A157898 Triangle read by rows: inverse binomial transform of A059576.

1, 0, 1, 1, 1, 2, 0, 2, 2, 4, 1, 2, 6, 4, 8, 0, 3, 6, 16, 8, 16, 1, 3, 12, 16, 40, 16, 32, 0, 4, 12, 40, 40, 96, 32, 64, 1, 4, 20, 40, 120, 96, 224, 64, 128, 0, 5, 20, 80, 120, 336, 224, 512, 128, 256

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Mar 08 2009

Keywords

• nonn
• tabl
• easy

Comments

The inverse binomial transform of the triangle A059576 is given by multiplying the triangle with A130595 from the left.

Examples

			First few rows of the triangle =
  1;
  0, 1;
  1, 1,  2;
  0, 2,  2,  4;
  1, 2,  6,  4,   8;
  0, 3,  6, 16,   8,  16;
  1, 3, 12, 16,  40,  16,  32;
  0, 4, 12, 40,  40,  96,  32,  64;
  1, 4, 20, 40, 120,  96, 224,  64, 128;
  0, 5, 20, 80, 120, 336, 224, 512, 128, 256;
  ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A059576, A097076 (row sums), A130595.

Cf. A004526, A008805, A011782, A058187, A059841, A189976.

Programs

Magma

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
function t(n, k) // t = A059576
  if k eq 0 or k eq n then return A011782(n);
  else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);
  end if; return t;
end function;
A157898:= func< n,k | (&+[(-1)^(n-j)*Binomial(n,j)*t(j,k): j in [k..n]]) >;
[A157898(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 03 2022

Maple

A059576 := proc (n, k)
    if n = 0 then
        return 1;
    end if;
    if k <= n and k >= 0 then
        add((-1)^j*2^(n-j-1)*binomial(k, j)*binomial(n-j, k), j = 0 .. min(k, n-k))
    else
        0 ;
    end if
end proc:
A157898 := proc(n,k)
    add ( A130595(n,j)*A059576(j,k),j=k..n) ;
end proc: # R. J. Mathar, Feb 13 2013

Mathematica

t[n_, k_]:= t[n, k]= If[k==0 || k==n, 2^(n-1) +Boole[n==0]/2, 2*t[n-1, k-1] + 2*t[n-1, k] -(2 -Boole[n==2])*t[n-2, k-1]]; (* t= A059576 *)
A157898[n_, k_]:= A157898[n, k]= Sum[(-1)^(n-j)*Binomial[n,j]*t[j,k], {j,k,n}];
Table[A157898[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 03 2022 *)

SageMath

@CachedFunction
def t(n, k): # t = A059576
    if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782
    else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)
def A157898(n,k): return sum((-1)^(n-j)*binomial(n,j)*t(j,k) for j in (k..n))
flatten([[A157898(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 03 2022

Formula

Sum_{k=0..n} T(n, k) = A097076(n+1).

From G. C. Greubel, Sep 03 2022: (Start)

T(n, k) = Sum_{j=k..n} (-1)^(n-j)*binomial(n,j)*A059576(j,k).

T(n, 0) = A059841(n).

T(n, 1) = A004526(n-1).

T(n, 2) = 2*A008805(n-2).

T(n, 3) = 4*A058187(n-3).

T(n, 4) = 8*A189976(n+4).

T(n, n) = A011782(n).

T(n, n-1) = A011782(n) - [n==0]. (End)