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%I A157147 #10 Jan 10 2022 03:06:46
%S A157147 1,1,1,1,5,1,1,15,15,1,1,37,110,37,1,1,83,568,568,83,1,1,177,2415,
%T A157147 5534,2415,177,1,1,367,9137,41027,41027,9137,367,1,1,749,32104,255155,
%U A157147 498814,255155,32104,749,1,1,1515,107442,1409814,4845540,4845540,1409814,107442,1515,1
%N A157147 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 1, read by rows.
%H A157147 G. C. Greubel, <a href="/A157147/b157147.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157147 T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 1.
%F A157147 T(n, n-k) = T(n, k).
%e A157147   1;
%e A157147   1,    1;
%e A157147   1,    5,      1;
%e A157147   1,   15,     15,       1;
%e A157147   1,   37,    110,      37,       1;
%e A157147   1,   83,    568,     568,      83,       1;
%e A157147   1,  177,   2415,    5534,    2415,     177,       1;
%e A157147   1,  367,   9137,   41027,   41027,    9137,     367,      1;
%e A157147   1,  749,  32104,  255155,  498814,  255155,   32104,    749,    1;
%e A157147   1, 1515, 107442, 1409814, 4845540, 4845540, 1409814, 107442, 1515, 1;
%p A157147 A157147:= proc(n,k)
%p A157147     option remember;
%p A157147     if k < 0 or k> n then 0;
%p A157147     elif k = 0 or k = n then 1;
%p A157147     else (n-k+1)*procname(n-1,k-1) +(k+1)*procname(n-1,k) +k*(n-k)*procname(n-2,k-1);
%p A157147     end if;
%p A157147 end proc:
%p A157147 seq(seq(A157147(n,k),k=0..n),n=0..10); # _R. J. Mathar_, Feb 06 2015
%t A157147 T[n_, k_, m_]:= T[n,k,m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1, k-1, m] + (m*k+1)*T[n-1, k, m] + m*k*(n-k)*T[n-2, k-1, m]];
%t A157147 Table[T[n,k,1], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 09 2022 *)
%o A157147 (Sage)
%o A157147 def T(n,k,m): # A157147
%o A157147     if (k==0 or k==n): return 1
%o A157147     else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*k*(n-k)*T(n-2,k-1,m)
%o A157147 flatten([[T(n,k,1) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Jan 09 2022
%Y A157147 Cf. A007318 (m=0), A157147 (m=1), A157148 (m=2), A157149 (m=3), A157150 (m=4), A157151 (m=5).
%Y A157147 Cf. A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
%K A157147 nonn,tabl,easy
%O A157147 0,5
%A A157147 _Roger L. Bagula_, Feb 24 2009
%E A157147 Edited by _G. C. Greubel_, Jan 09 2022

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