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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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: A104732 A132108 A210489 * A210552 A193376 A185095

Adjacent sequences:  A125172 A125173 A125174 * A125176 A125177 A125178

KEYWORD

nonn,tabl,easy

AUTHOR

Gary W. Adamson, Nov 22 2006

STATUS

approved

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Last modified July 10 03:36 EDT 2020. Contains 335570 sequences. (Running on oeis4.)