cp's OEIS Frontend

A157152 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.

%I A157152 #13 Jan 10 2022 03:06:06
%S A157152 1,1,1,1,3,1,1,7,7,1,1,15,30,15,1,1,31,108,108,31,1,1,63,359,594,359,
%T A157152 63,1,1,127,1145,2875,2875,1145,127,1,1,255,3568,12985,19246,12985,
%U A157152 3568,255,1,1,511,10966,56306,116640,116640,56306,10966,511,1
%N A157152 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 A157152 G. C. Greubel, <a href="/A157152/b157152.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157152 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 A157152 T(n, n-k, m) = T(n, k, m).
%F A157152 T(n, 1, 1) = A000225(n). - _G. C. Greubel_, Jan 09 2022
%e A157152 Triangle begins as:
%e A157152   1;
%e A157152   1,    1;
%e A157152   1,    3,     1;
%e A157152   1,    7,     7,      1;
%e A157152   1,   15,    30,     15,      1;
%e A157152   1,   31,   108,    108,     31,      1;
%e A157152   1,   63,   359,    594,    359,     63,      1;
%e A157152   1,  127,  1145,   2875,   2875,   1145,    127,      1;
%e A157152   1,  255,  3568,  12985,  19246,  12985,   3568,    255,     1;
%e A157152   1,  511, 10966,  56306, 116640, 116640,  56306,  10966,   511,    1;
%e A157152   1, 1023, 33417, 238024, 665702, 918530, 665702, 238024, 33417, 1023, 1;
%t A157152 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 A157152 Table[T[n,k,1], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 09 2022 *)
%o A157152 (Sage)
%o A157152 @CachedFunction
%o A157152 def T(n,k,m):  # A157152
%o A157152     if (k==0 or k==n): return 1
%o A157152     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 A157152 flatten([[T(n,k,1) for k in (0..n)] for n in (0..20)]) # _G. C. Greubel_, Jan 09 2022
%Y A157152 Cf. A007318 (m=0), this sequence (m=1), A157153 (m=2), A157154 (m=3), A157155 (m=4), A157156 (m=5).
%Y A157152 Cf. A157147, A157148, A157149, A157150, A157151, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
%K A157152 nonn,tabl,easy
%O A157152 0,5
%A A157152 _Roger L. Bagula_, Feb 24 2009
%E A157152 Edited by _G. C. Greubel_, Jan 09 2022