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A162246 Swinging polynomials, coefficients read by rows. 7
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 (list; graph; refs; listen; history; text; internal format)
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, ...

REFERENCES

Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008.

LINKS

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

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-Fran├žois Alcover, Jun 28 2013 *)

CROSSREFS

Cf. A056040, A001792.

Sequence in context: A144431 A053821 A076545 * A277447 A213126 A118400

Adjacent sequences:  A162243 A162244 A162245 * A162247 A162248 A162249

KEYWORD

easy,nonn,tabf

AUTHOR

Peter Luschny, Jun 28 2009

STATUS

approved

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Last modified October 15 05:56 EDT 2018. Contains 316202 sequences. (Running on oeis4.)