This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A129689 #20 Mar 28 2020 06:37:00
%S A129689 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 A129689 127,121,105,79,49,23,7,1,255,248,226,184,128,72,30,8,1,511,503,474,
%U A129689 410,312,200,102,38,9,1
%N A129689 A007318 * A129688.
%C A129689 Row sums = A057711: (1, 2, 6, 16, 40, 96, ...). A129690 = A129688 * A007318.
%C A129689 Riordan array ( (1-2*x+2*x^2)/((1-x)*(1-2*x)), x/(1-x) ). - _Peter Bala_, Mar 21 2018
%H A129689 Muniru A Asiru, <a href="/A129689/b129689.txt">Table of n, a(n) for n = 1..5151</a>
%H A129689 Peter Bala, <a href="/A081577/a081577.pdf">A note on the diagonals of a proper Riordan Array</a>
%F A129689 Binomial transform of A129688. A007318 * A129688 as infinite lower triangular matrices.
%F A129689 From _Peter Bala_, Mar 21 2018: (Start)
%F A129689 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 A129689 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 A129689 (End)
%e A129689 First few rows of the triangle are:
%e A129689 1;
%e A129689 1, 1;
%e A129689 3, 2, 1;
%e A129689 7, 5, 3, 1;
%e A129689 15, 12, 8, 4, 1;
%e A129689 31, 27, 20, 12, 5, 1;
%e A129689 63, 58, 47, 32, 17, 6, 1;
%e A129689 ...
%p A129689 C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if;
%p A129689 end proc:
%p A129689 for n from 0 to 12 do
%p A129689 seq(C(n, n-k) + add(2^(i-1)*C(n-i, n-k-i), i = 2..n), k = 0..n)
%p A129689 end do; # _Peter Bala_, Mar 21 2018
%t A129689 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 A129689 (Sage) # uses[riordan_array from A256893]
%o A129689 # After _Peter Bala_.
%o A129689 riordan_array((1-2*x+2*x^2)/((1-x)*(1-2*x)), x/(1-x), 8) # _Peter Luschny_, Mar 21 2018
%o A129689 (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 A129689 Cf. A129688, A007318, A057711, A129690.
%K A129689 nonn,tabl,easy
%O A129689 1,4
%A A129689 _Gary W. Adamson_, Apr 28 2007