A245334
A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.
Original entry on oeis.org
1, 2, 1, 3, 4, 2, 4, 9, 12, 6, 5, 16, 36, 48, 24, 6, 25, 80, 180, 240, 120, 7, 36, 150, 480, 1080, 1440, 720, 8, 49, 252, 1050, 3360, 7560, 10080, 5040, 9, 64, 392, 2016, 8400, 26880, 60480, 80640, 40320, 10, 81, 576, 3528, 18144, 75600, 241920, 544320
Offset: 0
. 0: 1;
. 1: 2, 1;
. 2: 3, 4, 2;
. 3: 4, 9, 12, 6;
. 4: 5, 16, 36, 48, 24;
. 5: 6, 25, 80, 180, 240, 120;
. 6: 7, 36, 150, 480, 1080, 1440, 720;
. 7: 8, 49, 252, 1050, 3360, 7560, 10080, 5040;
. 8: 9, 64, 392, 2016, 8400, 26880, 60480, 80640, 40320;
. 9: 10, 81, 576, 3528, 18144, 75600, 241920, 544320, 725760, 362880.
Cf.
A000142,
A001715,
A001720,
A001725,
A001730,
A049388,
A049389,
A049398,
A051431,
A052849,
A070960.
-
a245334 n k = a245334_tabl !! n !! k
a245334_row n = a245334_tabl !! n
a245334_tabl = iterate (\row@(h:_) -> (h + 1) : map (* h) row) [1]
-
Table[(n!)/((n - k)!)*(n + 1 - k), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 10 2017 *)
A002691
a(n) = (n+2) * (2n+1) * (2n-1)! / (n-1)!.
Original entry on oeis.org
1, 9, 120, 2100, 45360, 1164240, 34594560, 1167566400, 44108064000, 1843717075200, 84475764172800, 4209708914611200, 226676633863680000, 13114862387827200000, 811372819726909440000, 53449184499510159360000, 3735154775612827607040000
Offset: 0
- 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).
-
with(combstruct): a:=n-> add((count(Permutation(n*2+1), size=n+1)), j=0..n+1)/2: seq(a(n), n=0..16); # Zerinvary Lajos, May 03 2007
-
Join[{1},Table[(n+2)(2n+1)(2n-1)!/(n-1)!,{n,15}]] (* Harvey P. Dale, Jun 09 2011 *)
-
a(n)=(n+2)*(2*n+1)*(2*n-1)!/(n-1)!
A278071
Triangle read by rows, coefficients of the polynomials P(n,x) = (-1)^n*hypergeom( [n,-n], [], x), powers in descending order.
Original entry on oeis.org
1, 1, -1, 6, -4, 1, 60, -36, 9, -1, 840, -480, 120, -16, 1, 15120, -8400, 2100, -300, 25, -1, 332640, -181440, 45360, -6720, 630, -36, 1, 8648640, -4656960, 1164240, -176400, 17640, -1176, 49, -1, 259459200, -138378240, 34594560, -5322240, 554400, -40320, 2016, -64, 1
Offset: 0
Triangle starts:
. 1,
. 1, -1,
. 6, -4, 1,
. 60, -36, 9, -1,
. 840, -480, 120, -16, 1,
. 15120, -8400, 2100, -300, 25, -1,
. 332640, -181440, 45360, -6720, 630, -36, 1,
...
- H. L. Krall and O. Fink, A New Class of Orthogonal Polynomials: The Bessel Polynomials, Trans. Amer. Math. Soc. 65, 100-115, 1949.
- Herbert E. Salzer, Orthogonal Polynomials Arising in the Numerical Evaluation of Inverse Laplace Transforms, Mathematical Tables and Other Aids to Computation, Vol. 9, No. 52 (Oct., 1955), pp. 164-177, (see p.174 and footnote 7).
T(n,0) = Pochhammer(n, n) (cf.
A000407).
T(n,1) = -(n+1)*(2n)!/n! (cf.
A002690).
T(n,2) = (n+2)*(2n+1)*(2n-1)!/(n-1)! (cf.
A002691).
T(n,n-1) = (-1)^(n+1)*n^2 for n>=1 (cf.
A000290).
T(n,n-2) = n^2*(n^2-1)/2 for n>=2 (cf.
A083374).
-
p := n -> (-1)^n*hypergeom([n, -n], [], x):
ListTools:-Flatten([seq(PolynomialTools:-CoefficientList(simplify(p(n)), x, termorder=reverse), n=0..8)]);
# Alternatively the polynomials by recurrence:
P := proc(n,x) if n=0 then return 1 fi; if n=1 then return x-1 fi;
((((4*n-2)*(2*n-3)*x+2)*P(n-1,x)+(2*n-1)*P(n-2,x))/(2*n-3));
sort(expand(%)) end: for n from 0 to 6 do lprint(P(n,x)) od;
# Or by generalized Laguerre polynomials:
P := (n,x) -> n!*(-x)^n*LaguerreL(n,-2*n,-1/x):
for n from 0 to 6 do simplify(P(n,x)) od;
-
row[n_] := CoefficientList[(-1)^n HypergeometricPFQ[{n, -n}, {}, x], x] // Reverse;
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)
(* T(n,k)= *) t={};For[n=8,n>-1,n--,For[j=n+1,j>0,j--,PrependTo[t,(-1)^(j-n+1-Mod[n,2])*Product[(2*n-k)*k/(n-k+1),{k,j,n}]]]];t (* Detlef Meya, Aug 02 2023 *)
Showing 1-3 of 3 results.
Comments