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A098593
A triangle of Krawtchouk coefficients.
15
1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -2, -2, 1, 1, -3, -2, -2, -3, 1, 1, -4, -1, 0, -1, -4, 1, 1, -5, 1, 3, 3, 1, -5, 1, 1, -6, 4, 6, 6, 6, 4, -6, 1, 1, -7, 8, 8, 6, 6, 8, 8, -7, 1, 1, -8, 13, 8, 2, 0, 2, 8, 13, -8, 1, 1, -9, 19, 5, -6, -10, -10, -6, 5, 19, -9, 1, 1, -10, 26, -2, -17, -20, -20, -20, -17, -2, 26, -10, 1, 1, -11, 34, -14, -29, -25
OFFSET
0,12
COMMENTS
Row sums are A009545(n+1), with e.g.f. exp(x)(cos(x)+sin(x)). Diagonal sums are A077948.
The rows are the diagonals of the Krawtchouk matrices. Coincides with the Riordan array (1/(1-x),(1-2x)/(1-x)). - Paul Barry, Sep 24 2004
Corresponds to Pascal-(1,-2,1) array, read by antidiagonals. The Pascal-(1,-2,1) array has n-th row generated by (1-2x)^n/(1-x)^(n+1). - Paul Barry, Sep 24 2004
A modified version (different signs) of this triangle is given by T(n,k) = Sum_{j=0..n} C(n-k,j)*C(k,j)*cos(Pi*(k-j)). - Paul Barry, Jun 14 2007
REFERENCES
P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum walks, Contemporary Mathematics, 287 2001, pp. 83-96.
LINKS
Paul Barry, A note on Krawtchouk Polynomials and Riordan Arrays, JIS 11 (2008) 08.2.2
P. Feinsilver and J. Kocik, Krawtchouk polynomials and Krawtchouk matrices, arxiv:quant-ph/0702073, 2007.
FORMULA
T(n, k) = Sum_{i=0..k} binomial(n-k, k-i)*binomial(k, i)*(-1)^(k-i), k<=n.
T(n, k) = T(n-1, k) + T(n-1, k-1) - 2*T(n-2, k-1) (n>0). - Paul Barry, Sep 24 2004
T(n, k) = [k<=n]*Hypergeometric2F1(-k,k-n;1;-1). - Paul Barry, Jan 24 2011
E.g.f. for the n-th subdiagonal: exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} (-1)^k*binomial(n,k)* x^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 - 2*x + x^2/2) = 1 - x - 2*x^2/2! - 2*x^3/3! - x^4/4! + x^5/5! + .... - Peter Bala, Mar 05 2017
EXAMPLE
Rows begin {1}, {1,1}, {1,0,1}, {1,-1,-1,1}, {1,-2,-2,-2,1}, ...
From Paul Barry, Oct 05 2010: (Start)
Triangle begins
1,
1, 1,
1, 0, 1,
1, -1, -1, 1,
1, -2, -2, -2, 1,
1, -3, -2, -2, -3, 1,
1, -4, -1, 0, -1, -4, 1,
1, -5, 1, 3, 3, 1, -5, 1,
1, -6, 4, 6, 6, 6, 4, -6, 1,
1, -7, 8, 8, 6, 6, 8, 8, -7, 1,
1, -8, 13, 8, 2, 0, 2, 8, 13, -8, 1
Production matrix (related to large Schroeder numbers A006318) begins
1, 1,
0, -1, 1,
0, -2, -1, 1,
0, -6, -2, -1, 1,
0, -22, -6, -2, -1, 1,
0, -90, -22, -6, -2, -1, 1,
0, -394, -90, -22, -6, -2, -1, 1,
0, -1806, -394, -90, -22, -6, -2, -1, 1,
0, -8558, -1806, -394, -90, -22, -6, -2, -1, 1
Production matrix of inverse is
-1, 1,
-2, 1, 1,
-4, 2, 1, 1,
-8, 4, 2, 1, 1,
-16, 8, 4, 2, 1, 1,
-32, 16, 8, 4, 2, 1, 1,
-64, 32, 16, 8, 4, 2, 1, 1,
-128, 64, 32, 16, 8, 4, 2, 1, 1,
-256, 128, 64, 32, 16, 8, 4, 2, 1, 1 (End)
MATHEMATICA
T[n_, k_] := Sum[Binomial[n - k, k - j]*Binomial[k, j]*(-1)^(k - j), {j, 0, n}]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 15 2017 *)
PROG
(PARI) for(n=0, 10, for(k=0, n, print1(sum(i=0, k, binomial(n-k, k-i) *binomial(k, i)*(-1)^(k-i)), ", "))) \\ G. C. Greubel, Oct 15 2017
CROSSREFS
Cf. Pascal (1,m,1) array: A123562 (m = -3), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).
Sequence in context: A245514 A104754 A206827 * A144431 A053821 A076545
KEYWORD
easy,sign,tabl
AUTHOR
Paul Barry, Sep 17 2004
STATUS
approved