A129689 A007318 * A129688.

1, 1, 1, 3, 2, 1, 7, 5, 3, 1, 15, 12, 8, 4, 1, 31, 27, 20, 12, 5, 1, 63, 58, 47, 32, 17, 6, 1, 127, 121, 105, 79, 49, 23, 7, 1, 255, 248, 226, 184, 128, 72, 30, 8, 1, 511, 503, 474, 410, 312, 200, 102, 38, 9, 1

Offset: 1

Author

Gary W. Adamson, Apr 28 2007

Keywords

• nonn
• tabl
• easy

Comments

Row sums = A057711: (1, 2, 6, 16, 40, 96, ...). A129690 = A129688 * A007318.

Riordan array ( (1-2*x+2*x^2)/((1-x)*(1-2*x)), x/(1-x) ). - Peter Bala, Mar 21 2018

Examples

			First few rows of the triangle are:
   1;
   1,  1;
   3,  2,  1;
   7,  5,  3,  1;
  15, 12,  8,  4,  1;
  31, 27, 20, 12,  5,  1;
  63, 58, 47, 32, 17,  6,  1;
  ...
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..5151
• Peter Bala, A note on the diagonals of a proper Riordan Array

Crossrefs

Cf. A129688, A007318, A057711, A129690.

Programs

GAP

Flat(List([0..12],n->List([0..n],k->Binomial(n,k)+Sum([2..n],i->2^(i-1)*Binomial(n-i,n-k-i))))); # Muniru A Asiru, Mar 22 2018

Maple

C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if;
end proc:
for n from 0 to 12 do
   seq(C(n, n-k) + add(2^(i-1)*C(n-i, n-k-i), i = 2..n), k = 0..n)
end do; #  Peter Bala, Mar 21 2018

Mathematica

T[n_, k_] := Binomial[n, n-k] + Sum[2^(i-1) Binomial[n-i, n-k-i], {i, 2, n}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 10 2019 *)

Sage

# uses[riordan_array from A256893]
# After Peter Bala.
riordan_array((1-2*x+2*x^2)/((1-x)*(1-2*x)), x/(1-x), 8) # Peter Luschny, Mar 21 2018

Formula

Binomial transform of A129688. A007318 * A129688 as infinite lower triangular matrices.

From Peter Bala, Mar 21 2018: (Start)

T(n,k) = C(n, n-k) + Sum_{i = 2..n} 2^(i-1)*C(n-i, n-k-i), where C(n,k) = n!/(k!*(n-k)!) for 0 <= k <= n, otherwise 0.

Exp(x) * the e.g.f. for row n = the e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(7 + 5*x + 3*x^2/2! + x^3/3!) = 7 + 12*x + 20*x^2/2! + 32*x^3/3! + 49*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1-x) ).

(End)