cp's OEIS Frontend

A157155 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 = 4, read by rows.

%I A157155 #6 Jan 10 2022 16:38:41
%S A157155 1,1,1,1,6,1,1,31,31,1,1,156,462,156,1,1,781,5442,5442,781,1,1,3906,
%T A157155 57263,124860,57263,3906,1,1,19531,566153,2335435,2335435,566153,
%U A157155 19531,1,1,97656,5396164,38814088,71413750,38814088,5396164,97656,1
%N A157155 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 = 4, read by rows.
%H A157155 G. C. Greubel, <a href="/A157155/b157155.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157155 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 = 4.
%F A157155 T(n, n-k, m) = T(n, k, m).
%F A157155 T(n, 1, 4) = A003463(n). - _G. C. Greubel_, Jan 10 2022
%e A157155 Triangle begins as:
%e A157155   1;
%e A157155   1,     1;
%e A157155   1,     6,       1;
%e A157155   1,    31,      31,        1;
%e A157155   1,   156,     462,      156,        1;
%e A157155   1,   781,    5442,     5442,      781,        1;
%e A157155   1,  3906,   57263,   124860,    57263,     3906,       1;
%e A157155   1, 19531,  566153,  2335435,  2335435,   566153,   19531,     1;
%e A157155   1, 97656, 5396164, 38814088, 71413750, 38814088, 5396164, 97656, 1;
%t A157155 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 A157155 Table[T[n,k,4], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 10 2022 *)
%o A157155 (Sage)
%o A157155 @CachedFunction
%o A157155 def T(n,k,m):  # A157155
%o A157155     if (k==0 or k==n): return 1
%o A157155     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 A157155 flatten([[T(n,k,4) for k in (0..n)] for n in (0..20)]) # _G. C. Greubel_, Jan 10 2022
%Y A157155 Cf. A007318 (m=0), A157152 (m=1), A157153 (m=2), A157154 (m=3), this sequence (m=4), A157156 (m=5).
%Y A157155 Cf. A157147, A157148, A157149, A157150, A157151, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
%Y A157155 Cf. A003463.
%K A157155 nonn,tabl
%O A157155 0,5
%A A157155 _Roger L. Bagula_, Feb 24 2009
%E A157155 Edited by _G. C. Greubel_, Jan 10 2022