A157169 - OEIS

%I #6 Sep 08 2022 08:45:41

%S 1,1,1,1,5,1,1,9,9,1,1,13,26,13,1,1,17,52,52,17,1,1,21,87,134,87,21,1,

%T 1,25,131,275,275,131,25,1,1,29,184,491,670,491,184,29,1,1,33,246,798,

%U 1386,1386,798,246,33,1,1,37,317,1212,2562,3262,2562,1212,317,37,1

%N Triangle, read by rows, T(n,k,m) = (m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1), with m=1.

%H G. C. Greubel, <a href="/A157169/b157169.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n,k,m) = (m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1), with m=1.

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1,  5,   1;

%e   1,  9,   9,    1;

%e   1, 13,  26,   13,    1;

%e   1, 17,  52,   52,   17,    1;

%e   1, 21,  87,  134,   87,   21,    1;

%e   1, 25, 131,  275,  275,  131,   25,    1;

%e   1, 29, 184,  491,  670,  491,  184,   29,   1;

%e   1, 33, 246,  798, 1386, 1386,  798,  246,  33,  1;

%e   1, 37, 317, 1212, 2562, 3262, 2562, 1212, 317, 37, 1;

%p T(n,k,m):= (m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1); seq(seq( T(n,k,1), k=0..n), n=0..10); # _G. C. Greubel_, Nov 29 2019

%t T[n_, k_, m_]:= (m*(n-k)+1)*Binomial[n-1, k-1] + (m*k+1)*Binomial[n-1, k] + m*k*(n-k)*Binomial[n-2, k-1]; Table[T[n, k, 1], {n,0,10}, {k,0,n}]//Flatten

%o (PARI) T(n,k,m) = (m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1); \\ _G. C. Greubel_, Nov 29 2019

%o (Magma) m:=1; [(m*(n-k)+1)*Binomial(n-1, k-1) + (m*k+1)* Binomial(n-1, k) + m*k*(n-k)*Binomial(n-2, k-1): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 29 2019

%o (Sage) m=1; [[(m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1) for k in (0..n)] for n in [0..10]] # _G. C. Greubel_, Nov 29 2019

%o (GAP) m:=1;; Flat(List([0..10], n-> List([0..n], k-> (m*(n-k)+1)*Binomial(n-1, k-1) + (m*k+1)* Binomial(n-1, k) + m*k*(n-k)*Binomial(n-2, k-1) ))); # _G. C. Greubel_, Nov 29 2019

%Y Cf. this sequence (m=1), A157170 (m=2), A157171 (m=3).

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 24 2009