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A157156 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 = 5, read by rows.

%I A157156 #6 Jan 10 2022 16:42:58
%S A157156 1,1,1,1,7,1,1,43,43,1,1,259,806,259,1,1,1555,11720,11720,1555,1,1,
%T A157156 9331,151215,338770,151215,9331,1,1,55987,1828221,7892635,7892635,
%U A157156 1828221,55987,1,1,335923,21286168,162474781,304389070,162474781,21286168,335923,1
%N A157156 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 = 5, read by rows.
%H A157156 G. C. Greubel, <a href="/A157156/b157156.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157156 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 = 5.
%F A157156 T(n, n-k, m) = T(n, k, m).
%F A157156 T(n, 1, 5) = A003464(n). - _G. C. Greubel_, Jan 10 2022
%e A157156 Triangle begins as:
%e A157156   1;
%e A157156   1,      1;
%e A157156   1,      7,        1;
%e A157156   1,     43,       43,         1;
%e A157156   1,    259,      806,       259,         1;
%e A157156   1,   1555,    11720,     11720,      1555,         1;
%e A157156   1,   9331,   151215,    338770,    151215,      9331,        1;
%e A157156   1,  55987,  1828221,   7892635,   7892635,   1828221,    55987,      1;
%e A157156   1, 335923, 21286168, 162474781, 304389070, 162474781, 21286168, 335923, 1;
%t A157156 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 A157156 Table[T[n,k,5], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 10 2022 *)
%o A157156 (Sage)
%o A157156 @CachedFunction
%o A157156 def T(n,k,m):  # A157156
%o A157156     if (k==0 or k==n): return 1
%o A157156     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 A157156 flatten([[T(n,k,5) for k in (0..n)] for n in (0..20)]) # _G. C. Greubel_, Jan 10 2022
%Y A157156 Cf. A007318 (m=0), A157152 (m=1), A157153 (m=2), A157154 (m=3), A157155 (m=4), A157156 (m=5).
%Y A157156 Cf. A157147, A157148, A157149, A157150, A157151, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
%Y A157156 Cf. A003464.
%K A157156 nonn,tabl
%O A157156 0,5
%A A157156 _Roger L. Bagula_, Feb 24 2009
%E A157156 Edited by _G. C. Greubel_, Jan 10 2022