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A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).

%I A161198 #27 Jun 02 2025 01:43:31
%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(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.
%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.
%C A161198 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.
%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)!!.
%F A161198 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]
%F A161198 T(n,k) = 2^k * A028338(n,k). - _Philippe Deléham_, May 14 2015
%e A161198 From _Gary W. Adamson_, Jul 19 2011: (Start)
%e A161198 The first few rows of matrix M are:
%e A161198   1, 2,  0,  0, 0, ...
%e A161198   1, 3,  2,  0, 0, ...
%e A161198   1, 4,  5,  2, 0, ...
%e A161198   1, 5,  9,  7, 2, ...
%e A161198   1, 6, 14, 16, 9, ... (End)
%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: 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);
%p A161198 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);
%p A161198 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:
%p A161198 seq(print(seq(A161198(n,k), k = 0..n)), n = 0..6);  # _Peter Luschny_, May 09 2013
%t A161198 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 *)
%o A161198 (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
%o A161198 (Sage)
%o A161198 @CachedFunction
%o A161198 def A161198(n,k):
%o A161198     if k > n or k < 0 : return 0
%o A161198     if n == 0 and k == 0: return 1
%o A161198     return 2*A161198(n-1,k-1)+(2*n-1)*A161198(n-1,k)
%o A161198 for n in (0..6): [A161198(n,k) for k in (0..n)]  # _Peter Luschny_, May 09 2013
%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 is not always equals the second left hand column.
%Y A161198 Cf. A029635. [_Gary W. Adamson_, Jul 19 2011]
%K A161198 easy,nonn,tabl
%O A161198 0,3
%A A161198 _Johannes W. Meijer_, Jun 08 2009, Jul 22 2011