OFFSET
0,3
LINKS
Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
FORMULA
T(n, 0) = A000217(n).
T(n, n) = A046092(n).
T(2*n, n) = A062725(n) (central terms).
Sum_{k=0..n} T(n, k) = A245301(n).
From G. C. Greubel, Apr 01 2021: (Start)
T(n, 2) = A152948(n+4), n >= 2.
T(n, 3) = A152950(n+4), n >= 3.
T(n, 4) = A145018(n+5), n >= 4.
T(n, 5) = A167499(n+4), n >= 5.
T(n, 6) = A166136(n+5), n >= 6.
T(n, 7) = A167487(n+6), n >= 7.
T(n, n-1) = A056220(n), n >= 1.
T(n, n-2) = A142463(n-1), n >= 2.
T(n, n-3) = A054000(n-1), n >= 3.
T(n, n-4) = A090288(n-3), n >= 4.
T(n, n-5) = A268581(n-4), n >= 5.
T(n, n-6) = A059993(n-4), n >= 6.
T(n, n-7) = (-1)*A147973(n), n >= 7.
T(n, n-8) = A139570(n-5), n >= 8.
T(n, n-9) = A271625(n-5), n >= 9.
T(n, n-10) = A222182(n-4), n >= 10.
T(2*n, n-1) = A081270(n-1), n >= 1.
T(2*n, n+1) = A117625(n+1), n >= 1. (End)
EXAMPLE
First rows and their row sums (A245301):
0 0;
1, 4 5;
3, 7, 12 22;
6, 11, 17, 24 58;
10, 16, 23, 31, 40 120;
15, 22, 30, 39, 49, 60 215;
21, 29, 38, 48, 59, 71, 84 350;
28, 37, 47, 58, 70, 83, 97, 112 532;
36, 46, 57, 69, 82, 96, 111, 127, 144 768;
45, 56, 68, 81, 95, 110, 126, 143, 161, 180 1065;
55, 67, 80, 94, 109, 125, 142, 160, 179, 199, 220 1430;
66, 79, 93, 108, 124, 141, 159, 178, 198, 219, 241, 264 1870;
78, 92, 107, 123, 140, 158, 177, 197, 218, 240, 263, 287, 312 2392.
MATHEMATICA
Table[k + Binomial[n+k+1, 2], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)
PROG
(Haskell)
a245300 n k = (n + k) * (n + k + 1) `div` 2 + k
a245300_row n = map (a245300 n) [0..n]
a245300_tabl = map a245300_row [0..]
a245300_list = concat a245300_tabl
(Magma) [k + Binomial(n+k+1, 2): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 01 2021
(Sage) flatten([[k + binomial(n+k+1, 2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 01 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, Jul 17 2014
STATUS
approved