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Square array read by antidiagonals upwards: T(n,k) = n*((k-2)*(-1)^n+k+2)/4, n >= 0, k >= 0.
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%I #20 Oct 26 2023 10:10:06

%S 0,1,0,0,1,0,3,1,1,0,0,3,2,1,0,5,2,3,3,1,0,0,5,4,3,4,1,0,7,3,5,6,3,5,

%T 1,0,0,7,6,5,8,3,6,1,0,9,4,7,9,5,10,3,7,1,0,0,9,8,7,12,5,12,3,8,1,0,

%U 11,5,9,12,7,15,5,14,3,9,1,0,0,11,10,9,16,7

%N Square array read by antidiagonals upwards: T(n,k) = n*((k-2)*(-1)^n+k+2)/4, n >= 0, k >= 0.

%C Also square array T(n,k) read by antidiagonals in which column k lists the multiples of k and the odd numbers interleaved, n>=0, k>=0. Also square array T(n,k) read by antidiagonals in which if n is even then row n lists the multiples of (n/2), otherwise if n is odd then row n lists a constant sequence: the all n's sequence. Partial sums of the numbers of column k give the column k of A195152. Note that if k >= 1 then partial sums of the numbers of the column k give the generalized m-gonal numbers, where m = k + 4.

%C All columns are multiplicative. - _Andrew Howroyd_, Jul 23 2018

%e Array begins:

%e . 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...

%e . 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,...

%e . 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,...

%e . 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,...

%e . 0, 2, 4, 6, 8, 10, 12, 14, 16, 18,...

%e . 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,...

%e . 0, 3, 6, 9, 12, 15, 18, 21, 24, 27,...

%e . 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,...

%e . 0, 4, 8, 12, 16, 20, 24, 28, 32, 36,...

%e . 9, 9, 9, 9, 9, 9, 9, 9, 9, 9,...

%e . 0, 5, 10, 15, 20, 25, 30, 35, 40, 45,...

%e ...

%o (PARI) T(n,k) = n*((k-2)*(-1)^n+k+2)/4 \\ _Andrew Howroyd_, Jul 23 2018

%Y Rows: A000004, A000012, A001477, A010701, A005843, A010716, A008585, A010727, A008586, A010734, A008587.

%Y Columns k: A026741 (k=1), A001477 (k=2), zero together with A080512 (k=3), A022998 (k=4), A195140 (k=5), zero together with A165998 (k=6), A195159 (k=7), A195161 (k=8), A195312 k=(9), A195817 (k=10), A317311 (k=11), A317312 (k=12), A317313 (k=13), A317314 k=(14), A317315 (k=15), A317316 (k=16), A317317 (k=17), A317318 (k=18), A317319 k=(19), A317320 (k=20), A317321 (k=21), A317322 (k=22), A317323 (k=23), A317324 k=(24), A317325 (k=25), A317326 (k=26).

%K nonn,tabl,mult

%O 0,7

%A _Omar E. Pol_, Sep 14 2011