A002772
Number of terms in a bordered skew determinant.
Original entry on oeis.org
2, 6, 22, 101, 546, 3502, 25586, 214062, 1987516, 20599076, 232482372, 2876191276, 38228128472, 549706132536, 8408517839416, 137788390312712, 2383879842920976, 43846851982943152, 846470648320690736, 17266870434276713616, 367937854493289655072
Offset: 2
- T. Muir, The expression of any bordered skew determinant as a sum of products of Pfaffians, Proc. Roy. Soc. Edinburgh, 21 (1896), 342-359.
- T. Muir, The Theory of Determinants in the Historical Order of Development. 4 vols., Macmillan, NY, 1906-1923, Vol. 4, p. 278.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. Muir, The expression of any bordered skew determinant as a sum of products of Pfaffians, Proc. Roy. Soc. Edinburgh, 21 (1896), 342-359. [Annotated scan of pages 354-357 only]
- T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 4.
A243107
Number of terms in a bordered skew determinant.
Original entry on oeis.org
1, 1, 2, 4, 13, 41, 226, 1072, 9059, 58123, 657766, 5268836, 73980787, 707506879, 11823958238, 131277234376, 2542107619081, 32122718085497, 706963537444114, 10015472595953908, 246853433179370621, 3874536631479770761, 105709617658879558402
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..450
- T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 282.
- J. J. Sylvester, Note on determinants and duadic disynthemes, American J of Math, ii, (1879), 89-90, 214-222.
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a:= proc(n) option remember; `if`(n<4, [1$2, 2, 4][n+1],
(2*a(n-1)+2*(n-1)^2*a(n-2)-2*(n-1)*(n-2)*a(n-3)
-(n-1)*(n-2)*(n-3)*a(n-4))/2)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Aug 19 2014
-
b[n_] := Gamma[n+1/2] HypergeometricPFQ[{1/4, -n}, {}, -4]/Sqrt[Pi];
a[n_] := Sum[Binomial[n, n-2k] b[k], {k, 0, n/2}];
a /@ Range[0, 30]
(* Second program: *)
a[n_] := a[n] = If[n < 4, {1, 1, 2, 4}[[n+1]], (2a[n-1] + 2(n-1)^2 a[n-2] - 2(n-1)(n-2)a[n-3] - (n-1)(n-2)(n-3) a[n-4])/2];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 13 2020, after Alois P. Heinz *)
-
my(x='x+O('x^66)); Vec(serlaplace(exp(x+x^2/4) / (1-x^2)^(1/4))) \\ Joerg Arndt, Aug 20 2014
A246257
Triangular array read by rows: T(n, k) = S(n, [n/2]-k) and S(n,k) = C(n, 2*k)*(2*k-1)!!*((2*k-1)!! + 1)/2, n>=0, 0<=k<=[n/2].
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 6, 6, 1, 30, 10, 1, 120, 90, 15, 1, 840, 210, 21, 1, 5565, 3360, 420, 28, 1, 50085, 10080, 756, 36, 1, 446985, 250425, 25200, 1260, 45, 1, 4916835, 918225, 55440, 1980, 55, 1, 54033210, 29501010, 2754675, 110880, 2970, 66, 1
Offset: 0
Triangle starts:
[ 0] 1,
[ 1] 1,
[ 2] 1, 1,
[ 3] 3, 1,
[ 4] 6, 6, 1,
[ 5] 30, 10, 1,
[ 6] 120, 90, 15, 1,
[ 7] 840, 210, 21, 1,
[ 8] 5565, 3360, 420, 28, 1,
[ 9] 50085, 10080, 756, 36, 1,
[10] 446985, 250425, 25200, 1260, 45, 1.
-
T := proc(n, k) local j; j := iquo(n,2) - k;
(n!/(j!*(n-2*j)!))*(2^(-j-1)+GAMMA(j+1/2)/sqrt(4*Pi)) end:
seq(print(seq(T(n,k), k=0..iquo(n,2))), n=0..10);
-
row[n_] := FunctionExpand[HypergeometricPFQ[{-n/2, (1-n)/2}, {}, 2z] + HypergeometricPFQ[{1/2, -n/2, (1-n)/2}, {}, 4z]]/2 // CoefficientList[#, z]& // Reverse;
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)
-
from sage.functions.hypergeometric import closed_form
def A246257_row(n):
R. = ZZ[]
h = hypergeometric([-n/2,(1-n)/2], [], 2*z)
g = hypergeometric([1/2,-n/2,(1-n)/2], [], 4*z)
T = R(((closed_form(h)+closed_form(g))/2)).coefficients()
return T[::-1]
for n in range(13): A246257_row(n)
Showing 1-3 of 3 results.
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