0,3

The series expansion of (1-x)^((-1-2*n)/2) = sum(b(p)*x^p, p=0..infinity) for n = 0, 1, 2 , .. can be described with b(p) = (F(p,n)/ (2*n-1)!!)*(binomial(2*p,p)/4^(p)) with F(p,n) = ((2^(n)* (product((p+(2*k-1)/2), k=1..n)))). The roots of the F(p,n) polynomials can be found at p = (1-2*k)/2 with k from 1 to n for n = 0, 1, 2, .. . The coefficients of the F(p,n) polynomials lead to the triangle given above. The triangle row sums lead to A001147.

Quite surprisingly we discovered that sum(b(p)*x^p, p=0..infinity) = (1-x)^(-1-2*n)/2, for n = -1, -2, .. . We assume that if m = n+1 then the value returned for product(f(k), k = m..n) is 1 and if m> n+1 then 1/product(f(k), k=n+1..m-1) is the value returned. Furthermore (1-2*n)!! = (-1)^(n+1)/(2*n-3)!! for n = 1, 2, 3 .. . This leads to b(p) = ((-1-2*n)!!/ G(p,n))*(binomial(2*p,p) /4^(p)) for n = -1, -2, .. . For the G(p,n) polynomials we found that G(p,n) = F(-p,-n). The roots of the G(p,n) polynomials can be found at p=(2*k-1)/2 with k from 1 to (-n) for n = -1, -2, .. . The coefficients of the G(p,n) polynomials lead to a second triangle that stands with its head on top of the first one. It is remarkable that the row sums lead once again to A001147.

These two triangles together look like an hourglass so we propose to call the F(p,n) and the G(p,n) polynomials the hourglass polynomials.

a(n,m) := coeff(2^(n)*product((x+(2*k-1)/2),k=1..n), x, m) for n = 0, 1, .. ; m = 0, 1, .. .

a(n, m) = 2*a(n-1,m-1)+(2*n-1)*a(n-1,m) with a(n, n) = 2^n and a(n, 0) = (2*n-1)!!.

The first few G(p,n) polynomials are:

G(p,-3) = 15 - 46*p + 36*p^2 - 8*p^3

G(p,-2) = 3 - 8*p + 4*p^2

G(p,-1) = 1 - 2*p

The first few F(p,n) polynomials are:

F(p,0) = 1

F(p,1) = 1 + 2*p

F(p,2) = 3 + 8*p + 4*p^2

F(p,3) = 15 + 46*p + 36*p^2 + 8*p^3

The first few rows of the upper and lower hourglass triangles are:

[15, -46, 36, -8]

[3, -8 , 4]

[1, -2]

[1]

[1, 2]

[3, 8 , 4]

[15, 46, 36, 8]

nmax:=7; for n from 0 to nmax do a(n, n):=2^n od: for n from 0 to nmax do a(n, 0):=doublefactorial(2*n-1) od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := 2*a(n-1, m-1)+(2*n-1)*a(n-1, m) od: od: T:=0: for n from 0 to nmax do for m from 0 to n do a(T):=a(n, m): T:=T+1: od: od: seq(a(n), n=0..T-1);

Cf. A001790 [(1-x)^(-1/2)], A001803 [(1-x)^(-3/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/2)].

Cf. A002596 [(1-x)^(1/2)], A161200 [(1-x)^(3/2)] and A161202 [(1-x)^(5/2)].

A046161 gives the denominators of the series expansions of all (1-x)^((-1-2*n)/2).

A028338 is a scaled triangle version, A039757 is a scaled signed triangle version and A109692 is a transposed scaled triangle version.

A001147 is the first left hand column and equals the row sums.

A004041 is the second left hand column divided by 2, A028339 is the third left hand column divided by 4, A028340 is the fourth left hand column divided by 8, A028341 is the fifth left hand column divided by 16.

A000012, A000290, A024196, A024197 and A024198 are the first (n-m=0), second (n-m=1), third (n-m=2), fourth (n-m=3) and fifth (n-m=4) right hand columns divided by 2^m.

A074599 * A025549 not always equals the second left hand column.

Sequence in context: A110142 A158928 A098514 this_sequence A093898 A084110 A100208

Adjacent sequences: A161195 A161196 A161197 this_sequence A161199 A161200 A161201

easy,nonn,tabl

Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 2009