A162246 Swinging polynomials, coefficients read by rows.

1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 6, 3, 3, 1, 1, 4, 4, 12, 6, 12, 4, 4, 1, 1, 5, 5, 20, 10, 30, 10, 20, 5, 5, 1, 1, 6, 6, 30, 15, 60, 20, 60, 15, 30, 6, 6, 1, 1, 7, 7, 42, 21, 105, 35, 140, 35, 105, 21, 42, 7, 7, 1

Offset

0,6

Comments

Let p(n,x) = (1+x^2)^n+n*x*(1+x^2)^(n-1), then T(n,k) are the coefficients of these polynomials, read by rows, n = 0,1,...

The central numbers of the rows, i.e., the coefficients of x^n of p(n,x), are the swinging factorial numbers A056040(n).

Row sums: sum_{k=0..2n} T(n,k) = A001792(n).

sum_{k=0..2n} isodd(n+k)T(n,k) = 2^n(isodd(n)+(n/2)isodd(n+1))

= 0, 2, 4, 8, 32, 32, 192, 128, 1024, 512, 5120, ...

sum_{k=0..2n} iseven(n+k)T(n,k) = 2^n(isodd(n)(n/2)+isodd(n+1))

= 1, 1, 4, 12, 16, 80, 64, 448, 256, 2304, 1024, ...

Links

Table of n, a(n) for n=0..63.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Formula

T(n,k) = n!/((n-ceiling(k/2))!*floor(k/2)!).

Example

The central coefficients are marked by [].
[1]
1,[1],1
1,2,[2],2,1
1,3,3,[6],3,3,1
1,4,4,12,[6],12,4,4,1
1,5,5,20,10,[30],10,20,5,5,1
1,6,6,30,15,60,[20],60,15,30,6,6,1
1,7,7,42,21,105,35,[140],35,105,21,42,7,7,1
p(0,x) = 1
p(1,x) = x^2+x+1
p(2,x) = x^4+2x^3+2x^2+2x+1
p(3,x) = x^6+3x^5+3x^4+6x^3+3x^2+3x+1
p(4,x) = x^8+4x^7+4x^6+12x^5+6x^4+12x^3+4x^2+4x+1
p(5,x) = x^10+5x^9+5x^8+20x^7+10x^6+30x^5+10x^4+20x^3+5x^2+5x+1

Maple

p := (n, x) -> (1+x^2)^n+n*x*(1+x^2)^(n-1):
seq(print(seq(coeff(expand(p(n, x)), x, i), i=0..2*n)), n=0..7);
T := (n, k) -> n!/((n-ceil(k/2))!*floor(k/2)!);
seq(print(seq(T(n, k), k=0..2*n)), n=0..7);

Mathematica

t[n_, k_] := If[EvenQ[k], Binomial[n, k/2], Binomial[n, (k-1)/2]*(n-(k-1)/2)]; Table[t[n, k], {n, 0, 7}, {k, 0, 2*n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

Crossrefs

Cf. A056040, A001792.

Sequence in context: A144431 A053821 A076545 * A277447 A333698 A213126

Adjacent sequences: A162243 A162244 A162245 * A162247 A162248 A162249

Keyword

easy,nonn,tabf

Author

Peter Luschny, Jun 28 2009

Status

approved