A131298 -id:A131298 - cp's OEIS frontend

A104730 Triangle read by rows: T(n,k)=C(n+1,k)-C(k,n-k+1).

1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 5, 10, 7, 1, 1, 6, 15, 19, 9, 1, 1, 7, 21, 35, 31, 11, 1, 1, 8, 28, 56, 69, 46, 13, 1, 1, 9, 36, 84, 126, 121, 64, 15, 1, 1, 10, 45, 120, 210, 251, 195, 85, 17, 1, 1, 11, 55, 165, 330, 462, 456, 295, 109, 19, 1, 1, 12, 66

Offset: 1

Gary W. Adamson, Mar 20 2005

Row sums are A027934: 1, 2, 5, 11, 24, 51, 107... Diagonal sums are A131298.

			The first few rows of the triangle are:
1;
1, 1;
1, 3, 1;
1, 4, 5, 1;
1, 5, 10, 7, 1;
1, 6, 15, 19, 9, 1;
1, 7, 31, 35, 31, 11, 1;
...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A027934, A026729, A007318.

Table[Binomial[n+1,k]-Binomial[k,n-k+1],{n,0,20},{k,0,n}]//Flatten (* Harvey P. Dale, Jan 16 2024 *)

Perform the operation A - B; then extract the triangle after deleting all zeros. P = infinite lower triangular Pascal's triangle matrix (A007318); B = A026729, as an infinite lower triangular matrix: [1; 0, 1;, 0, 1, 1; 0, 0, 2, 1; 0, 0, 1, 3, 1;...].

Better definition from Paul Barry, Jun 26 2007

More terms from Harvey P. Dale, Jan 16 2024

A291311 Expansion of (1-x^2)/((1-x-x^2)*(1-x-x^4)).

Original entry on oeis.org

1, 2, 3, 5, 9, 16, 27, 45, 75, 125, 207, 341, 560, 918, 1502, 2453, 4000, 6515, 10601, 17235, 28000, 45461, 73773, 119665, 194033, 314519, 509685, 825768, 1337612, 2166360, 3508085, 5680122, 9196043, 14886981, 24097953, 39005540, 63131935, 102176733, 165362855, 267614381

Offset: 0

Roger L. Bagula, Aug 21 2017

Index entries for linear recurrences with constant coefficients, signature (2,0,-1,1,-1,-1).

Cf. A131298.

LinearRecurrence[{2,0,-1,1,-1,-1},{1,2,3,5,9,16},40] (* Harvey P. Dale, Apr 18 2018 *)

my(x='x+O('x^50)); Vec((1 - x^2)/((1-x-x^2)*(1-x-x^4))) \\ Michel Marcus, Jun 25 2023

G.f.: (1-x^2)/((1-x-x^2)*(1-x-x^4)).

cp's OEIS Frontend

A104730 Triangle read by rows: T(n,k)=C(n+1,k)-C(k,n-k+1).

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