A077948 -id:A077948 - cp's OEIS frontend

A098593 A triangle of Krawtchouk coefficients.

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

Paul Barry, Sep 17 2004

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

			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)

P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum walks, Contemporary Mathematics, 287 2001, pp. 83-96.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
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.

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).

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 *)

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

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

A077971 Expansion of 1/(1+x-x^2-2*x^3).

Original entry on oeis.org

1, -1, 2, -1, 1, 2, -3, 7, -6, 7, 1, -6, 21, -25, 34, -17, 1, 50, -83, 135, -118, 87, 65, -214, 453, -537, 562, -193, -319, 1250, -1955, 2567, -2022, 679, 2433, -5798, 9589, -10521, 8514, 143, -12671, 29842, -42227, 46727, -29270, -8457, 72641, -139638, 195365, -189721, 105810, 95199, -368831

Offset: 0

N. J. A. Sloane, Nov 17 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,1,2).

Cf. A077948.

First differences of A077901.

a:=[1,-1,2];; for n in [4..60] do a[n]:=-a[n-1]+a[n-2]+2*a[n-3]; od; a; # G. C. Greubel, Jun 24 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/(1+x-x^2-2*x^3) )); // G. C. Greubel, Jun 24 2019

LinearRecurrence[{-1,1,2}, {1,-1,2}, 60] (* or *) CoefficientList[Series[ 1/(1 +x-x^2-2*x^3), {x,0,60}], x] (* G. C. Greubel, Jun 24 2019 *)

Vec(1/(1+x-x^2-2*x^3)+O(x^60)) \\ Charles R Greathouse IV, Sep 26 2012

(1/(1+x-x^2-2*x^3)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Jun 24 2019

a(n) = (-1)^n * A077948(n). - G. C. Greubel, Jun 24 2019

A078011 Expansion of (1-x)/(1-x-x^2+2*x^3).

Original entry on oeis.org

1, 0, 1, -1, 0, -3, -1, -4, 1, -1, 8, 5, 15, 4, 9, -17, -16, -51, -33, -52, 17, 31, 152, 149, 239, 84, 25, -369, -512, -931, -705, -612, 545, 1343, 3112, 3365, 3791, 932, -2007, -8657, -12528, -17171, -12385, -4500, 17457, 37727, 64184, 66997, 55727, -5644, -83911, -201009, -273632, -306819

Offset: 0

N. J. A. Sloane, Nov 17 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-2).

Cf. A077948.

a:=[1,0,1];; for n in [4..60] do a[n]:=a[n-1]+a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Jun 28 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x)/(1-x-x^2+2*x^3) )); // G. C. Greubel, Jun 28 2019

CoefficientList[Series[(1-x)/(1-x-x^2+2x^3),{x,0,60}],x] (* or *) LinearRecurrence[ {1,1,-2},{1,0,1},60] (* Harvey P. Dale, May 04 2013 *)

my(x='x+O('x^60)); Vec((1-x)/(1-x-x^2+2*x^3)) \\ G. C. Greubel, Jun 28 2019

((1-x)/(1-x-x^2+2*x^3)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Jun 28 2019

a(0)=1, a(1)=0, a(2)=1, a(n) = a(n-1) + a(n-2) - 2*a(n-3). - Harvey P. Dale, May 04 2013

a(n) = A077948(n) - A077948(n-1). - R. J. Mathar, Nov 07 2015

A099038 Diagonal sums of a Krawtchouk triangle.

Original entry on oeis.org

1, 1, 0, 1, 5, 6, 3, 13, 42, 55, 55, 162, 413, 591, 810, 2001, 4451, 6900, 11091, 24795, 51030, 84337, 147253, 309666, 610695, 1058041, 1928646, 3903175, 7528741, 13480380, 25126093, 49640405, 94739568, 173440389, 326974495, 636424008

Offset: 0

Paul Barry, Sep 23 2004

Diagonal sums of A099037.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A077948.

Table[Sum[Binomial[n - k, k]*Sum[(-1)^i*Binomial[k, i]*Binomial[n - 2*k, k - i], {i, 0, n}], {k, 0, Floor[n/2]}], {n,0,50}] (* G. C. Greubel, Dec 31 2017 *)

for(n=0,30, print1(sum(k=0,floor(n/2), binomial(n-k,k)*sum(i=0,n,(-1)^i*binomial(k,i)*binomial(n-2*k,k-i))), ", ")) \\ G. C. Greubel, Dec 31 2017

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*Sum_{i=0..n} (-1)^i*C(k, i) * C(n-2k, k-i).

Conjecture: n*a(n) -n*a(n-1) +n*a(n-2) +3*(-n+1)*a(n-3) +(-5*n+13)*a(n-4) +(n-3)*a(n-5)=0. - R. J. Mathar, Dec 21 2014

A117355 Riordan array (1/(1-x^2),x(1-2x^2)/(1-x^2)).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, -1, 0, 1, 0, -1, 0, -2, 0, 1, 1, 0, -2, 0, -3, 0, 1, 0, -2, 0, -2, 0, -4, 0, 1, 1, 0, -2, 0, -1, 0, -5, 0, 1, 0, -3, 0, 0, 0, 1, 0, -6, 0, 1, 1, 0, -1, 0, 3, 0, 4, 0, -7, 0, 1, 0, -4, 0, 3, 0, 6, 0, 8, 0, -8, 0, 1, 1, 0, 1, 0, 6, 0, 8, 0, 13, 0, -9, 0, 1

Offset: 0

Paul Barry, Mar 09 2006

Row sums are A077948.

Diagonal sums are the aerated version of A009545 with g.f. 1/(1-2x^2+2x^4).

			Triangle begins
1,
0, 1,
1, 0, 1,
0, 0, 0, 1,
1, 0, -1, 0, 1,
0, -1, 0, -2, 0, 1,
1, 0, -2, 0, -3, 0, 1,
0, -2, 0, -2, 0, -4, 0, 1

Number triangle T(n,k)=sum{j=0..n-k, C(j-(n-k)/2-1,j)C(k,j)(1+(-1)^(n-k))/2}

T(n,k)=T(n-1,k-1)+T(n-2,k)-2*T(n-3,k-1), T(0,0)=T(1,1)=T(2,0)=T(2,2)=1, T(1,1)=T(2,1)=0, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 16 2014

cp's OEIS Frontend

A098593 A triangle of Krawtchouk coefficients.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A077971 Expansion of 1/(1+x-x^2-2*x^3).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A078011 Expansion of (1-x)/(1-x-x^2+2*x^3).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A099038 Diagonal sums of a Krawtchouk triangle.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A117355 Riordan array (1/(1-x^2),x(1-2x^2)/(1-x^2)).

Views

Author

Keywords

Comments

Examples

Formula