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%I #7 Sep 08 2022 08:45:41
%S 1,1,1,1,17,1,1,289,289,1,1,4913,83521,4913,1,1,83521,24137569,
%T 24137569,83521,1,1,1419857,6975757441,118587876497,6975757441,
%U 1419857,1,1,24137569,2015993900449,582622237229761,582622237229761,2015993900449,24137569,1
%N Triangle T(n, k, m) = (m+2)^(k*(n-k)) with m = 15, read by rows.
%H G. C. Greubel, <a href="/A156581/b156581.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k, m) = b(n, m)/(b(k, m)*b(n-k, m)) with b(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} binomial(j-1, i)*(k+1)^i ), b(n, 0) = n!, and m = 15.
%F T(n, k, m) = (m+2)^(k*(n-k)) with m = 15. - _G. C. Greubel_, Jun 28 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 17, 1;
%e 1, 289, 289, 1;
%e 1, 4913, 83521, 4913, 1;
%e 1, 83521, 24137569, 24137569, 83521, 1;
%e 1, 1419857, 6975757441, 118587876497, 6975757441, 1419857, 1;
%t (* First program *)
%t b[n_, k_]:= b[n, k]= If[k==0, n!, Product[Sum[Binomial[j-1, i]*(k+1)^i, {i, 0, j-1}], {j, n}]];
%t T[n_, k_, m_]:= T[n, k, m]= b[n, m]/(b[k, m]*b[n-k, m]);
%t Table[T[n, k, 15], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 28 2021 *)
%t (* Second program *)
%t T[n_, k_, m_]:= (m+2)^(k*(n-k)); Table[T[n,k,15], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 28 2021 *)
%o (Magma)
%o A156581:= func< n,k,m | (m+2)^(k*(n-k)) >;
%o [A156581(n,k,15): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 28 2021
%o (Sage)
%o def A156581(n,k,m): return (m+2)^(k*(n-k))
%o flatten([[A156581(n,k,15) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 28 2021
%Y Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), this sequence (m=15).
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 10 2009
%E Edited by _G. C. Greubel_, Jun 28 2021
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