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A124051 Quasi-mirror of A062196 formatted as a triangular array.

%I A124051 #12 Feb 07 2025 16:35:17
%S A124051 3,6,8,10,30,15,15,80,90,24,21,175,350,210,35,28,336,1050,1120,420,48,
%T A124051 36,588,2646,4410,2940,756,63,45,960,5880,14112,14700,6720,1260,80,55,
%U A124051 1485,11880,38808,58212,41580,13860,1980,99,66,2200,22275,95040,194040,199584,103950,26400,2970,120
%N A124051 Quasi-mirror of A062196 formatted as a triangular array.
%H A124051 G. C. Greubel, <a href="/A124051/b124051.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A124051 From _G. C. Greubel_, Feb 07 2025: (Start)
%F A124051 T(n, k) = binomial(n+1, n-k+1)*binomial(n+3, n-k+1).
%F A124051 T(2*n, n) = (1/2)*A000894(n) + (5/2)*[n=0].
%F A124051 Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*( (1+(-1)^n)*(-1)^(n/2)*A286033((n+4)/2) + (1-(-1)^n)*((-1)^((n+1)/2)*A000108((n+1)/2) - 1) ). (End)
%e A124051 Triangle begins as:
%e A124051    3;
%e A124051    6,    8;
%e A124051   10,   30,    15;
%e A124051   15,   80,    90,    24;
%e A124051   21,  175,   350,   210,     35;
%e A124051   28,  336,  1050,  1120,    420,     48;
%e A124051   36,  588,  2646,  4410,   2940,    756,     63;
%e A124051   45,  960,  5880, 14112,  14700,   6720,   1260,    80;
%e A124051   55, 1485, 11880, 38808,  58212,  41580,  13860,  1980,   99;
%e A124051   66, 2200, 22275, 95040, 194040, 199584, 103950, 26400, 2970, 120;
%p A124051 for n from 0 to 10 do seq(binomial(n,i-1)*binomial(n+2,n+1-i), i=1..n ) od;
%t A124051 A124051[n_, k_]:= Binomial[n+1,n-k+1]*Binomial[n+3,n-k+1];
%t A124051 Table[A124051[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 07 2025 *)
%o A124051 (Magma)
%o A124051 A124051:= func< n,k | Binomial(n+1,n-k+1)*Binomial(n+3,n-k+1) >;
%o A124051 [A124051(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 07 2025
%o A124051 (SageMath)
%o A124051 def A124051(n,k): return binomial(n+1,n-k+1)*binomial(n+3,n-k+1)
%o A124051 print(flatten([[A124051(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Feb 07 2025
%Y A124051 Cf. A000108, A000894, A062196, A286033.
%Y A124051 Columns k: A000217(n+2) (k=0), A002417(n+1) (k=1), A001297(n) (k=2), A105946(n-2) (k=3), A105947(n-3) (k=4), A105948(n-4) (k=5), A107319(n-5) (k=6).
%Y A124051 Diagonals: A005563(n+1) (k=n), A033487(n) (k=n-1), A027790(n) (k=n-2), A107395(n-3) (k=n-3), A107396(n-4) (k=n-4), A107397(n-5) (k=n-5), A107398(n-6) (k=n-6), A107399(n-7) (k=n-7).
%Y A124051 Sums: A322938(n+1) (row).
%K A124051 nonn,tabl,easy
%O A124051 0,1
%A A124051 _Zerinvary Lajos_, Nov 03 2006