A146990 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.

1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 68, 134, 68, 1, 1, 630, 1885, 1885, 630, 1, 1, 7782, 31119, 46676, 31119, 7782, 1, 1, 117656, 588266, 1176525, 1176525, 588266, 117656, 1, 1, 2097160, 12582940, 31457336, 41943110, 31457336, 12582940, 2097160, 1

Offset: 0

Author

Roger L. Bagula, Nov 04 2008

Keywords

• nonn
• tabl

Comments

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

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    4,     1;
  1,   12,    12,     1;
  1,   68,   134,    68,     1;
  1,  630,  1885,  1885,   630,    1;
  1, 7782, 31119, 46676, 31119, 7782, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A028262.

Programs

GAP

T:= function(n,k)
    if n<2 then return Binomial(n,k);
    else return Binomial(n,k) + n^(n-1)*Binomial(n-2,k-1);
    fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jan 09 2020

Magma

T:= func< n,k | n lt 2 select Binomial(n,k) else Binomial(n,k) + n^(n-1)*Binomial(n-2,k-1) >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 09 2020

Maple

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

Mathematica

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[%]

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

Sage

@CachedFunction
def T(n, k):
    if (n<2): return binomial(n,k)
    else: return binomial(n,k) + n^(n-1)*binomial(n-2,k-1)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 09 2020

Formula

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

Extensions

Edited by G. C. Greubel, Jan 09 2020