cp's OEIS Frontend

A157207 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) = k if k <= floor(n/2) otherwise n-k, and m = 1, read by rows.

%I A157207 #8 Jan 11 2022 02:18:57
%S A157207 1,1,1,1,5,1,1,14,14,1,1,33,94,33,1,1,72,442,442,72,1,1,151,1752,3818,
%T A157207 1752,151,1,1,310,6306,25358,25358,6306,310,1,1,629,21390,144524,
%U A157207 268852,144524,21390,629,1,1,1268,69822,746744,2312836,2312836,746744,69822,1268,1
%N A157207 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) = k if k <= floor(n/2) otherwise n-k, and m = 1, read by rows.
%H A157207 G. C. Greubel, <a href="/A157207/b157207.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157207 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) = k if k <= floor(n/2) otherwise n-k, and m = 1.
%F A157207 T(n, n-k, m) = T(n, k, m).
%F A157207 T(n, 1, 1) = A094002(n-1). - _G. C. Greubel_, Jan 10 2022
%e A157207 Triangle begins as:
%e A157207   1;
%e A157207   1,    1;
%e A157207   1,    5,     1;
%e A157207   1,   14,    14,      1;
%e A157207   1,   33,    94,     33,       1;
%e A157207   1,   72,   442,    442,      72,       1;
%e A157207   1,  151,  1752,   3818,    1752,     151,      1;
%e A157207   1,  310,  6306,  25358,   25358,    6306,    310,     1;
%e A157207   1,  629, 21390, 144524,  268852,  144524,  21390,   629,    1;
%e A157207   1, 1268, 69822, 746744, 2312836, 2312836, 746744, 69822, 1268, 1;
%t A157207 f[n_,k_]:= If[k<=Floor[n/2], k, n-k];
%t A157207 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 A157207 Table[T[n,k,1], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 10 2022 *)
%o A157207 (Sage)
%o A157207 def f(n,k): return k if (k <= n//2) else n-k
%o A157207 @CachedFunction
%o A157207 def T(n,k,m):  # A157207
%o A157207     if (k==0 or k==n): return 1
%o A157207     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 A157207 flatten([[T(n,k,1) for k in (0..n)] for n in (0..20)]) # _G. C. Greubel_, Jan 10 2022
%Y A157207 Cf. A007318 (m=0), this sequence (m=1), A157208 (m=2), A157209 (m=3).
%Y A157207 Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
%Y A157207 Cf. A094002.
%K A157207 nonn,tabl
%O A157207 0,5
%A A157207 _Roger L. Bagula_, Feb 25 2009
%E A157207 Edited by _G. C. Greubel_, Jan 10 2022