Search: id:A161198
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%I A161198
%S A161198 1,1,2,3,8,4,15,46,36,8,105,352,344,128,16,945,3378,3800,1840,400,32,
%T A161198 10395,39048,48556,27840,8080,1152,64,135135,528414,709324,459032,
%U A161198 160720,31136,3136,128
%N A161198 Triangle of polynomial coefficients related to the series expansions 
               of (1-x)^((-1-2*n)/2)
%C A161198 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.
%C A161198 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.
%C A161198 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.
%F A161198 a(n,m) := coeff(2^(n)*product((x+(2*k-1)/2),k=1..n), x, m) for n = 0, 
               1, .. ; m = 0, 1, .. .
%F A161198 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)!!.
%e A161198 The first few G(p,n) polynomials are:
%e A161198 G(p,-3) = 15 - 46*p + 36*p^2 - 8*p^3
%e A161198 G(p,-2) = 3 - 8*p + 4*p^2
%e A161198 G(p,-1) = 1 - 2*p
%e A161198 The first few F(p,n) polynomials are:
%e A161198 F(p,0) = 1
%e A161198 F(p,1) = 1 + 2*p
%e A161198 F(p,2) = 3 + 8*p + 4*p^2
%e A161198 F(p,3) = 15 + 46*p + 36*p^2 + 8*p^3
%e A161198 The first few rows of the upper and lower hourglass triangles are:
%e A161198 [15, -46, 36, -8]
%e A161198 [3, -8 , 4]
%e A161198 [1, -2]
%e A161198 [1]
%e A161198 [1, 2]
%e A161198 [3, 8 , 4]
%e A161198 [15, 46, 36, 8]
%p A161198 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);
%Y A161198 Cf. A001790 [(1-x)^(-1/2)], A001803 [(1-x)^(-3/2)], A161199 [(1-x)^(-5/
               2)] and A161201 [(1-x)^(-7/2)].
%Y A161198 Cf. A002596 [(1-x)^(1/2)], A161200 [(1-x)^(3/2)] and A161202 [(1-x)^(5/
               2)].
%Y A161198 A046161 gives the denominators of the series expansions of all (1-x)^((-1-2*n)/
               2).
%Y A161198 A028338 is a scaled triangle version, A039757 is a scaled signed triangle 
               version and A109692 is a transposed scaled triangle version.
%Y A161198 A001147 is the first left hand column and equals the row sums.
%Y A161198 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.
%Y A161198 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.
%Y A161198 A074599 * A025549 not always equals the second left hand column.
%Y A161198 Sequence in context: A110142 A158928 A098514 this_sequence A093898 A084110 
               A100208
%Y A161198 Adjacent sequences: A161195 A161196 A161197 this_sequence A161199 A161200 
               A161201
%K A161198 easy,nonn,tabl
%O A161198 0,3
%A A161198 Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 2009

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