 - sinh(X)^2*sinh(3*X)^2), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f.: G(x) * (cosh(3*x) - sinh(3*x)*cosh(x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2), where G(x) is the e.g.f. of A245155.
a(n) ~ sqrt(2) * n^n / (5^(1/4) * exp(n) * (log((1+sqrt(5))/2))^(n+1/2)). - Vaclav Kotesovec, Nov 04 2014

A245154 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2).

Original entry on oeis.org

1, 3, 9, 36, 189, 2148, 26109, 371136, 5407929, 95795568, 1832049009, 41428038336, 972380766069, 25736128903488, 705111069908709, 21600790506395136, 683861855417706609, 23836956839153265408, 853476673589938069209, 33263825890074489025536
Offset: 0

Views

Author

Paul D. Hanna, Jul 12 2014

Keywords

Comments

Limit (a(n)/n!)^(-1/n) = log( (1+sqrt(5))/2 ) = 0.4812118250596...

Examples

			E.g.f.: A(x) = 1 + 3*x + 9*x^2/2! + 36*x^3/3! + 189*x^4/4! + 2148*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 9*x^2/2! + 189*x^4/4! + 26109*x^6/6! + 5407929*x^8/8! +...
A1(x) = 3*x + 36*x^3/3! + 2148*x^5/5! + 371136*x^7/7! + 95795568*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm of the e.g.f. is an odd function:
Log(A(x)) = 3*x + 9*x^3/3! + 1095*x^5/5! + 119469*x^7/7! + 28399275*x^9/9! + 11494484529*x^11/11! + 6432743099055*x^13/13! +...
thus A(x)*A(-x) = 1.

Crossrefs

Cf. A245153, A245155, A245139, A245165.

Programs

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

Formula

E.g.f.: G(x) * (cosh(x) - sinh(x)*cosh(3*x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2), where G(x) is the e.g.f. of A245155.
a(n) ~ 2*sqrt(2) * n^n / (5^(1/4) * exp(n) * (log((1+sqrt(5))/2))^(n+1/2)). - Vaclav Kotesovec, Nov 04 2014
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