A125175 Triangle T(n,k) = |A053123(n/2+k/2,k)| for even n+k, T(n,k)= A082985((n+k-1)/2,k) for odd n+k; read by rows, 0<=k<=n.

1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 10, 7, 5, 1, 6, 14, 20, 9, 6, 1, 7, 21, 30, 35, 11, 7, 1, 8, 27, 56, 55, 56, 13, 8, 1, 9, 36, 77, 126, 91, 84, 15, 9, 1, 10, 44, 120, 182, 252, 140, 120, 17, 10, 1, 11, 55, 156, 330, 378, 462, 204, 165, 19, 11

Offset

0,3

Links

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

Formula

T(n,k) = binomial(n+1,k) if n+k even. T(n,k) = binomial(n-1,k)*(n+k)/(n-k) if n+k odd. - R. J. Mathar, Sep 08 2013

Example

First few rows of the triangle are:
  1;
  1, 2;
  1, 3,  3;
  1, 4,  5,  4;
  1, 5, 10,  7,   5;
  1, 6, 14, 20,   9,  6;
  1, 7, 21, 30,  35, 11,  7;
  1, 8, 27, 56,  55, 56, 13,  8;
  1, 9, 36, 77, 126, 91, 84, 15, 9; ...

Maple

A125175 := proc(n, k)
        if type(n+k, 'even') then
                binomial(n+1, k) ;
        else
                binomial(n-1, k)*(n+k)/(n-k) ;
        end if;
end proc: # R. J. Mathar, Sep 08 2013

Mathematica

Table[If[EvenQ[n+k], Binomial[n+1, k], Binomial[n-1, k]*(n+k)/(n-k)], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 05 2019 *)

Prog

(PARI) {T(n, k) = if((n+k)%2==0, binomial(n+1, k), binomial(n-1, k)* (n+k)/(n-k))}; \\ G. C. Greubel, Jun 05 2019
(Magma) [[ k eq n select n+1 else (n+k mod 2) eq 0 select Binomial(n+1, k) else Binomial(n-1, k)*(n+k)/(n-k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 05 2019
(Sage)
def T(n, k):
    if (mod(n+k, 2)==0): return binomial(n+1, k)
    else: return binomial(n-1, k)* (n+k)/(n-k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 05 2019

Crossrefs

Cf. A053123, A082985, A125176 (row sums).

Sequence in context: A377000 A210489 A344821 * A210552 A193376 A185095

Adjacent sequences: A125172 A125173 A125174 * A125176 A125177 A125178

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Nov 22 2006

Status

approved