Dmitry Efimov - cp's OEIS frontend

User: Dmitry Efimov

Dmitry Efimov has authored 4 sequences.

A336400 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,2,1,...,1,2); a(0)=1, a(1)=2.

1, 2, 9, 44, 303, 2697, 29438, 380529, 5683359, 96290588, 1824544857, 38229811083, 877643031554, 21906313145979, 590665804363833, 17109084307014332, 529833078045763263, 17468521692479218209, 610901505126064857854, 22586913755160674065113

Offset: 0

Dmitry Efimov, Jul 22 2020

nonn

Number of perfect matchings of a chord diagram with 2*n vertices, where neighboring vertices are joined by two chords, and any other pair of vertices is joined by one chord.

			A symmetric 4x4 Toeplitz matrix A with the first row (0,2,1,2) has the form:
0 2 1 2
2 0 2 1
1 2 0 2
2 1 2 0.
Its hafnian equals Hf(A)=a12*a34+a13*a24+a14*a23=2*2+1*1+2*2=9.

Georg Fischer, Table of n, a(n) for n = 0..200
Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2020.

Cf. A001515, A336114, A336286.

I:=[1,2,9]; [n le 3 select I[n] else ( (22*n^2-118*n+139)*Self(n-1) + (10*n^2-74*n+141)*Self(n-2) + 5*(n-5)*Self(n-3) )/(11*n-40): n in [1..30]]; // G. C. Greubel, Sep 28 2023

a:= proc(n) option remember; `if`(n<3, [1, 2, 9][n+1],
     ((22*n^2-74*n+43)*a(n-1)+(10*n^2-54*n+77)*a(n-2)
      +5*(n-4)*a(n-3))/(11*n-29))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 28 2020

Join[{1,2},Table[2^(n+1)*HypergeometricU[n,1+2 n,2],{n,2,19}]] (* Stefano Spezia, Jul 22 2020 *)
(* or *) Join[{1, 2}, Table[n*Sum[(n+k-1)!/(k!*(n-k)!*2^(k-1)),{k,0,n}],{n,2,200}]] (* Georg Fischer, Jul 28 2020 *)

def A336400(n): return n+1 if n<2 else (2/factorial(n-1))*sum(binomial(n,k)*gamma(n+k)/2^k for k in range(n+1))
[A336400(n) for n in range(31)] # G. C. Greubel, Sep 28 2023

a(n) = n*Sum_{k=0..n} (n+k-1)!/(k!*(n-k)!*2^(k-1)), n>=2.
a(n) = A001515(n) + A001515(n-1), n>=2.
a(n) ~ (2*n)!*e/(n!*2^n).
a(n) = 2^(n+1)*U(n, 1+2*n, 2) for n >= 2, where U(a, b, c) is the confluent hypergeometric function. - Stefano Spezia, Jul 22 2020
a(n) = ((22*n^2-74*n+43)*a(n-1) + (10*n^2-54*n+77)*a(n-2) + 5*(n-4)*a(n-3)) / (11*n-29) for n >= 3. - Alois P. Heinz, Jul 28 2020
E.g.f.: - 1 - x + (1 + 1/sqrt(1-2*x))*exp(1 - sqrt(1-2* x)). - G. C. Greubel, Sep 28 2023

Incorrect recurrences removed and a(18) corrected by Georg Fischer, Jul 28 2020

A336286 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,1,2,...,2,0); a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 5, 57, 859, 16087, 362781, 9593105, 291347603, 9998539791, 382732896853, 16169762600329, 747423640472235, 37523173542935207, 2033249827596197549, 118278700627740322977, 7352204062275501662371, 486343759162888783503775, 34112193002666850227154213

Offset: 0

Dmitry Efimov, Jul 16 2020

Number of perfect matchings of an arc diagram with 2*n vertices, where neighboring vertices are joined by one arc, the vertices 1 and 2*n are not adjacent if n>=2, and all other pairs of vertices are joined by two arcs.

			A symmetric 4 X 4 Toeplitz matrix A with the first row (0,1,2,0) has the form:
  0 1 2 0
  1 0 1 2
  2 1 0 1
  0 2 1 0.
Its hafnian equals Hf(A) = a12*a34 + a13*a24 + a14*a23 = 1*1 + 2*2 + 0*1 = 5 = a(2).

Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2020.

Cf. A002119, A336114, A336400.

[1,1,seq(add((-1)^(n-k-1)*(n+k-1)!*(-3*n+k)/(k!*(n-k)!),k=0..n),n=2..32)] # Georg Fischer, Jun 05 2021

Join[{1,1},RecurrenceTable[{a[n+1] == (4*n+4)*a[n]-(8*n-13)*a[n-1]-2*a[n-2], a[2]==5, a[3]==57, a[4]==859}, a[n], {n,2,32}]] (* Georg Fischer, Jun 05 2021 *)

a(n) = Sum_{k=0..n} (-1)^(n-k)*(n+k-1)!*(3*n-k)/(k!*(n-k)!), n>=2.
D-finite with recurrence a(n+1) = (4n+4)*a(n) - (8n-13)*a(n-1) - 2*a(n-2), n>=4.
D-finite with recurrence a(n+1) = ((32*n^2-12*n+2)*a(n) + (8*n+1)*a(n-1))/(8*n-7), n>=3.
a(n) = |A002119(n)| - 2*|A002119(n-1)|, n>=2.
a(n) ~ (2*n)!/sqrt(e)*n!.

A336114 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,1,2,...,2,1); a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 6, 64, 930, 17088, 380870, 9992064, 301738626, 10310669440, 393355695942, 16573741095360, 764401360062626, 38304552622588224, 2072335759298438790, 120390122318741003008, 7474705606285243345410, 493940966313183768532224, 34613731176130328980714886

Offset: 0

Dmitry Efimov, Jul 21 2020

Number of perfect matchings of a chord diagram with 2*n vertices, where neighboring vertices are joined by one chord, and any other pair of vertices is joined by two chords.

			A symmetric 4x4 Toeplitz matrix A with the first row (0,1,2,1) has the form:
0 1 2 1
1 0 1 2
2 1 0 1
1 2 1 0.
Its hafnian equals Hf(A) = a12*a34+a13*a24+a14*a23 = 1*1+2*2+1*1 = 6 = a(2).

Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2020.

Cf. A002119, A336286, A336400.

Join[{1,1},Table[2 HypergeometricU[n,1+2 n,-1],{n,2,16}]] (* Stefano Spezia, Jul 22 2020 *)

a(n) = 2*n*Sum_{k=0..n} (-1)^(n-k)*(n+k-1)!/(k!*(n-k)!), n>=2.
D-finite with recurrence a(n+1) = (4*n+3)*a(n)-(4*n-7)*a(n-1)-a(n-2), n>=4.
D-finite with recurrence a(n+1) = (8*n^2*a(n)+(2*n+1)*a(n-1))/(2*n-1), n>=3.
a(n) = |A002119(n)|-|A002119(n-1)|, n>=2.
a(n) ~ (2*n)!/(sqrt(e)*n!).
a(n) = U(n,1+2*n,-1) for n >= 2, where U(a,b,c) is the confluent hypergeometric function of the second kind. - Stefano Spezia, Jul 22 2020

A268306 The number of even permutations p of 1,2,...,n such that -1<=p(i)-i<=2 for i=1,2,...,n.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 41, 75, 137, 252, 464, 853, 1568, 2884, 5305, 9757, 17945, 33006, 60708, 111659, 205372, 377738, 694769, 1277879, 2350385, 4323032, 7951296, 14624713, 26899040, 49475048, 90998801, 167372889, 307846737, 566218426

Offset: 1

Dmitry Efimov, Jan 31 2016

			There exist two even permutations p of 1,2,3 such that -1<=p(i)-i<=2 for i=1,2,3: (123) and (312), therefore a(3)=2.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-1,0,-1).

Cf. A133872, A000073, A001710, A003221.

CoefficientList[Series[(1 - x + x^2 - x^3 - x^5)/((1 - x) (1 + x^2) (1 - x - x^2 - x^3)), {x, 0, 34}], x] (* Michael De Vlieger, Feb 01 2016 *)

t1:0$ t2:0$ t3:1$  for i:1 thru 100 do (a:1/2+sin((2*i+1)*%pi/4)/sqrt(2),
t:t1+t2+t3, t1:t2, t2:t3, t3:t, e:(t+a)/2, print(e));

Vec(x*(1-x+x^2-x^3-x^5)/((1-x)*(1+x^2)*(1-x-x^2-x^3)) + O(x^50)) \\ Colin Barker, Jan 31 2016

a(n) = A133872(n)/2 + A000073(n+2)/2.

G.f.: x*(1-x+x^2-x^3-x^5) / ((1-x)*(1+x^2)*(1-x-x^2-x^3)). - Colin Barker, Jan 31 2016

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User: Dmitry Efimov

A336400 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,2,1,...,1,2); a(0)=1, a(1)=2.

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A336286 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,1,2,...,2,0); a(0)=a(1)=1.

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A336114 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,1,2,...,2,1); a(0)=a(1)=1.

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A268306 The number of even permutations p of 1,2,...,n such that -1<=p(i)-i<=2 for i=1,2,...,n.

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