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A157118 Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A001263(n*k+1, n-k+1) if k <= n otherwise A001263(n*(n-k)+1, k+1) and T(1, k) = 1, read by rows.

%I A157118 #7 Sep 08 2022 08:45:41
%S A157118 2,1,1,1,6,1,1,27,27,1,1,88,672,88,1,1,225,9150,9150,225,1,1,486,
%T A157118 98385,395352,98385,486,1,1,931,1126951,11748681,11748681,1126951,931,
%U A157118 1,1,1632,14600320,402703120,588593280,402703120,14600320,1632,1,1,2673,201755880,16093941435,32251030119,32251030119,16093941435,201755880,2673,1
%N A157118 Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A001263(n*k+1, n-k+1) if k <= n otherwise A001263(n*(n-k)+1, k+1) and T(1, k) = 1, read by rows.
%H A157118 G. C. Greubel, <a href="/A157118/b157118.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157118 T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A001263(n*k+1, n-k+1) if k <= n otherwise A001263(n*(n-k)+1, k+1) and T(1, k) = 1.
%F A157118 T(n, n-k) = T(n, k).
%e A157118 Triangle begins as:
%e A157118   2;
%e A157118   1,    1;
%e A157118   1,    6,        1;
%e A157118   1,   27,       27,         1;
%e A157118   1,   88,      672,        88,         1;
%e A157118   1,  225,     9150,      9150,       225,         1;
%e A157118   1,  486,    98385,    395352,     98385,       486,        1;
%e A157118   1,  931,  1126951,  11748681,  11748681,   1126951,      931,    1;
%e A157118   1, 1632, 14600320, 402703120, 588593280, 402703120, 14600320, 1632, 1;
%t A157118 A001263[n_, k_]:= Binomial[n-1,k-1]*Binomial[n,k]/(n-k+1);
%t A157118 f[n_, k_]:= If[k<=n, A001263[n*k+1,n-k+1], A001263[n*(n-k)+1,k+1]];
%t A157118 T[n_, k_]:= If[n==1, 1, f[n,k] + f[n,n-k]];
%t A157118 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 11 2022 *)
%o A157118 (Magma)
%o A157118 A001263:= func< n,k | Binomial(n-1,k-1)*Binomial(n,k)/(n-k+1) >;
%o A157118 f:= func< n,k | k le n select A001263(n*k+1,n-k+1) else A001263(n*(n-k)+1, k+1) >;
%o A157118 A157118:= func< n,k | n eq 1 select 1 else f(n,k) + f(n,n-k) >;
%o A157118 [A157118(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 11 2022
%o A157118 (Sage)
%o A157118 def A001263(n,k): return binomial(n-1,k-1)*binomial(n,k)/(n-k+1)
%o A157118 def f(n,k): return A001263(n*k+1,n-k+1) if (k<n+1) else A001263(n*(n-k)+1, k+1)
%o A157118 def A157118(n,k): return 1 if (n==1) else f(n,k) + f(n,n-k)
%o A157118 flatten([[A157118(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jan 11 2022
%Y A157118 Cf. A001263, A157114, A157117.
%K A157118 nonn,tabl
%O A157118 0,1
%A A157118 _Roger L. Bagula_, Feb 23 2009
%E A157118 Edited by _G. C. Greubel_, Jan 11 2022