A131252 -id:A131252 - cp's OEIS frontend

A131253 Row sums of triangle A131252.

1, 3, 8, 17, 34, 64, 117, 209, 368, 641, 1108, 1904, 3257, 5551, 9432, 15985, 27030, 45616, 76845, 129245, 217056, 364033, 609768, 1020192, 1705009, 2846619, 4748072, 7912529, 13174858, 21919456, 36440613, 60538409, 100503632, 166744961, 276476092, 458151440

Offset: 0

Gary W. Adamson, Jun 23 2007

nonn

			a(3) = 17 = sum of row 3 terms of A131252: (7 + 6 + 3 + 1).

Andrew Howroyd, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4, -4, -2, 4, 0, -1)

Row sums of A131252.

I:=[1,3,8,17,34,64]; [n le 6 select I[n] else 4*Self(n-1)- 4*Self(n-2)-2*Self(n-3)+4*Self(n-4)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Aug 10 2018

LinearRecurrence[{4, -4, -2, 4, 0, -1}, {1, 3, 8, 17, 34, 64}, 40] (* Vincenzo Librandi, Aug 10 2018 *)

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

a(n)={sum(k=0, n, (k+1)*sum(i=0, k, binomial(n-k, k-i)))} \\ Andrew Howroyd, Aug 09 2018

From Andrew Howroyd, Aug 09 2018: (Start)
a(n) = Sum_{k=0..n} (k+1)*(Sum_{i=0..k} binomial(n-k, k-i)).
a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) - a(n-6).
G.f.: (1 - x - x^3)/((1 - x)^2*(1 - x - x^2)^2).
(End)

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

A131251 A000012 * A052509.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 3, 1, 5, 10, 7, 3, 1, 6, 15, 14, 7, 3, 1, 7, 21, 25, 15, 7, 3, 1, 8, 28, 41, 30, 15, 7, 3, 1, 9, 36, 63, 56, 31, 15, 7, 3, 1, 10, 45, 92, 98, 62, 31, 15, 7, 3, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A001924: (1, 3, 7, 14, 26, 46, 79, ...). A131252 = A052509 * A000012.
From Clark Kimberling, Feb 07 2011: (Start)
When formatted as a rectangle R with northwest corner
  1, 2, 3,  4,  5,  6, ...
  1, 3, 6, 10, 15, 21, ...
  1, 3, 7, 14, 25, 41, ...
  1, 3, 7, 15, 30, 56, ...
  1, 3, 7, 15, 31, 62, ...
  ...
the following properties hold:
R is the accumulation array of the transpose of A052553 (a version of Pascal's triangle); see A144112 for the definition of accumulation array.
row 1: A000027
row 2: A000217
row 3: A004006
row 4: A055795
row 5: A057703
row 6: A115567
limiting row:  A000225
antidiagonal sums: A001924.
(End)

			First few rows of the triangle:
  1;
  2,  1;
  3,  3,  1;
  4,  6,  3,  1;
  5, 10,  7,  3,  1;
  6, 15, 14,  7,  3,  1;
  7, 21, 25, 15,  7,  3,  1;
  ...

Cf. A052509, A000012, A001924, A131252.

A000012 * A052509 as infinite lower triangular matrices.

cp's OEIS Frontend

A131253 Row sums of triangle A131252.

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