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A104562
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Inverse of the Motzkin triangle A064189.
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13
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1, -1, 1, 0, -2, 1, 1, 1, -3, 1, -1, 2, 3, -4, 1, 0, -4, 2, 6, -5, 1, 1, 2, -9, 0, 10, -6, 1, -1, 3, 9, -15, -5, 15, -7, 1, 0, -6, 3, 24, -20, -14, 21, -8, 1, 1, 3, -18, -6, 49, -21, -28, 28, -9, 1, -1, 4, 18, -36, -35, 84, -14, -48, 36, -10, 1, 0, -8, 4, 60, -50, -98, 126, 6, -75, 45, -11, 1, 1, 4, -30, -20, 145, -36, -210, 168, 45, -110, 55, -12, 1
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OFFSET
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0,5
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COMMENTS
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Or, triangle read by rows: T(0, 0) = 1; for n >= 1 T(n, k) is the coefficient of x^k in the monic characteristic polynomial of the n X n tridiagonal matrix with 1's on the main, sub- and superdiagonal (0 <= k <= n). The characteristic polynomial has a root 1 + 2*cos(Pi/(n + 1)). - Gary W. Adamson, Nov 19 2006
Row sums have g.f. 1/(1 + x^2); diagonal sums are (-1)^n. Riordan array (1/(1 + x + x^2), x/(1 + x + x^2)).
Or, triangle read by rows in which row n gives coefficients of characteristic polynomial of the n X n tridiagonal matrix with 1's on the main diagonal and -1's on the two adjacent diagonals. For example: M(3) = {{1, -1, 0}, {-1, 1, -1}, {0, -1, 1}}. - Roger L. Bagula, Mar 15 2008
Subtriangle of the triangle given by [0,-1,1,-1,0,0,0,0,0,0,0,...) DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 27 2010
Triangle of coefficients of Chebyshev's S(n, x-1) polynomials (exponents of x in increasing order). - Philippe Deléham, Feb 19 2012
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REFERENCES
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Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.
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LINKS
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FORMULA
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T(n, k) = Sum_{j=0..n} (-1)^(k-j)*(-1)^((n-j)/2) C((n+j)/2, j)(1+(-1)^(n+j))C(j, k)/2.
T(n, k) = (-1)^(n - k)*C(n, k)*hypergeom([(k - n)/2, (k - n + 1)/2], [-n], 4)) for n >= 1. - Peter Luschny, Apr 25 2016
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EXAMPLE
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Triangle starts:
[0] 1;
[1] -1, 1;
[2] 0, -2, 1;
[3] 1, 1, -3, 1;
[4] -1, 2, 3, -4, 1;
[5] 0, -4, 2, 6, -5, 1;
[6] 1, 2, -9, 0, 10, -6, 1;
[7] -1, 3, 9, -15, -5, 15, -7, 1;
[8] 0, -6, 3, 24, -20, -14, 21, -8, 1;
[9] 1, 3, -18, -6, 49, -21, -28, 28, -9, 1.
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Triangle [0,-1,1,-1,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] begins:
1;
0, 1;
0, -1, 1;
0, 0, -2, 1;
0, 1, 1, -3, 1;
0, -1, 2, 3, -4, 1;
... (End)
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MAPLE
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with(linalg): m:=proc(i, j) if abs(i-j)<=1 then 1 else 0 fi end: T:=(n, k)->coeff(charpoly(matrix(n, n, m), x), x, k): 1; for n from 1 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
# Alternatively:
T := (n, k) -> `if`(n=0, 1, (-1)^(n-k)*binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], 4)): seq(seq(simplify(T(n, k)), k=0..n), n=0..10); # Peter Luschny, Apr 25 2016
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MATHEMATICA
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nmax = 12;
M[n_, k_] := Binomial[n, k] Hypergeometric2F1[(k-n)/2, (k-n+1)/2, k+2, 4];
invM = Inverse@Table[M[n, k], {n, 0, nmax}, {k, 0, nmax}];
T[n_, k_] := invM[[n+1, k+1]];
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PROG
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(Sage)
@CachedFunction
if n< 0: return 0
if n==0: return 1 if k == 0 else 0
(Sage) # Alternatively as coefficients of polynomials:
def S(n, x):
if n==0: return 1
if n==1: return x-1
return (x-1)*S(n-1, x)-S(n-2, x)
for n in (0..7): print(S(n, x).list()) # Peter Luschny, Jun 23 2015
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CROSSREFS
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Apart from signs identical to A101950.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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