A098593 -id:A098593 - cp's OEIS frontend

A143683 Pascal-(1,8,1) array.

1, 1, 1, 1, 10, 1, 1, 19, 19, 1, 1, 28, 118, 28, 1, 1, 37, 298, 298, 37, 1, 1, 46, 559, 1540, 559, 46, 1, 1, 55, 901, 4483, 4483, 901, 55, 1, 1, 64, 1324, 9856, 21286, 9856, 1324, 64, 1, 1, 73, 1828, 18388, 67006, 67006, 18388, 1828, 73, 1, 1, 82, 2413, 30808, 164242, 304300, 164242, 30808, 2413, 82, 1

Offset: 0

Paul Barry, Aug 28 2008

			Square array begins as:
  1,  1,    1,     1,      1,       1,        1, ... A000012;
  1, 10,   19,    28,     37,      46,       55, ... A017173;
  1, 19,  118,   298,    559,     901,     1324, ...
  1, 28,  298,  1540,   4483,    9856,    18388, ...
  1, 37,  559,  4483,  21286,   67006,   164242, ...
  1, 46,  901,  9856,  67006,  304300,  1004590, ...
  1, 55, 1324, 18388, 164242, 1004590,  4443580, ...
Antidiagonal triangle begins as:
  1;
  1,  1;
  1, 10,   1;
  1, 19,  19,    1;
  1, 28, 118,   28,    1;
  1, 37, 298,  298,   37,   1;
  1, 46, 559, 1540,  559,  46,  1;
  1, 55, 901, 4483, 4483, 901, 55, 1;

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

Cf.Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), 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).

Cf. A003683, A017173, A143680.

a143683 n k = a143683_tabl !! n !! k
a143683_row n = a143683_tabl !! n
a143683_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) (map (* 8) ([0] ++ us ++ [0])) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
-- Reinhard Zumkeller, Mar 16 2014

A143683:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A143683(n,k,8): k in [0..n], n in [0..12]]; // G. C. Greubel, May 27 2021

Table[Hypergeometric2F1[-k, k-n, 1, 9], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 9).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2021

Square array: T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 8*T(n-1, k-1) + T(n-1, k).
Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*9^j.
Rows are the expansions of (1+8*x)^k/(1-x)^(k+1).
Riordan array (1/(1-x), x*(1+8*x)/(1-x)).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 9). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(9*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 18*x + 81*x^2/2) = 1 + 19*x + 118*x^2/2! + 298*x^3/3! + 559*x^4/4! + 901*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n,k) = A003683(n+1). - G. C. Greubel, May 27 2021

A307090 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2j) binomial(n-k,2*j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, -2, -2, 1, 1, 1, 1, -5, -8, -5, 1, 1, 1, 1, -9, -17, -17, -9, 1, 1, 1, 1, -14, -29, -34, -29, -14, 1, 1, 1, 1, -20, -44, -54, -54, -44, -20, 1, 1, 1, 1, -27, -62, -74, -74, -74, -62, -27, 1, 1, 1, 1, -35, -83, -90, -74, -74, -90, -83, -35, 1, 1

Offset: 0

Seiichi Manyama, Mar 24 2019

			Triangle begins:
n\k | 0  1    2    3    4    5    6  7  8
----+-------------------------------------
0   | 1;
1   | 1, 1;
2   | 1, 1,   1;
3   | 1, 1,   1,   1;
4   | 1, 1,   0,   1,   1;
5   | 1, 1,  -2,  -2,   1,   1;
6   | 1, 1,  -5,  -8,  -5,   1,   1;
7   | 1, 1,  -9, -17, -17,  -9,   1, 1;
8   | 1, 1, -14, -29, -34, -29, -14, 1, 1;

Seiichi Manyama, Rows n = 0..139, flattened

Row sums give A099587(n+1).
T(2*n,n) gives A307091.
Cf. A007318, A098593, A119326, A119335.

T[n_, k_] := Sum[(-1)^j * Binomial[k, 2*j] * Binomial[n - k, 2*j], {j, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, May 20 2021 *)

A099037 Triangle of diagonals of symmetric Krawtchouk matrices.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 3, -3, -1, 1, 8, -12, 8, 1, 1, 15, -20, 20, -15, -1, 1, 24, -15, 0, -15, 24, 1, 1, 35, 21, -105, 105, -21, -35, -1, 1, 48, 112, -336, 420, -336, 112, 48, 1, 1, 63, 288, -672, 756, -756, 672, -288, -63, -1, 1, 80, 585, -960, 420, 0, 420, -960, 585, 80, 1, 1, 99, 1045, -825, -1980, 4620, -4620, 1980, 825, -1045, -99, -1

Offset: 0

Paul Barry, Sep 23 2004

Row sums have e.g.f. BesselI(0,2*x) (A000984 with interpolated zeros).

Diagonal sums are A099038.

			Triangle begins as:
1.
1, -1.
1,  0,  1.
1,  3, -3,  1.
1,  8, -12, 8, 1. ...

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 100 rows, flattened
P. Feinsilver and R. Fitzgerald, The Spectrum of Symmetric Krawtchouk Matrices, Linear Algebra and Its Applications, Vol. 235 (1996), pp. 121-139.

Cf. A000984, A099038, A098593,

T[n_, k_]:= If[k <= n, Binomial[n, k]*Sum[(-1)^j*Binomial[k, j]*Binomial[n - k, k - j], {j, 0, n}], 0]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 31 2017 *)

{T(n, k) = binomial(n, k)*sum(j=0,n, (-1)^j*binomial(k, j)*binomial(n-k, k-j))};
for(n=0,20, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 31 2017

Triangle T(n, k) = if(k<=n, C(n, k)*Sum_{i=0..n} (-1)^i*C(k, i)C(n-k, k-i), 0).

Triangle T(n, k) = Sum_{j=0..n} (-1)^(n-j)*C(n,j)*C(j,k)*C(k,j-k) = C(n,k)*A098593(n,k).

A143685 Pascal-(1,9,1) array.

Original entry on oeis.org

1, 1, 1, 1, 11, 1, 1, 21, 21, 1, 1, 31, 141, 31, 1, 1, 41, 361, 361, 41, 1, 1, 51, 681, 1991, 681, 51, 1, 1, 61, 1101, 5921, 5921, 1101, 61, 1, 1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1, 1, 81, 2241, 24681, 96201, 96201, 24681, 2241, 81, 1, 1, 91, 2961, 41511, 239241, 460251, 239241, 41511, 2961, 91, 1

Offset: 0

Paul Barry, Aug 28 2008

			Square array begins as:
  1,  1,    1,     1,      1,       1,        1, ... A000012;
  1, 11,   21,    31,     41,      51,       61, ... A017281;
  1, 21,  141,   361,    681,    1101,     1621, ...
  1, 31,  361,  1991,   5921,   13151,    24681, ...
  1, 41,  681,  5921,  29761,   96201,   239241, ...
  1, 51, 1101, 13151,  96201,  460251,  1565301, ...
  1, 61, 1621, 24681, 239241, 1565301,  7272861, ...
Antidiagonal triangle begins as:
  1;
  1,  1;
  1, 11,    1;
  1, 21,   21,     1;
  1, 31,  141,    31,     1;
  1, 41,  361,   361,    41,     1;
  1, 51,  681,  1991,   681,    51,    1;
  1, 61, 1101,  5921,  5921,  1101,   61,  1;
  1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1;

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Cf. A002534, A143680, A143682.

Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), 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), this sequence (m = 9).

A143685:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A143685(n,k,9): k in [0..n], n in [0..12]]; // G. C. Greubel, May 29 2021

Table[Hypergeometric2F1[-k, k-n, 1, 10], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 10).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 29 2021

Square array: T(n, k) = T(n, k-1) + 9*T(n-1, k-1) + T(n-1, k) with T(n, 0) = T(0, k) = 1.
Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*10^j.
Riordan array (1/(1-x), x*(1+9*x)/(1-x)).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 10). - Jean-François Alcover, May 24 2013
Sum_{k=0..n} T(n, k) = A002534(n+1). - G. C. Greubel, May 29 2021

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A143683 Pascal-(1,8,1) array.

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A307090 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2*j) * binomial(n-k,2*j).

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A099037 Triangle of diagonals of symmetric Krawtchouk matrices.

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A143685 Pascal-(1,9,1) array.

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Magma

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A307090 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2j) binomial(n-k,2*j).