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A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2) 23
1, 1, 2, 3, 8, 4, 15, 46, 36, 8, 105, 352, 344, 128, 16, 945, 3378, 3800, 1840, 400, 32, 10395, 39048, 48556, 27840, 8080, 1152, 64, 135135, 528414, 709324, 459032, 160720, 31136, 3136, 128 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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(x,n) = 2^n * product( x+(2*k-1)/2, k=1..n). The roots of the F(x,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(x,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.

LINKS

Table of n, a(n) for n=0..35.

FORMULA

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)!!.

a(n,m) = the (m+1)-th term in the top row of M^n, where M is an infinite square production matrix; M[i,j] = A029635(i,j-1) = binomial(i, j-1) + binomial(i-1, j-2) with A029635 the (1.2)-Pascal triangle, see the examples and second Maple program. [Gary W. Adamson, Jul 19 2011]

EXAMPLE

From Gary W. Adamson, Jul 19 2011: (Start)

The first few rows of matrix M are:

1, 2,  0,  0, 0, ...

1, 3,  2,  0, 0, ...

1, 4,  5,  2, 0, ...

1, 5,  9,  7, 2, ...

1, 6, 14, 16, 9, ... (End)

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]

MAPLE

nmax:=7; for n from 0 to nmax do a(n, n):=2^n: 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: seq(seq(a(n, k), k=0..n), n=0..nmax);

nmax:=7: M := Matrix(1..nmax+1, 1..nmax+1): A029635 := proc(n, k): binomial(n, k) + binomial(n-1, k-1) end: for i from 1 to nmax do for j from 1 to i+1 do M[i, j] := A029635(i, j-1) od: od: for n from 0 to nmax do B := M^n: for m from 0 to n do a(n, m):= B[1, m+1] od: od: seq(seq(a(n, m), m=0..n), n=0..nmax);

A161198 := proc(n, k) option remember; if k > n or k < 0 then 0 elif n = 0 and k = 0 then 1 else 2*A161198(n-1, k-1) + (2*n-1)*A161198(n-1, k) fi end:

seq(print(seq(A161198(n, k), k = 0..n)), n = 0..6);  # Peter Luschny, May 09 2013

MATHEMATICA

nmax = 7; a[n_, 0] := (2*n-1)!!; a[n_, n_] := 2^n; a[n_, m_] := a[n, m] = 2*a[n-1, m-1]+(2*n-1)*a[n-1, m]; Table[a[n, m], {n, 0, nmax}, {m, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Feb 25 2014, after Maple *)

PROG

(PARI) for(n=0, 9, print(Vec(Ser( 2^n*prod( k=1, n, x+(2*k-1)/2 ), , n+1)))))  \\ - M. F. Hasler, Jul 23 2011

(Sage)

@CachedFunction

def A161198(n, k):

    if k > n or k < 0 : return 0

    if n == 0 and k == 0: return 1

    return 2*A161198(n-1, k-1)+(2*n-1)*A161198(n-1, k)

for n in (0..6): [A161198(n, k) for k in (0..n)]  # Peter Luschny, May 09 2013

CROSSREFS

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.

Cf. A029635 [Gary W. Adamson, Jul 19 2011]

Sequence in context: A193975 A224665 A098514 * A195232 A093898 A194931

Adjacent sequences:  A161195 A161196 A161197 * A161199 A161200 A161201

KEYWORD

easy,nonn,tabl

AUTHOR

Johannes W. Meijer, Jun 08 2009, Jul 22 2011

STATUS

approved

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Last modified December 20 02:13 EST 2014. Contains 252240 sequences.