A113953 -id:A113953 - cp's OEIS frontend

A114188 Riordan array (1/(1-x),x(1+x)/(1-x)^2).

1, 1, 1, 1, 4, 1, 1, 9, 7, 1, 1, 16, 26, 10, 1, 1, 25, 70, 52, 13, 1, 1, 36, 155, 190, 87, 16, 1, 1, 49, 301, 553, 403, 131, 19, 1, 1, 64, 532, 1372, 1462, 736, 184, 22, 1, 1, 81, 876, 3024, 4446, 3206, 1216, 246, 25, 1, 1, 100, 1365, 6084, 11826, 11584, 6190, 1870, 317

Offset: 0

Paul Barry, Nov 16 2005

Product of A007318 and A113953, that is, (1/(1-x),x/(1-x))*(1,x(1+2x)).
Row sums are A025192. Diagonal sums are A052980.
Inverse is A114189. A signed version is A110511.

			Triangle begins
1;
1, 1;
1, 4, 1;
1, 9, 7, 1;
1, 16, 26, 10, 1;
1, 25, 70, 52, 13, 1;
1, 36,155,190, 87, 16, 1;

P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.

Cf. A007318, A025192, A052980, A110511, A113953, A114189.

T(n, k) = Sum_{j=0..n} C(n, j)*C(k, j-k)2^(j-k).
T(n, k) = Sum_{j=0..n-k} C(k, j)*C(n+k-j, 2k).
T(n,k) = 2*T(n-1,k)+T(n-1,k-1)-T(n-2,k)+T(n-2,k-1), T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 11 2014
G.f.: 1/(1-y-x*(1+y)/(1-y)). - Vladimir Kruchinin, Apr 21 2015

A329918 Coefficients of orthogonal polynomials related to the Jacobsthal numbers A152046, triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 4, 0, 1, 0, 4, 0, 6, 0, 1, 0, 0, 12, 0, 8, 0, 1, 0, 8, 0, 24, 0, 10, 0, 1, 0, 0, 32, 0, 40, 0, 12, 0, 1, 0, 16, 0, 80, 0, 60, 0, 14, 0, 1, 0, 0, 80, 0, 160, 0, 84, 0, 16, 0, 1, 0, 32, 0, 240, 0, 280, 0, 112, 0, 18, 0, 1

Offset: 0

Peter Luschny, Nov 28 2019

			Triangle starts:
  [0] 1;
  [1] 0,  1;
  [2] 0,  0,  1;
  [3] 0,  2,  0,  1;
  [4] 0,  0,  4,  0,  1;
  [5] 0,  4,  0,  6,  0,  1;
  [6] 0,  0, 12,  0,  8,  0,  1;
  [7] 0,  8,  0, 24,  0, 10,  0,  1;
  [8] 0,  0, 32,  0, 40,  0, 12,  0, 1;
  [9] 0, 16,  0, 80,  0, 60,  0, 14, 0, 1;
The first few polynomials:
  p(0,x) = 1;
  p(1,x) = x;
  p(2,x) = x^2;
  p(3,x) = 2*x + x^3;
  p(4,x) = 4*x^2 + x^4;
  p(5,x) = 4*x + 6*x^3 + x^5;
  p(6,x) = 12*x^2 + 8*x^4 + x^6;

Row sums are A001045 starting with 1, which is A152046. These are in signed form also the alternating row sums. Diagonal sums are aerated A133494.

Cf. A110509, A113953, A114192, A167431, A322942.

using Nemo # Returns row n.
function A329918(row)
    R, x = PolynomialRing(ZZ, "x")
    function p(n)
        n < 3 && return x^n
        x*p(n-1) + 2*p(n-2)
    end
    p = p(row)
    [coeff(p, k) for k in 0:row]
end
for row in 0:9 println(A329918(row)) end # prints triangle

T := (n, k) -> `if`((n+k)::odd, 0, 2^((n-k)/2)*binomial((n+k)/2-1, (n-k)/2)):
seq(seq(T(n, k), k=0..n), n=0..11);

p(n) = x*p(n-1) + 2*p(n-2) for n >= 3; p(0) = 1, p(1) = x, p(2) = x^2.
T(n, k) = [x^k] p(n).
T(n, k) = 2^((n-k)/2)*binomial((n+k)/2-1, (n-k)/2) if n+k is even otherwise 0.

A099092 Riordan array (1,2+4x).

Original entry on oeis.org

1, 0, 2, 0, 4, 4, 0, 0, 16, 8, 0, 0, 16, 48, 16, 0, 0, 0, 96, 128, 32, 0, 0, 0, 64, 384, 320, 64, 0, 0, 0, 0, 512, 1280, 768, 128, 0, 0, 0, 0, 256, 2560, 3840, 1792, 256, 0, 0, 0, 0, 0, 2560, 10240, 10752, 4096, 512, 0, 0, 0, 0, 0, 1024, 15360, 35840, 28672, 9216, 1024, 0, 0, 0

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A063727. Diagonal sums are A052907.

The Riordan array (1, s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).

			Rows begin
  {1},
  {0,  2},
  {0,  4,  4},
  {0,  0, 16,  8},
  {0,  0, 16, 48, 16}, ...

Cf. A026729, A038210, A052907, A063727, A113953.

Number triangle T(n,k) = binomial(k, n-k)*2^n; columns have g.f. (2x+4x^2)^k.

T(n,k) = A113953(n,k)*2^k = A026729(n,k)*2^n. - Philippe Deléham, Dec 11 2008

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A114188 Riordan array (1/(1-x),x(1+x)/(1-x)^2).

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A329918 Coefficients of orthogonal polynomials related to the Jacobsthal numbers A152046, triangle read by rows, T(n, k) for 0 <= k <= n.

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A099092 Riordan array (1,2+4x).

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