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%I #15 Jan 07 2019 12:31:19
%S 1,-1,2,0,-6,4,6,-8,-2,4,4,0,0,-20,24,-6,40,-96,54,-40,144,-126,20,
%T -96,114,-4,24,-36,8,0,0,0,-56,96,-48,8,168,-576,624,-216,-280,1440,
%U -2400,1296,280,-1920,4320,-3184,-168,1440,-4080,3816,56,-576,1968,-2232,-8,96,-384,512
%N The sequence b(n,m) of the m polynomial coefficients of the n-th order B-spline wavelet scaled by n!2^n, read by rows, with n in {0,1,2,...} and m in {1,2,3,...,2(n+1)^2}.
%C The n-th order B-spline wavelet function B_n(x) may be calculated with the expression B_n(x) = (1/(2^n)) Sum_{k=0..n+1} (-1)^k binomial(n+1,k) N_n(2x-k)
%C where
%C * N_n(x) is the n-th order B-spline and as such (see A289358)
%C N_n(x) = (1/n!)Sum_{m=0..n+1} (-1)^m binomial(n+1,m)(x-m)^n step(x-m),
%C * step(x) is the step function defined as step(x) = {1 for x >= 0
%C {0 otherwise.
%C From these definitions, it is apparent that the coefficients of the polynomials induced by n!2^n*B_n(x) are integers and can be "flattened" (as in the Pascal triangle A007318) to form an integer sequence, part of which is listed above.
%D Daniel J. Greenhoe, "Wavelet Structure and Design" (2013), isbn13: 9780983801139, page 318, Theorem H.1.
%D Ole Christensen, "An Introduction to Frames and Riesz Bases" (2003), isbn13: 9780817642952, pages 335--338, equation (14.41) and Figure 14.4.
%D Ole Christensen, "Frames and bases: An Introductory Course" (2008), isbn13: 9780817646776, pages 283--284.
%H Daniel J. Greenhoe, <a href="/A290826/a290826.pdf">Technical report</a> for this sequence.
%H Daniel J. Greenhoe, <a href="/A290826/a290826.max.txt">Maxima script</a> supporting this sequence.
%H Daniel J. Greenhoe, <a href="https://www.researchgate.net/publication/318110604">B-splines and B-spline wavelets</a>, Technical Report [version 0.20], July 2017.
%H Ole Christensen, <a href="http://books.google.com/books?vid=ISBN0817642951&pg=PA335">equation (14.41)</a>, "An Introduction to Frames and Riesz Bases" (2003), ISBN 9780817646783.
%H Ole Christensen, <a href="http://books.google.com/books?vid=ISBN0817646787&pg=PA283">equation (11.63)</a>, "Frames and bases: An Introductory Course" (2008), ISBN 9780817646783.
%H Daniel J. Greenhoe, <a href="https://www.researchgate.net/publication/312529555">Wavelet Structure and Design</a>, [version 1.20], January 2017, "Mathematical Structure and Design" series, volume 3, Theorem H.1, pages 267--268.
%H Daniel J. Greenhoe, <a href="https://www.researchgate.net/publication/312201065">Structure and Analysis of Mathematical Spaces</a>, [version 0.30], January 2017, "Mathematical Structure and Design" series, volume 2, Proposition 11.4 (Power Sums), pages 186--187.
%F The n-th order B-spline B_n(x) may be expressed as a set of polynomials as described in the COMMENTS section.
%e The m=2(n+1)^2 coefficients for the n-th order B-spline wavelet B_n(x) begin at the sequence index value p=2Sum_{k=0..n}k^2=2(1/6)n(n+1)(2n+1) and end at index value p+2(n+1)^2-1.
%e Each set of m=2(n+1)^2 coefficients for n=0,1,2,... can be written in the form of an (2n+2)X(n+1) matrix A_n as...
%e A_0 = [ 1] for n=0 (index values 0 to 1)
%e [-1]
%e [ 2 0]
%e A_1 = [-6 4] for n=1 (index values 2 to 9)
%e [ 6 -8]
%e [-2 4]
%e [ 4 0 0]
%e [-20 24 -6]
%e A_2 = [ 40 -96 54] for n=2 (index values 10 to 27)
%e [-40 144 -126]
%e [ 20 -96 114]
%e [ -4 24 -36]
%e That is, the sequence of integers induces a sequence of (2n+2)X(n+1) matrices (A_0, A_1, A_2, ...).
%e Taking the n=2 case for example, the coefficients for B_2(x) begin at index value p=2(0+1+4)=10, end at index value p+2(3^2)-1=27, and are scaled by 2!2^2=8. Using this data yields the following expression for B_2(x):
%e [ 4 0 0 : for 0 <= x < 0.5] [x^2]
%e [-20 24 -6 : for 0.5 <= x < 1.0] [ x ]
%e 8B_2(x) = [ 40 -96 54 : for 1.0 <= x < 1.5] [ 1 ]
%e [-40 144 -126 : for 1.5 <= x < 2.0]
%e [ 20 -96 114 : for 2.0 <= x < 2.5]
%e [ -4 24 -36 : for 2.5 <= x < 3.0]
%e [ 0 0 0 : otherwise ]
%e { 4x^3 : for 0 <= x < 0.5}
%e {-20x^3 +24x^2 -6 : for 0.5 <= x < 1.0}
%e = { 40x^3 -96x^2 +54 : for 1.0 <= x < 1.5}
%e {-40x^3 +144x^2 -126 : for 1.5 <= x < 2.0}
%e { 20x^3 -96x^2 114 : for 2.0 <= x < 2.5}
%e { -4x^3 +24x^2 -36 : for 2.5 <= x < 3.0}
%e { 0 : otherwise }
%e Note: Sum_{k=1..n}k^2 is called a "power sum".
%e For proof that p=Sum_{k=0..n}k^2=(1/6)n(n+1)(2n+1) (as indicated above),
%e see the "Structure and Analysis of Mathematical Spaces" link.
%o (Maxima)
%o reset();
%o kill(all);
%o load(orthopoly);
%o n:2;
%o Bnx:(1/n!)*(1/2^n)*sum((-1)^k*binomial(n+1,k)
%o *sum((-1)^m*binomial(n+1,m)*(2*x-m-k)^n*unit_step(2*x-m-k),m,0,n+1),k,0,n+1);
%o assume(x<=0); print(n!*2^n,"B(x)= ",expand(n!*2^n*Bnx)," for x<=0"); forget(x<=0);
%o for i:0 thru (n+0.5) step 0.5 do(
%o assume(x>i,x<(i+0.5)),
%o print(n!*2^n,"B(x)= ",expand(n!*2^n*Bnx)," for ",i,"<x<",i+0.5),
%o forget(x>i,x<(i+0.5))
%o );
%o assume(x>(n+1)); print(n!*2^n,"B(x)= ",expand(n!*2^n*Bnx)," for x>",n+1); forget(x>(n+1));
%Y Cf. A289358.
%K sign,tabf
%O 0,3
%A _Daniel J. Greenhoe_, Aug 11 2017