A336114 -id:A336114 - cp's OEIS frontend

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!.

A338456 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).

Original entry on oeis.org

1, 1, 4, 45, 968, 34265, 1799748, 131572357, 12770710096, 1589142683313, 246658484353100

Offset: 0

Stefano Spezia, Oct 28 2020

			a(2) = 4 because the hafnian of
0  1  1  2
1  0  1  1
1  1  0  1
2  1  1  0
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 4.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix
Wikipedia, Toeplitz Matrix

Cf. A004526.
Cf. A002378 (conjectured determinant of M(2n+1)), A083392 (conjectured determinant of M(n+1)), A332566 (permanent of M(n)), A333119 (k-th super- and subdiagonal sums of the matrix M(n)).
Cf. A202038, A336114, A336286, A336400.

k[i_]:=Floor[i/2]; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Array[k, n]], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 5, 0]

tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j\2, if (j==1, i\2)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
a(n) = {my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]);); s/(n!*2^n);} \\ Michel Marcus, Nov 11 2020

a(5) from Michel Marcus, Nov 11 2020

a(6)-a(10) from Pontus von Brömssen, Oct 14 2023

A356483 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of prime(1), prime(2), ..., prime(2n).

Original entry on oeis.org

1, 3, 55, 2999, 347391, 69702479, 22441691645, 10776262328919, 7190279422736061, 6439969796874334809, 7447188585071730451961

Offset: 0

Stefano Spezia, Aug 09 2022

			a(2) = 55 because the hafnian of
    2  3  5  7
    3  2  3  5
    5  3  2  3
    7  5  3  2
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 55.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix
Wikipedia, Toeplitz Matrix

Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356481, A356482, A356484.
Cf. A356490 (determinant of M(n)), A356491 (permanent of M(n)).

k[i_]:=Prime[i]; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Array[k, n]], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = my(m = matrix(n, n, i, j, if (i==1, prime(j), if (j==1, prime(i))))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 14 2023

A356484 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of prime(2n), prime(2*n-1), ..., prime(1).

Original entry on oeis.org

1, 2, 44, 5210, 1368900, 604109562, 535920536336, 728155179271474, 1103827431509790216, 2651375713654260218986, 7537958658258053003685636

Offset: 0

Stefano Spezia, Aug 09 2022

a(n) is even for n >= 1. - Robert Israel, Oct 13 2023

			a(2) = 44 because the hafnian of
    7  5  3  2
    5  7  5  3
    3  5  7  5
    2  3  5  7
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 44.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix
Wikipedia, Toeplitz Matrix

Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356481, A356482, A356483.
Cf. A356492 (determinant of M(n)), A356493 (permanent of M(n)).

haf:= proc(A)
    local n, s, Pairpart, p;
    Pairpart := proc(L) local j, t; if L = {} then return {{}}; end if; {seq(seq({{L[1], L[j]}} union t, t = procname(L minus {L[1], L[j]})), j = 2 .. nops(L))}; end proc;
    n := LinearAlgebra:-Dimension(A);
    if n[1] <> n[2] then
        error "must be square matrix";
    end if;
    n := n[1];
    if n::odd then
        error "dimension of matrix must be even";
    end if;
    add(mul(A[s[1], s[2]], s = p), p = Pairpart({$ (1 .. n)}));
end proc:
f:= proc(n) local i; haf(LinearAlgebra:-ToeplitzMatrix([seq(ithprime(i),i=2*n..1,-1)],symmetric)) end proc:
f(0):= 1:
map(f, [$0..7]); # Robert Israel, Oct 13 2023

k[i_]:=Prime[i]; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Reverse[Array[k, n]]], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = my(m = matrix(n, n, i, j, if (i==1, prime(n-j+1), if (j==1, prime(n-i+1))))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

a(6) from Michel Marcus, May 02 2023
a(7)-a(9) from Robert Israel, Oct 13 2023
a(10) from Pontus von Brömssen, Oct 14 2023

A356481 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of 1, 2, ..., 2n.

Original entry on oeis.org

1, 2, 21, 532, 24845, 1856094, 203076097, 30633787976, 6097546660185, 1548899852221210, 489114616743840461

Offset: 0

Stefano Spezia, Aug 09 2022

			a(2) = 21 because the hafnian of
    1  2  3  4
    2  1  2  3
    3  2  1  2
    4  3  2  1
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 21.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix
Wikipedia, Toeplitz Matrix

Cf. A001792 (absolute value of the determinant of M(n)), A204235 (permanent of M(n)).
Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356482, A356483, A356484.

k[i_]:=i; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Array[k, n]], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = my(m = matrix(n, n, i, j, if (i==1, j, if (j==1, i)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 14 2023

A356482 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of 2n, 2*n-1, ..., 1.

Original entry on oeis.org

1, 1, 16, 714, 62528, 9056720, 1960138560, 592615689904, 238560786221056, 123358665203311104, 79683847063011614720

Offset: 0

Stefano Spezia, Aug 09 2022

			a(2) = 16 because the hafnian of
    4  3  2  1
    3  4  3  2
    2  3  4  3
    1  2  3  4
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 16.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix
Wikipedia, Toeplitz Matrix

Cf. A001792 (determinant of M(n)), A307783.
Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356481, A356483, A356484.

k[i_]:=i; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Reverse[Array[k, n]]], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = my(m = matrix(n, n, i, j, if (i==1, n-j+1, if (j==1, n-i+1)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 14 2023

A098985 Denominators in series expansion of log( Sum_{m=-oo,oo} q^(m^2) ).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 5, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 5, 31, 1, 11, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 15, 23, 47, 3, 49, 25, 17, 13, 53, 27, 55, 7, 57, 29, 59, 5, 61, 31, 63, 1, 65, 11, 67, 17, 23, 35, 71, 9, 73, 37, 75, 19, 77, 39, 79

Offset: 0

N. J. A. Sloane, Oct 24 2004

For n>0, a(n) is the denominator of Sum_{odd d|n} 1/d. See Sumit Kumar Jha link. - Michel Marcus, Jul 21 2020

			2*q-2*q^2+8/3*q^3-2*q^4+12/5*q^5-8/3*q^6+16/7*q^7-2*q^8+26/9*q^9-...

Sumit Kumar Jha, An identity for the sum of inverses of odd divisors of n in terms of the number of representations of n as a sum of r squares, arXiv:2007.04243 [math.GM], 2020.

Cf. A000122, A098984, A098986.

Cf. A336113 (numerators).

A098985_list := proc(n::integer)
    local q,m,nsq ;
    nsq := floor(sqrt(n)) ;
    add(q^(m^2),m=-nsq-1..nsq+1) ;
    log(%) ;
    taylor(%,q=0,n+1) ;
    [seq( denom(coeftayl(%,q=0,i)) ,i=1..n) ] ;
end proc:
A098985_list(200) ; # R. J. Mathar, Jul 16 2020
A336114 := proc(n::integer)
    local a ;
    for d in numtheory[divisors](n) do
        if type(d,'odd') then
            a := a+1/d ;
        end if;
    end do;
    denom(a) ;
end proc:
seq(A336114(n),n=1..70) ; # R. J. Mathar, Jul 16 2020

Denominator[CoefficientList[Series[Log[Sum[q^m^2, {m, -Infinity, Infinity}]], {q, 0, 79}], q]] (* L. Edson Jeffery, Jul 14 2014 *)
a[n_] := Denominator @ DivisorSum[n, 1/# &, OddQ[#] &]; Array[a, 100] (* Amiram Eldar, Jul 09 2020 *)

lista(nn) = {my(k = sqrtint(nn), s = sum(m=-k-1, k+1, x^(m^2)) + O(x^nn)); apply(x->denominator(x), Vec(log(s)));} \\ Michel Marcus, Jul 17 2020

a(n) = if (n==0, 1, denominator(sumdiv(n, d, if (d%2, 1/d)))); \\ Michel Marcus, Jul 21 2020; corrected Jun 13 2022

A357279 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = i + j - 1.

Original entry on oeis.org

1, 2, 43, 2610, 312081, 61825050, 18318396195, 7586241152490, 4184711271725985, 2965919152834367730, 2626408950849351178875

Offset: 0

Stefano Spezia, Sep 25 2022

The n X n matrix M is the n-th principal submatrix of A002024 considered as an array, and it is singular for n > 2.

			a(2) = 43 because the hafnian of
    1  2  3  4
    2  3  4  5
    3  4  5  6
    4  5  6  7
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 43.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A002024, A002415 (absolute value of the coefficient of x^(n-2) in the characteristic polynomial of M(n)), A095833 (k-th super- and subdiagonal sums of the matrix M(n)), A204248 (permanent of M(n)).
Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356481, A356482, A356483, A356484.

M[i_, j_, n_]:=Part[Part[Table[r+c-1,{r,n},{c,n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = matrix(n, n, i, j, i+j-1);
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 14 2023

A357419 a(n) is the hafnian of the 2n X 2n symmetric Pascal matrix defined by M[i, j] = A007318(i + j - 2, i - 1).

Original entry on oeis.org

1, 1, 17, 4929, 23872137, 1901611778409, 2469317979267366913, 52019468048773355156225921, 17726418489020770628047341494927089, 97518325438289444681986165275143492027985129, 8648473129650550498122567373327602114148485950241817345

Offset: 0

Stefano Spezia, Sep 27 2022

			a(2) = 17 because the hafnian of
    1,  1,  1,   1
    1,  2,  3,   4
    1,  3,  6,  10
    1,  4, 10,  20
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 17.

Eric Weisstein's World of Mathematics, Pascal Matrix
Wikipedia, Hafnian
Wikipedia, Pascal matrix
Wikipedia, Symmetric matrix

Cf. A007318.
Cf. A006134 (trace of M(n)), A095833 (k-th super- and subdiagonal sums of M(n)), A320845 (permanent of M(n)).
Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356481, A356482, A356483, A356484.

M[i_, j_, n_]:=Part[Part[Table[Binomial[r+c-2,r-1], {r, n}, {c, n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

a(6)-a(10) from Pontus von Brömssen, Oct 14 2023

cp's OEIS Frontend

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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A338456 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).

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A356483 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of prime(1), prime(2), ..., prime(2*n).

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A356484 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of prime(2*n), prime(2*n-1), ..., prime(1).

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A356481 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of 1, 2, ..., 2*n.

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A356482 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of 2*n, 2*n-1, ..., 1.

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A356483 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of prime(1), prime(2), ..., prime(2n).

A356484 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of prime(2n), prime(2*n-1), ..., prime(1).

A356481 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of 1, 2, ..., 2n.

A356482 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of 2n, 2*n-1, ..., 1.