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A062707 Table by antidiagonals of n*k*(k+1)/2.

%I A062707 #16 Sep 08 2022 08:45:03
%S A062707 0,0,0,0,1,0,0,3,2,0,0,6,6,3,0,0,10,12,9,4,0,0,15,20,18,12,5,0,0,21,
%T A062707 30,30,24,15,6,0,0,28,42,45,40,30,18,7,0,0,36,56,63,60,50,36,21,8,0,0,
%U A062707 45,72,84,84,75,60,42,24,9,0,0,55,90,108,112,105,90,70,48,27,10,0
%N A062707 Table by antidiagonals of n*k*(k+1)/2.
%H A062707 G. C. Greubel, <a href="/A062707/b062707.txt">Antidiagonals n = 0..100, flattened</a>
%F A062707 T(n, k) = T(n, 1)*T(1, k) = A001477(n)*A000217(k).
%F A062707 T(n, k) = A057145(n+2, k+1)-(k+1).
%e A062707   0   0   0   0   0   0   0   0   0
%e A062707   0   1   3   6  10  15  21  28  36
%e A062707   0   2   6  12  20  30  42  56  72
%e A062707   0   3   9  18  30  45  63  84 108
%e A062707   0   4  12  24  40  60  84 112 144
%e A062707   0   5  15  30  50  75 105 140 180
%e A062707   0   6  18  36  60  90 126 168 216
%e A062707   0   7  21  42  70 105 147 196 252
%e A062707   0   8  24  48  80 120 168 224 288
%p A062707 seq(seq(k*binomial(n-k+1,2), k=0..n), n=0..12); # _G. C. Greubel_, Sep 02 2019
%t A062707 Table[k*Binomial[n-k+1, 2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 02 2019 *)
%o A062707 (PARI) T(n,k) = k*binomial(n-k+1,2);
%o A062707 for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Sep 02 2019
%o A062707 (Magma) [k*Binomial(n-k+1,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 02 2019
%o A062707 (Sage) [[k*binomial(n-k+1,2) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Sep 02 2019
%o A062707 (GAP) Flat(List([0..12], n-> List([0..n], k-> k*Binomial(n-k+1,2)))); # _G. C. Greubel_, Sep 02 2019
%Y A062707 Main diagonal is A002411. Sum of antidiagonals is A000332.
%K A062707 easy,nonn,tabl
%O A062707 0,8
%A A062707 _Henry Bottomley_, Jul 11 2001