A131403 -id:A131403 - cp's OEIS frontend

A131402 2*A007318 - (A046854 + A065941 - A000012).

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 7, 6, 1, 1, 7, 14, 14, 7, 1, 1, 9, 20, 33, 20, 9, 1, 1, 10, 31, 56, 56, 31, 10, 1, 1, 12, 40, 97, 111, 97, 40, 12, 1, 1, 13, 55, 142, 217, 217, 142, 55, 13, 1, 1, 15, 67, 213, 358, 463, 358, 213, 67, 15, 1, 1, 16, 86, 287, 590, 841, 841, 590, 287, 86, 16, 1

Offset: 0

Gary W. Adamson, Jul 07 2007

Row sums = A131403: (1, 2, 5, 10, 21, 44, 93, ...).

			First few rows of the triangle are:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  4,  1;
  1,  6,  7,  6,  1;
  1,  7, 14, 14,  7,  1;
  1,  9, 20, 33, 20,  9,  1;
  1, 10, 31, 56, 56, 31, 10,  1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Row sums are A131403.

Cf. A046854, A065941.

T(n,k) = if(k <= n, 2*binomial(n, k) + 1 - binomial((n + k)\2, k) - binomial(n-(k+1)\2, k\2), 0) \\ Andrew Howroyd, Aug 09 2018

2*A007318 - (A046854 + A065941 - A000012) as infinite lower triangular matrices.

T(n,k) = 2*binomial(n, k) + 1 - binomial(floor((n + k)/2), k) - binomial(n-floor((k+1)/2), floor(k/2)). - Andrew Howroyd, Aug 09 2018

Missing terms inserted and a(55) and beyond from Andrew Howroyd, Aug 09 2018

A131405 Row sums of triangle A131404.

Original entry on oeis.org

1, 2, 7, 16, 37, 82, 179, 384, 813, 1702, 3531, 7272, 14889, 30342, 61603, 124700, 251825, 507582, 1021535, 2053388, 4123481, 8274002, 16591767, 33254356, 66623317, 133432082, 267164239, 534814024, 1070413693, 2142098602, 4286254091, 8575836312, 17157057669

Offset: 0

Gary W. Adamson, Jul 07 2007

nonn

			a(4) = 37 = sum of row 4 terms of triangle A131404: (1 + 11 + 13 + 11 + 1).

Andrew Howroyd, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5, -8, 3, 3, -2)

Row sums of A131404.

Cf. A131403.

Vec((1 - 3*x + 5*x^2 - 6*x^3 + 4*x^4)/((1 - x)^2*(1 - 2*x)*(1 - x - x^2)) + O(x^40)) \\ Andrew Howroyd, Aug 09 2018

From Andrew Howroyd, Aug 09 2018: (Start)
a(n) = 5*a(n-1) - 8*a(n-2) + 3*a(n-3) + 3*a(n-4) - 2*a(n-5).
G.f.: (1 - 3*x + 5*x^2 - 6*x^3 + 4*x^4)/((1 - x)^2*(1 - 2*x)*(1 - x - x^2)).
(End)

Terms a(10) and beyond from Andrew Howroyd, Aug 09 2018

cp's OEIS Frontend

A131402 2*A007318 - (A046854 + A065941 - A000012).

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