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A289358 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}. 2
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, 12, -48, 64, 1, 0, 0, 0, 0, -4, 20, -30, 20, -5, 6, -60, 210, -300, 155, -4, 60, -330, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

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

where

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

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

     binomial(n,k) = n!/((n-k)!k!),

*  step(x) is the step function defined as step(x) = {1 for x >= 0

                                                     {0 otherwise.

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.

REFERENCES

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

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

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

LINKS

Daniel J. Greenhoe, Values for orders n=0..8.

Daniel J. Greenhoe, Technical report for this sequence.

Daniel J. Greenhoe, Maxima script supporting this sequence.

Daniel J. Greenhoe, B-splines and B-spline wavelets, Technical Report [version 0.20], July 2017.

Daniel J. Greenhoe, Wavelet Structure and Design, [version 1.20], January 2017, "Mathematical Structure and Design" series, volume 3, Theorem H.1, pages 267--268.

FORMULA

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

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

EXAMPLE

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.

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...

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

   A_0 = [1]

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

   A_1 = [ 1 0]

         [-1 2]

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

         [ 1  0  0 ]

   A_2 = [-2  6 -3 ]

         [ 1 -6  9 ]

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

         [  1   0   0   0]

   A_3 = [ -3  12 -12   4]

         [  3 -24  60 -44]

         [ -1  12 -48  64]

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

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.

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

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

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

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

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

             [ 0    0   0   0 : otherwise     ]

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

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

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

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

             {                    0 :otherwise

Note: Sum_{k=1..n}k^2 is called a "power sum".

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.

PROG

(Maxima)

n:2;

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

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

for i:0 thru n step 1 do(

  assume(x>i, x<(i+1)),

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

  );

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

CROSSREFS

Cf. A276321.

Sequence in context: A261630 A301503 A059431 * A271698 A113263 A063658

Adjacent sequences:  A289355 A289356 A289357 * A289359 A289360 A289361

KEYWORD

sign,tabf

AUTHOR

Daniel J. Greenhoe, Jul 03 2017

STATUS

approved

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Last modified July 28 17:15 EDT 2021. Contains 346335 sequences. (Running on oeis4.)