A135229 Triangle A000012(signed) * A103451 * A007318, read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 3, 6, 7, 4, 1, 1, 3, 9, 13, 11, 5, 1, 1, 4, 12, 22, 24, 16, 6, 1, 1, 4, 16, 34, 46, 40, 22, 7, 1, 1, 5, 20, 50, 80, 86, 62, 29, 8, 1

Offset: 0

Author

Gary W. Adamson, Nov 23 2007

Keywords

• nonn
• tabl

Comments

row sums = A005578 starting (1, 2, 3, 6, 11, 22, 43, 86, ...).

Examples

			First few rows of the triangle are:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,  1;
  1, 2,  4,  3,  1;
  1, 3,  6,  7,  4,  1;
  1, 3,  9, 13, 11,  5,  1;
  1, 4, 12, 22, 24, 16,  6, 1;
  1, 4, 16, 34, 46, 40, 22, 7, 1;
...
		

Links

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

Crossrefs

Cf. A005578, A007318, A103451.

Programs

Magma

function T(n,k)
  if k eq 0 then return 1;
  else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

Maple

T:= proc(n, k) option remember;
      if k=0 then 1
    else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

Mathematica

T[n_, k_]:= T[n, k]= If[k==0, 1, Sum[Binomial[n-1-2*j, k-1], {j, 0, Floor[(n-1)/2]}]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

PARI

T(n,k) = if(k==0, 1, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) ); \\ G. C. Greubel, Nov 20 2019

Sage

@CachedFunction
def T(n, k):
    if (k==0): 1
    else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

Formula

T(n,k) = A000012(signed) * A103451 * A007318 as infinite lower triangular matrices, where A000012(signed) = (1; -1,1; 1,-1,1; ...).

T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) = 1. - G. C. Greubel, Nov 20 2019

Extensions

Offset changed by G. C. Greubel, Nov 20 2019