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

Author

Peter Luschny, Jun 28 2009

Keywords

• easy
• nonn
• tabf

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, ...

Examples

			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
		

Links

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

Crossrefs

Cf. A056040, A001792.

Programs

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 *)

Formula

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