cp's OEIS Frontend

A157273 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+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 2, read by rows.

%I A157273 #6 Feb 05 2022 06:44:14
%S A157273 1,1,1,1,12,1,1,47,47,1,1,154,590,154,1,1,477,4498,4498,477,1,1,1448,
%T A157273 28323,71232,28323,1448,1,1,4363,162313,816503,816503,162313,4363,1,1,
%U A157273 13110,882764,7897486,15979230,7897486,882764,13110,1,1,39353,4654100,69030716,245382470,245382470,69030716,4654100,39353,1
%N A157273 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+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 2, read by rows.
%H A157273 G. C. Greubel, <a href="/A157273/b157273.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157273 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+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 2.
%F A157273 T(n, n-k, m) = T(n, k, m).
%e A157273 Triangle begins as:
%e A157273   1;
%e A157273   1,     1;
%e A157273   1,    12,       1;
%e A157273   1,    47,      47,        1;
%e A157273   1,   154,     590,      154,         1;
%e A157273   1,   477,    4498,     4498,       477,         1;
%e A157273   1,  1448,   28323,    71232,     28323,      1448,        1;
%e A157273   1,  4363,  162313,   816503,    816503,    162313,     4363,       1;
%e A157273   1, 13110,  882764,  7897486,  15979230,   7897486,   882764,   13110,     1;
%e A157273   1, 39353, 4654100, 69030716, 245382470, 245382470, 69030716, 4654100, 39353, 1;
%t A157273 f[n_,k_]:= If[k<=Floor[n/2], 2*k+1, 2*(n-k)+1];
%t A157273 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 A157273 Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 05 2022 *)
%o A157273 (Sage)
%o A157273 def f(n,k): return 2*k+1 if (k <= n//2) else 2*(n-k)+1
%o A157273 @CachedFunction
%o A157273 def T(n,k,m):  # A157207
%o A157273     if (k==0 or k==n): return 1
%o A157273     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 A157273 flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 05 2022
%Y A157273 Cf. A007318 (m=0), A157272 (m=1), this sequence (m=2), A157274 (m=3).
%Y A157273 Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157275, A157277, A157278.
%K A157273 nonn,tabl
%O A157273 0,5
%A A157273 _Roger L. Bagula_, Feb 26 2009
%E A157273 Edited by _G. C. Greubel_, Feb 05 2022