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%I #10 Jan 10 2022 13:28:33
%S 1,1,1,1,4,1,1,13,13,1,1,40,98,40,1,1,121,614,614,121,1,1,364,3519,
%T 6832,3519,364,1,1,1093,19179,64759,64759,19179,1093,1,1,3280,101368,
%U 558712,947038,558712,101368,3280,1,1,9841,525436,4538324,12078814,12078814,4538324,525436,9841,1
%N 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 = 2, read by rows.
%H G. C. Greubel, <a href="/A157153/b157153.txt">Rows n = 0..50 of the triangle, flattened</a>
%F 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 = 2.
%F T(n, n-k, m) = T(n, k, m) for m = 2.
%F T(n, 1, 2) = A003462(n). - _G. C. Greubel_, Jan 10 2022
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 13, 13, 1;
%e 1, 40, 98, 40, 1;
%e 1, 121, 614, 614, 121, 1;
%e 1, 364, 3519, 6832, 3519, 364, 1;
%e 1, 1093, 19179, 64759, 64759, 19179, 1093, 1;
%e 1, 3280, 101368, 558712, 947038, 558712, 101368, 3280, 1;
%e 1, 9841, 525436, 4538324, 12078814, 12078814, 4538324, 525436, 9841, 1;
%t 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 Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 10 2022 *)
%o (Sage)
%o @CachedFunction
%o def T(n,k,m): # A157153
%o if (k==0 or k==n): return 1
%o 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 flatten([[T(n,k,2) for k in (0..n)] for n in (0..20)]) # _G. C. Greubel_, Jan 10 2022
%Y Cf. A007318 (m=0), A157152 (m=1), this sequence (m=2), A157154 (m=3), A157155 (m=4), A157156 (m=5).
%Y Cf. A157147, A157148, A157149, A157150, A157151, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
%Y Cf. A003462.
%K nonn,tabl,easy
%O 0,5
%A _Roger L. Bagula_, Feb 24 2009
%E Edited by _G. C. Greubel_, Jan 10 2022
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