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A174102 Triangle read by rows: T(n, m) = floor(binomial(n+1, m)* binomial(n+2, m)/(2*m+2)), 1 <= m <= n.

%I A174102 #25 Sep 08 2022 08:45:51
%S A174102 1,3,3,5,10,5,7,25,25,7,10,52,87,52,10,14,98,245,245,98,14,18,168,588,
%T A174102 882,588,168,18,22,270,1260,2646,2646,1260,270,22,27,412,2475,6930,
%U A174102 9702,6930,2475,412,27,33,605,4537,16335,30492,30492,16335,4537,605,33
%N A174102 Triangle read by rows: T(n, m) = floor(binomial(n+1, m)* binomial(n+2, m)/(2*m+2)), 1 <= m <= n.
%C A174102 Row sums are {1, 6, 20, 64, 211, 714, 2430, 8396, 29390, 104004, 371448, 1337216, ...}.
%H A174102 G. C. Greubel, <a href="/A174102/b174102.txt">Rows n = 1..100 of triangle, flattened</a>
%F A174102 T(n, m) = floor(binomial(n+1, m-1)*binomial(n+2, m-1)/(2*m)).
%e A174102 Triangle begins as:
%e A174102    1;
%e A174102    3,   3;
%e A174102    5,  10,    5;
%e A174102    7,  25,   25,    7;
%e A174102   10,  52,   87,   52,   10;
%e A174102   14,  98,  245,  245,   98,   14;
%e A174102   18, 168,  588,  882,  588,  168,   18;
%e A174102   22, 270, 1260, 2646, 2646, 1260,  270,  22;
%e A174102   27, 412, 2475, 6930, 9702, 6930, 2475, 412, 27;
%t A174102 T[n_, k_] = Floor[Binomial[n+1, k]*Binomial[n+2, k]/(2*(k+1))];
%t A174102 Table[T[n, k], {n,1,12}, {k,1,n}]//Flatten (* modified by _G. C. Greubel_, Apr 13 2019 *)
%o A174102 (PARI) {T(n,k) = (binomial(n+1,k)*binomial(n+2,k)/(2*k+2))\1};
%o A174102 for(n=1,12, for(k=1,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Apr 13 2019
%o A174102 (Magma) [[Floor(Binomial(n+1, k)*Binomial(n+2, k)/(2*k+2)): k in [1..n]]: n in [1..12]]; // _G. C. Greubel_, Apr 13 2019
%o A174102 (Sage) [[floor(binomial(n+1,k)*binomial(n+2,k)/(2*k+2)) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Apr 13 2019
%Y A174102 Cf. A166454.
%Y A174102 Cf. A011848 (right diagonal).
%K A174102 nonn,tabl
%O A174102 1,2
%A A174102 _Roger L. Bagula_, Mar 07 2010
%E A174102 Partially edited by _Jon E. Schoenfield_, Dec 02 2013
%E A174102 Edited by _G. C. Greubel_, Apr 13 2019