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Swinging polynomials, coefficients read by rows.
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%I #11 May 08 2020 17:31:18

%S 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,

%T 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,

%U 35,105,21,42,7,7,1

%N Swinging polynomials, coefficients read by rows.

%C 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,...

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

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

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

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

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

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

%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.

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

%e The central coefficients are marked by [].

%e [1]

%e 1,[1],1

%e 1,2,[2],2,1

%e 1,3,3,[6],3,3,1

%e 1,4,4,12,[6],12,4,4,1

%e 1,5,5,20,10,[30],10,20,5,5,1

%e 1,6,6,30,15,60,[20],60,15,30,6,6,1

%e 1,7,7,42,21,105,35,[140],35,105,21,42,7,7,1

%e p(0,x) = 1

%e p(1,x) = x^2+x+1

%e p(2,x) = x^4+2x^3+2x^2+2x+1

%e p(3,x) = x^6+3x^5+3x^4+6x^3+3x^2+3x+1

%e p(4,x) = x^8+4x^7+4x^6+12x^5+6x^4+12x^3+4x^2+4x+1

%e p(5,x) = x^10+5x^9+5x^8+20x^7+10x^6+30x^5+10x^4+20x^3+5x^2+5x+1

%p p := (n,x) -> (1+x^2)^n+n*x*(1+x^2)^(n-1):

%p seq(print(seq(coeff(expand(p(n,x)),x,i),i=0..2*n)),n=0..7);

%p T := (n,k) -> n!/((n-ceil(k/2))!*floor(k/2)!);

%p seq(print(seq(T(n,k),k=0..2*n)),n=0..7);

%t 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 *)

%Y Cf. A056040, A001792.

%K easy,nonn,tabf

%O 0,6

%A _Peter Luschny_, Jun 28 2009