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Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + n^(n-1) * binomial(n-2, k-1) otherwise.
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%I #10 Feb 06 2025 07:38:19

%S 1,1,1,1,4,1,1,12,12,1,1,68,134,68,1,1,630,1885,1885,630,1,1,7782,

%T 31119,46676,31119,7782,1,1,117656,588266,1176525,1176525,588266,

%U 117656,1,1,2097160,12582940,31457336,41943110,31457336,12582940,2097160,1

%N Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + n^(n-1) * binomial(n-2, k-1) otherwise.

%C Row sums are: {1, 2, 6, 26, 272, 5032, 124480, 3764896, 134217984, 5509980800, 256000001024, ...}.

%H G. C. Greubel, <a href="/A146990/b146990.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + n^(n-1) * binomial(n-2, k-1) otherwise.

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 4, 1;

%e 1, 12, 12, 1;

%e 1, 68, 134, 68, 1;

%e 1, 630, 1885, 1885, 630, 1;

%e 1, 7782, 31119, 46676, 31119, 7782, 1;

%p seq(seq( `if`(n<2, binomial(n,k), binomial(n,k) + n^(n-1)*binomial(n-2,k-1)), k=0..n), n=0..10); # _G. C. Greubel_, Jan 09 2020

%t Table[If[n <2, Binomial[n, m], Binomial[n, m] + n^(n - 1)*Binomial[n - 2, m - 1]], {n, 0, 10}, {m, 0, n}]; Flatten[%]

%o (PARI) T(n,k) = if(n<2, binomial(n,k), binomial(n,k) + n^(n-1)*binomial(n-2,k-1) ); \\ _G. C. Greubel_, Jan 09 2020

%o (Magma) T:= func< n,k | n lt 2 select Binomial(n,k) else Binomial(n,k) + n^(n-1)*Binomial(n-2,k-1) >;

%o [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jan 09 2020

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o if (n<2): return binomial(n,k)

%o else: return binomial(n,k) + n^(n-1)*binomial(n-2,k-1)

%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Jan 09 2020

%o (GAP)

%o T:= function(n,k)

%o if n<2 then return Binomial(n,k);

%o else return Binomial(n,k) + n^(n-1)*Binomial(n-2,k-1);

%o fi; end;

%o Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Jan 09 2020

%Y Cf. A028262.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Nov 04 2008

%E Edited by _G. C. Greubel_, Jan 09 2020