%I #57 Sep 20 2017 11:19:29

%S 1,1,0,-1,2,1,0,0,-2,6,-3,1,-6,9,1,0,0,0,-3,12,-12,4,3,-24,60,-44,-1,

%T 12,-48,64,1,0,0,0,0,-4,20,-30,20,-5,6,-60,210,-300,155,-4,60,-330,

%U 780,-655,1,-20,150,-500,625,1,0,0,0,0,0,-5,30,-60,60,-30,6,10,-120,540,-1140,1170,-474,-10,180,-1260,4260

%N The sequence a(n,m) of the m polynomial coefficients of the n-th order B-spline scaled by n!, read by rows, with n in {0,1,2,...} and m in {1,2,3,...,(n+1)^2}.

%C The n-th order B-spline N_n(x) may be calculated with the expression

%C N_n(x) = (1/n!) Sum_{k=0..n+1} (-1)^k binomial(n+1,k) (x-k)^n step(x-k),

%C where

%C * n! is n factorial, which is defined as n! = n(n-1)(n-2)...(1),

%C * binomial(n,k) is the binomial coefficient, which can be defined as

%C binomial(n,k) = n!/((n-k)!k!),

%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!*N_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 Ole Christensen, Frames and bases: An Introductory Course, 2008, isbn13:9780817646776, page 142, Theorem 6.1.3.

%D Charles K. Chui, An Introduction to Wavelets, 1992, isbn13: 9780121745844, page 84, equation (4.1.12).

%D Daniel J. Greenhoe, Wavelet Structure and Design, 2013, isbn13: 9780983801139, page 318, Theorem H.1.

%H Daniel J. Greenhoe, <a href="/A289358/b289358.txt">Values for orders n=0..8.</a>

%H Daniel J. Greenhoe, <a href="/A289358/a289358.pdf">Technical report</a> for this sequence.

%H Daniel J. Greenhoe, <a href="/A289358/a289358.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 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.

%F The n-th order B-spline N_n(x) may be calculated with the expression

%F N_n(x) = (1/n!) Sum_{k=0..n+1} (-1)^k binomial(n+1,k) (x-k)^n step(x-k).

%e The m=(n+1)^2 coefficients for the n-th order B-spline N_n(x) begin at the sequence index value p=Sum_{k=0..n}k^2=(1/6)n(n+1)(2n+1) and end at index value p+(n+1)^2-1.

%e Each set of m=(n+1)^2 coefficients for n=0,1,2,... can be written in the form of an (n+1)X(n+1) matrix A_n as...

%e for n=0 (index values 0 to 0):

%e A_0 = [1]

%e for n=1 (index values 1 to 4):

%e A_1 = [ 1 0]

%e [-1 2]

%e for n=2 (index values 5 to 13):

%e [ 1 0 0 ]

%e A_2 = [-2 6 -3 ]

%e [ 1 -6 9 ]

%e for n=3 (index values 14 to 29):

%e [ 1 0 0 0]

%e A_3 = [ -3 12 -12 4]

%e [ 3 -24 60 -44]

%e [ -1 12 -48 64]

%e That is, the sequence of integers induces a sequence of (n+1)X(n+1) square matrices (A_0, A_1, A_2, ...).

%e Taking the specific case of n=3, for example, the coefficients for N_3(x) begin at index value p=0+1+4+9=14 and end at index value p+4^2-1=29.

%e Using the coefficients from this range of indices yields the following expression for N_3(x):

%e [ 1 0 0 0 : for 0 <= x < 1] [x^3]

%e 3!N_3(x) = [-3 12 -12 4 : for 1 <= x < 2] [x^2]

%e [ 3 -24 60 -44 : for 2 <= x < 3] [ x ]

%e [-1 12 -48 64 : for 3 <= x < 4] [ 1 ]

%e [ 0 0 0 0 : otherwise ]

%e { x^3 :for 0 <= x < 1

%e {-3x^3 +12x^2 -12x + 4 :for 1 <= x < 2

%e = { 3x^3 -24x^2 +60x -44 :for 2 <= x < 3

%e {- x^3 +12x^2 -48x +64 :for 3 <= x < 4

%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 stated above), see Appendix B of the Technical Report link.

%o (Maxima)

%o n:2;

%o Nnx:(1/n!)*sum((-1)^k*binomial(n+1,k)*(x-k)^n*unit_step(x-k),k,0,n+1);

%o assume(x<=0); print(n!,"N(x)= ",expand(n!*Nnx)," for x<=0"); forget(x<=0);

%o for i:0 thru n step 1 do(

%o assume(x>i,x<(i+1)),

%o print(n!,"N(x)= ",expand(n!*Nnx)," for ",i,"<x<",i+1), forget(x>i,x<(i+1))

%o );

%o assume(x>(n+1)); print(n!,"N(x)= ",expand(n!*Nnx)," for x>",n+1); forget(x>(n+1));

%Y Cf. A276321.

%K sign,tabf

%O 0,5

%A _Daniel J. Greenhoe_, Jul 03 2017