A224838 Triangle of falling diagonals of A011973 (with rows displayed as centered text).

1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 1, 4, 6, 5, 1, 1, 10, 10, 6, 1, 1, 5, 20, 15, 7, 1, 6, 15, 35, 21, 8, 1, 1, 21, 35, 56, 28, 9, 1, 1, 7, 56, 70, 84, 36, 10, 1, 8, 28, 126, 126, 120, 45, 11, 1, 1, 36, 84, 252, 210, 165, 55, 12, 1, 1, 9, 120, 210, 462, 330, 220, 66, 13, 1

Offset

1,4

Comments

Row sums are A005314 with offset = 1.

Links

Table of n, a(n) for n=1..75.

Formula

r(n) = binomial(n-floor((4n+15-6k+(-1)^k)/12), n-floor((4n+15-6k+(-1)^k)/12)-floor((2n-1)/3)+k-1), {k,1,floor((2n+2)/3)}.

R(n) = binomial(n-Floor((k+1)/2), n-Floor((3k-1)/2)), {k,1,floor((2n+2)/3)} - gives the terms of each row in reverse order.

Example

First 11 triangle rows are :
1
1,  1
2,  1
1,  3,  1
1,  3,  4,  1
4,  6,  5,  1
1, 10, 10,  6,  1
1,  5, 20, 15,  7,  1
6, 15, 35, 21,  8,  1
1, 21, 35, 56, 28,  9,  1
1,  7, 56, 70, 84, 36, 10,  1

Mathematica

Table[Reverse[Table[Binomial[n - Floor[(k + 1)/2], n - Floor[(3 k - 1)/2]], {k, Floor[(2 n + 2)/3]}]], {n, 13}] (* T. D. Noe, Jul 25 2013 *)
Column[Table[Binomial[n - Floor[(4 n + 15 - 6 k + (-1)^k)/12], n - Floor[(4 n + 15 - 6 k + (-1)^k)/12] - Floor[(2 n - 1)/3] + k - 1], {n, 1, 25}, {k, 1, Floor[(2 n + 2)/3]}]] (* John Molokach, Jul 25 2013 *)

Crossrefs

Cf. A005314, A227300, A001973, A000045, A004396.

Sequence in context: A181846 A305499 A210873 * A030272 A157128 A359899

Adjacent sequences: A224835 A224836 A224837 * A224839 A224840 A224841

Keyword

nonn,tabf

Author

John Molokach, Jul 21 2013

Status

approved