A184884 -id:A184884 - cp's OEIS frontend

A296320 T(n,k)=Number of nXk 0..1 arrays with each 1 horizontally, vertically or antidiagonally adjacent to 1 neighboring 1.

1, 2, 2, 3, 6, 3, 4, 11, 11, 4, 6, 27, 32, 27, 6, 9, 60, 96, 96, 60, 9, 13, 132, 295, 434, 295, 132, 13, 19, 301, 902, 1970, 1970, 902, 301, 19, 28, 669, 2747, 8470, 12547, 8470, 2747, 669, 28, 41, 1502, 8380, 37431, 77426, 77426, 37431, 8380, 1502, 41, 60, 3370, 25577

Offset: 1

R. H. Hardin, Dec 10 2017

Table starts
..1....2.....3......4........6.........9..........13...........19............28
..2....6....11.....27.......60.......132.........301..........669..........1502
..3...11....32.....96......295.......902........2747.........8380.........25577
..4...27....96....434.....1970......8470.......37431.......164807........723019
..6...60...295...1970....12547.....77426......490668......3078638......19343899
..9..132...902...8470....77426....676269.....6069953.....54182821.....482859661
.13..301..2747..37431...490668...6069953....78105580....994666167...12644605701
.19..669..8380.164807..3078638..54182821...994666167..18043360170..326902733082
.28.1502.25577.723019.19343899.482859661.12644605701.326902733082.8435284786616

			Some solutions for n=5 k=4
..0..0..0..0. .1..0..0..0. .0..0..0..0. .0..0..1..0. .0..0..0..0
..0..0..0..0. .1..0..0..0. .0..0..0..0. .0..0..1..0. .1..1..0..0
..1..0..0..0. .0..1..1..0. .0..0..1..0. .0..0..0..0. .0..0..1..0
..1..0..0..1. .0..0..0..1. .0..1..0..0. .0..1..0..1. .0..1..0..0
..0..0..0..1. .1..1..0..1. .0..0..0..0. .0..1..0..1. .0..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..544

Column 1 is A000930(n+1).

Column 2 is A184884(n+1).

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-3)
k=2: a(n) = a(n-1) +2*a(n-2) +2*a(n-3) -a(n-4) +a(n-5)
k=3: a(n) = a(n-1) +4*a(n-2) +6*a(n-3) +3*a(n-4) -3*a(n-6) +a(n-7) -3*a(n-9) +a(n-11)
k=4: [order 21]
k=5: [order 43]
k=6: [order 85]

A184883 Number triangle T(n,k) = [k<=n]*Hypergeometric2F1([-k, 2k-2n], [1], 2).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 9, 13, 1, 1, 13, 41, 25, 1, 1, 17, 85, 129, 41, 1, 1, 21, 145, 377, 321, 61, 1, 1, 25, 221, 833, 1289, 681, 85, 1, 1, 29, 313, 1561, 3649, 3653, 1289, 113, 1, 1, 33, 421, 2625, 8361, 13073, 8989, 2241, 145, 1, 1, 37, 545, 4089, 16641, 36365, 40081, 19825, 3649, 181, 1

Offset: 0

Paul Barry, Jan 24 2011

			Triangle begins
  1;
  1,  1;
  1,  5,   1;
  1,  9,  13,    1;
  1, 13,  41,   25,     1;
  1, 17,  85,  129,    41,     1;
  1, 21, 145,  377,   321,    61,     1;
  1, 25, 221,  833,  1289,   681,    85,     1;
  1, 29, 313, 1561,  3649,  3653,  1289,   113,    1;
  1, 33, 421, 2625,  8361, 13073,  8989,  2241,  145,   1;
  1, 37, 545, 4089, 16641, 36365, 40081, 19825, 3649, 181, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A099463 (row sums), A114123, A184884 (diagonal sums).

T:= func< n,k | (&+[Binomial(k,j)*Binomial(2*(n-k), j)*2^j: j in [0..k]]) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2021

A184883[n_, k_]:= Hypergeometric2F1[-k, 2*(k-n), 1, 2];
Table[A184883[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 19 2021 *)

def A184883(n,k): return simplify( hypergeometric([-k, 2*(k-n)], [1], 2) )
flatten([[A184883(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 19 2021

T(n,k) = [k<=n]*Sum_{j=0..k} C(2*n-2*k,j)*C(k,j)*2^j.
T(n, n-k) = A114123(n, k).
Sum_{k=0..n} T(n, k) = A099463(n+1).
Sum_{k=0..floor(n/2)} T(n, k) = A184884(n).
T(n, k) = Hypergeometric2F1([-k, 2*(k-n)], [1], 2). - G. C. Greubel, Nov 19 2021

cp's OEIS Frontend

A296320 T(n,k)=Number of nXk 0..1 arrays with each 1 horizontally, vertically or antidiagonally adjacent to 1 neighboring 1.

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A184883 Number triangle T(n,k) = [k<=n]*Hypergeometric2F1([-k, 2k-2n], [1], 2).

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