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
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.
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)).
-
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
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).
Original entry on oeis.org
1, 3, 55, 2999, 347391, 69702479, 22441691645, 10776262328919, 7190279422736061, 6439969796874334809, 7447188585071730451961
Offset: 0
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.
-
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
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).
Original entry on oeis.org
1, 2, 44, 5210, 1368900, 604109562, 535920536336, 728155179271474, 1103827431509790216, 2651375713654260218986, 7537958658258053003685636
Offset: 0
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.
-
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
A356481
a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of 1, 2, ..., 2*n.
Original entry on oeis.org
1, 2, 21, 532, 24845, 1856094, 203076097, 30633787976, 6097546660185, 1548899852221210, 489114616743840461
Offset: 0
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.
Cf.
A001792 (absolute value of the determinant of M(n)),
A204235 (permanent of M(n)).
-
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
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.
Original entry on oeis.org
1, 1, 16, 714, 62528, 9056720, 1960138560, 592615689904, 238560786221056, 123358665203311104, 79683847063011614720
Offset: 0
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.
-
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
A012259
Expansion of e.g.f. exp(arctanh(tan(x))).
Original entry on oeis.org
1, 1, 1, 5, 17, 121, 721, 6845, 58337, 698161, 7734241, 111973685, 1526099057, 25947503401, 419784870961, 8200346492525, 153563504618177, 3389281372287841, 72104198836466881, 1774459993676715365, 42270463533824671697, 1147649139272698443481
Offset: 0
Patrick Demichel (patrick.demichel(AT)hp.com)
exp(arctanh(tan(x))) = 1 + x + x^2/2! + 5*x^3/3! + 17*x^4/4! + 121*x^5/5! + ...
- G. C. Greubel, Table of n, a(n) for n = 0..425
- Shi-Mei Ma, Jun Ma, and Yeong-Nan Yeh, The alternating run polynomials of permutations, arXiv:1904.11437 [math.CO], 2019.
- Shi-Mei Ma, Jun Ma, and Yeong-Nan Yeh, David-Barton type identities and alternating run polynomials, Academia Sinica (Taipei, 2019).
- Mathematics Stack Exchange, Mystery regarding power series of 1/sqrt(1+x^x), Question 6939.
-
m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Sqrt((1+Tan(x))/(1-Tan(x))) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 06 2019
-
With[{nn=30}, CoefficientList[Series[Sqrt[(1+Tan[x])/(1-Tan[x])], {x, 0, nn}], x]*Range[0,nn]!] (* Vaclav Kotesovec, Oct 23 2013 *)
-
{a(n)=local(A=1); for(i=0, n, A = exp( intformal( (A^2 + subst(A^2, x, -x))/2 +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 04 2017
-
my(x='x+O('x^30)); Vec(serlaplace( sqrt((1+tan(x))/(1-tan(x))) )) \\ G. C. Greubel, Jun 06 2019
-
m = 30; T = taylor(sqrt((1+tan(x))/(1-tan(x))), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jun 06 2019
A185410
A decomposition of the double factorials A001147.
Original entry on oeis.org
1, 1, 0, 1, 2, 0, 1, 10, 4, 0, 1, 36, 60, 8, 0, 1, 116, 516, 296, 16, 0, 1, 358, 3508, 5168, 1328, 32, 0, 1, 1086, 21120, 64240, 42960, 5664, 64, 0, 1, 3272, 118632, 660880, 900560, 320064, 23488, 128, 0, 1, 9832, 638968, 6049744, 14713840, 10725184, 2225728, 95872, 256, 0
Offset: 0
Triangle begins:
1,
1, 0,
1, 2, 0,
1, 10, 4, 0,
1, 36, 60, 8, 0,
1, 116, 516, 296, 16, 0,
1, 358, 3508, 5168, 1328, 32, 0,
1, 1086, 21120, 64240, 42960, 5664, 64, 0,
1, 3272, 118632, 660880, 900560, 320064, 23488, 128, 0,
1, 9832, 638968, 6049744, 14713840, 10725184, 2225728, 95872, 256, 0,
...
In the Savage-Viswanathan paper, the coefficients appear as
1
1 2
1 10 4
1 36 60 8
1 116 516 296 16
1 358 3508 5168 1328 32
1 1086 21120 64240 42960 5664 64
...
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
- S.-M. Ma and T. Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, arXiv preprint arXiv:1409.6525 [math.CO], 2014.
- C. D. Savage and G. Viswanathan, The 1/k-Eulerian polynomials, Elec. J. of Comb., Vol. 19, Issue 1, #P9 (2012).
-
T[0, 0] := 1; T[n_, -1] := 0; T[n_, n_] := 0; T[n_, k_] := T[n, k] = (n - k)*T[n - 1, k - 1] + (2*k + 1)*T[n - 1, k]; Join[{1}, Table[If[k < 0, 0, If[k >= n, 0, 2^k*T[n, k]]], {n, 1, 5}, {k, 0, n}] // Flatten] (* G. C. Greubel, Jun 30 2017 *)
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
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.
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)).
-
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
A291207
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 + x/(1 - 2^k*x/(1 + 3^k*x/(1 - 4^k*x/(1 + 5^k*x/(1 - ...)))))).
Original entry on oeis.org
1, 1, -1, 1, -1, 0, 1, -1, -1, 1, 1, -1, -3, 5, 0, 1, -1, -7, 27, 17, -2, 1, -1, -15, 167, 441, -121, 0, 1, -1, -31, 1071, 10673, -11529, -721, 5, 1, -1, -63, 6815, 262305, -1337713, -442827, 6845, 0, 1, -1, -127, 42687, 6525377, -161721441, -297209047, 23444883, 58337, -14
Offset: 0
G.f. of column k: A_k(x) = 1 - x + (1 - 2^k)*x^2 + (2^(k + 1) - 4^k + 6^k - 1)*x^3 + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
-1, -1, -1, -1, -1, -1, ...
0, -1, -3, -7, -15, -31, ...
1, 5, 27, 167, 1071, 6815, ...
0, 17, 441, 10673, 262305, 6525377, ...
-2, -121, -11529, -1337713, -161721441, -19802585281, ...
-
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-(-1)^i i^k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 9}, {n, 0, j}] // Flatten
A331405
G.f.: 1/(1 - 1*2*x/(1 + 2*3*x/(1 - 3*4*x/(1 + 4*5*x/(1 - 5*6*x/(1 + ...)))))), a continued fraction.
Original entry on oeis.org
1, 2, -8, -112, 2176, 71936, -3163136, -196237312, 15258124288, 1531746516992, -185088737017856, -27405687884087296, 4747122204712370176, 973473732763710390272, -228670532983871365971968, -62056343388674412796444672, 18982531521384459634512756736
Offset: 0
-
nmax = 16; CoefficientList[Series[1/(1 + ContinuedFractionK[(-1)^k k (k + 1) x, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
Showing 1-10 of 13 results.
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