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A161198
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Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2)
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24
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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
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OFFSET
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0,3
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COMMENTS
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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.
Triangle T(n,k), read by rows, given by (1, 2, 3, 4, 5, 6, 7, 8, 9, ...) DELTA (2, 0, 2, 0, 2, 0, 2, 0, 2, ...) where DELTA is the operator defined in A084938. Philippe Deléham, May 14 2015.
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LINKS
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FORMULA
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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]
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EXAMPLE
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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]
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MAPLE
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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:
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MATHEMATICA
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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-François Alcover, Feb 25 2014, after Maple *)
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PROG
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(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
if k > n or k < 0 : return 0
if n == 0 and k == 0: return 1
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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STATUS
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approved
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