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A157637 Triangle, T(n, k, m) = 1 if (k=0 or k=n), otherwise (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*A157636(n, k)*T(n-2, k-1, m) for m = 1, read by rows.

%I A157637 #8 Dec 13 2021 19:14:34
%S A157637 1,1,1,1,5,1,1,16,16,1,1,42,136,42,1,1,99,816,816,99,1,1,219,3951,
%T A157637 10200,3951,219,1,1,466,16632,94827,94827,16632,466,1,1,968,63670,
%U A157637 716160,1601070,716160,63670,968,1,1,1981,228112,4657522,20836740,20836740,4657522,228112,1981,1
%N A157637 Triangle, T(n, k, m) = 1 if (k=0 or k=n), otherwise (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*A157636(n, k)*T(n-2, k-1, m) for m = 1, read by rows.
%C A157637 For the case of m = 0 the triangle becomes T(n, k, 0) = A007318(n, k). - _G. C. Greubel_, Dec 13 2021
%H A157637 G. C. Greubel, <a href="/A157637/b157637.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157637 T(n, k, m) = 1 if (k=0 or k=n), otherwise (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*A157636(n, k)*T(n-2, k-1, m) for m = 1.
%F A157637 T(n, k) = T(n, n-k). - _G. C. Greubel_, Dec 13 2021
%e A157637 Triangle begins as:
%e A157637   1;
%e A157637   1,    1;
%e A157637   1,    5,      1;
%e A157637   1,   16,     16,       1;
%e A157637   1,   42,    136,      42,        1;
%e A157637   1,   99,    816,     816,       99,        1;
%e A157637   1,  219,   3951,   10200,     3951,      219,       1;
%e A157637   1,  466,  16632,   94827,    94827,    16632,     466,      1;
%e A157637   1,  968,  63670,  716160,  1601070,   716160,   63670,    968,    1;
%e A157637   1, 1981, 228112, 4657522, 20836740, 20836740, 4657522, 228112, 1981, 1;
%t A157637 A157636[n_, k_]:= If[k==0||k==n, 1, n*k*(n-k)/2];
%t A157637 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*A157636[n, k]*T[n-2,k-1,m]];
%t A157637 Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten
%o A157637 (Sage)
%o A157637 @CachedFunction
%o A157637 def A157636(n,k): return 1 if (k==0 or k==n) else n*k*(n-k)/2
%o A157637 def T(n,k,q): return 1 if (k==0 or k==n) else (q*(n-k) +1)*T(n-1, k-1, q) + (q*k + 1)*T(n-1, k, q) + q*A157636(n, k)*T(n-2, k-1, q)
%o A157637 flatten([[T(n,k,1) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Dec 13 2021
%Y A157637 Cf. A007318, A157523, A157636.
%K A157637 nonn,tabl
%O A157637 0,5
%A A157637 _Roger L. Bagula_, Mar 03 2009
%E A157637 Edited by _G. C. Greubel_, Dec 13 2021