A129689 - OEIS

%I #21 Sep 22 2025 16:00:52

%S 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,

%T 127,121,105,79,49,23,7,1,255,248,226,184,128,72,30,8,1,511,503,474,

%U 410,312,200,102,38,9,1

%N A007318 * A129688.

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

%C Riordan array ( (1-2*x+2*x^2)/((1-x)*(1-2*x)), x/(1-x) ). - _Peter Bala_, Mar 21 2018

%H Muniru A Asiru, <a href="/A129689/b129689.txt">Table of n, a(n) for n = 1..5151</a>

%H Peter Bala, <a href="/A081577/a081577.pdf">A note on the diagonals of a proper Riordan Array</a>

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

%F From _Peter Bala_, Mar 21 2018: (Start)

%F 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.

%F 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) ).

%F (End)

%e First few rows of the triangle are:

%e    1;

%e    1,  1;

%e    3,  2,  1;

%e    7,  5,  3,  1;

%e   15, 12,  8,  4,  1;

%e   31, 27, 20, 12,  5,  1;

%e   63, 58, 47, 32, 17,  6,  1;

%e   ...

%p C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if;

%p end proc:

%p for n from 0 to 12 do

%p    seq(C(n, n-k) + add(2^(i-1)*C(n-i, n-k-i), i = 2..n), k = 0..n)

%p end do; #  _Peter Bala_, Mar 21 2018

%t 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 *)

%o (SageMath) # uses[riordan_array from A256893]

%o # After _Peter Bala_.

%o riordan_array((1-2*x+2*x^2)/((1-x)*(1-2*x)), x/(1-x), 8) # _Peter Luschny_, Mar 21 2018

%o (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

%Y Cf. A129688, A007318, A057711, A129690.

%K nonn,tabl,easy

%O 1,4

%A _Gary W. Adamson_, Apr 28 2007