cp's OEIS Frontend

A157277 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 2, read by rows.

%I A157277 #6 Feb 06 2022 02:06:38
%S A157277 1,1,1,1,10,1,1,39,39,1,1,128,470,128,1,1,397,3558,3558,397,1,1,1206,
%T A157277 22387,55452,22387,1206,1,1,3635,128377,632343,632343,128377,3635,1,1,
%U A157277 10924,698788,6107192,12269406,6107192,698788,10924,1,1,32793,3686880,53375112,187721254,187721254,53375112,3686880,32793,1
%N A157277 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 2, read by rows.
%H A157277 G. C. Greubel, <a href="/A157277/b157277.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157277 T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 2.
%F A157277 T(n, n-k, m) = T(n, k, m).
%e A157277 Triangle begins as:
%e A157277   1;
%e A157277   1,     1;
%e A157277   1,    10,      1;
%e A157277   1,    39,     39,       1;
%e A157277   1,   128,    470,     128,        1;
%e A157277   1,   397,   3558,    3558,      397,       1;
%e A157277   1,  1206,  22387,   55452,    22387,    1206,      1;
%e A157277   1,  3635, 128377,  632343,   632343,  128377,   3635,     1;
%e A157277   1, 10924, 698788, 6107192, 12269406, 6107192, 698788, 10924, 1;
%t A157277 f[n_,k_]:= If[k<=Floor[n/2], 2*k, 2*(n-k)];
%t A157277 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*f[n,k]*T[n-2,k-1,m]];
%t A157277 Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 05 2022 *)
%o A157277 (Sage)
%o A157277 def f(n,k): return 2*k if (k <= n//2) else 2*(n-k)
%o A157277 @CachedFunction
%o A157277 def T(n,k,m):  # A157277
%o A157277     if (k==0 or k==n): return 1
%o A157277     else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*f(n,k)*T(n-2,k-1,m)
%o A157277 flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 05 2022
%Y A157277 Cf. A007318 (m=0), A157275 (m=1), this sequence (m=2), A157278 (m=3).
%Y A157277 Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274.
%K A157277 nonn,tabl
%O A157277 0,5
%A A157277 _Roger L. Bagula_, Feb 26 2009
%E A157277 Edited by _G. C. Greubel_, Feb 05 2022