A185779 - OEIS

%I #8 Jul 12 2017 03:18:59

%S 1,4,4,10,17,10,20,45,45,20,35,95,126,95,35,56,175,281,281,175,56,84,

%T 294,546,662,546,294,84,120,462,966,1358,1358,966,462,120,165,690,

%U 1596,2534,2941,2534,1596,690,165,220,990,2502,4410,5790,5790,4410,2502,990,220,286,1375,3762,7272,10620,12021,10620,7272,3762,1375,286,364,1859,5467,11484,18432,23229,23229,18432,11484,5467,1859,364,455

%N Third accumulation array of Pascal's triangle (as a rectangle), by antidiagonals.

%C Using "Axxxxxx < Ayyyyyy" to mean that Ayyyyyy is the accumulation array of Axxxxxx, as defined at A144112:

%C A185779 < A144225 < A007318 < A014430 < A077023 < A185779, where each of these is formatted as a rectangle (e.g., A007318 is Pascal's triangle).  See A185778.

%C row 1: A000292

%C row 2: A095667

%H G. C. Greubel, <a href="/A185779/b185779.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F T(n,k) = C(n+k+4,n+2) - (k+3)*(k+4)/2 - (k+2)*n*(k*n+n+3*k+7)/4, for k>=1, n>=1.

%e Northwest corner:

%e 1....4...10...20...35

%e 4....17..45...95...175

%e 10...45..126..281..546

%e 20...95..281..662..1358

%t f[n_, k_] := Binomial[n + k + 4, n + 2] - (k + 3)*(k + 4)/2 - (k + 2)* n*(k*n + n + 3*k + 7)/4; TableForm[Table[f[n, k], {n, 1, 5}, {k, 1, 5}]]

%t Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* _G. C. Greubel_, Jul 12 2017 *)

%Y Cf. A007318, A014430, A077023, A144112, A144225, A185778.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Feb 03 2011