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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 =  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, "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.)