A110814 Inverse of a triangle of pyramidal numbers.

1, -3, 1, 7, -4, 1, -15, 11, -5, 1, 31, -26, 16, -6, 1, -63, 57, -42, 22, -7, 1, 127, -120, 99, -64, 29, -8, 1, -255, 247, -219, 163, -93, 37, -9, 1, 511, -502, 466, -382, 256, -130, 46, -10, 1, -1023, 1013, -968, 848, -638, 386, -176, 56, -11, 1, 2047, -2036, 1981, -1816, 1486, -1024, 562, -232, 67, -12, 1, -4095, 4083

Offset: 0

Author

Paul Barry, Aug 05 2005

Keywords

• easy
• sign
• tabl

Comments

Inverse of A110813. Array factors as (1/(1+2x),x)*(1/(1+x),x/(1+x)). Row sums are (-2)^n. Diagonal sums are (-1)^n*A008466(n+2). Signed version of A104709.

Examples

			Rows begin
    1;
   -3,   1;
    7,  -4,   1;
  -15,  11,  -5,   1;
   31, -26,  16,  -6,   1;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Programs

Maple

A110814_row := proc(n) add((-1)^k*add(binomial(n,n-i)*x^(n-k-1),i=0..k),k=0..n-1); coeffs(sort(%)) end; seq(print(A110814_row(n)),n=1..6); # Peter Luschny, Sep 29 2011

Mathematica

T[n_, k_] := Sum[(-2)^(n - j)*Binomial[j, k]*(-1)^(j - k), {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 19 2017 *)

PARI

for(n=0,10, for(k=0,n, print1(sum(j=0,n, (-2)^(n-j)*(-1)^(j-k)* binomial(j,k)), ", "))) \\ G. C. Greubel, Oct 19 2017

Formula

Number triangle T(n, k) = Sum_{j=0..n} (-2)^(n-j)*binomial(j, k)*(-1)^(j-k).

Riordan array (1/(1+3x+2x^2), x/(1+x)).

T(n,k) = -3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) + 2*T(n-2,k-1), T(0,0)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 30 2013