A236471 - OEIS

%I #16 Apr 19 2018 03:45:36

%S 1,1,1,2,4,1,4,13,7,1,8,38,33,10,1,16,104,129,62,13,1,32,272,450,304,

%T 100,16,1,64,688,1452,1289,590,147,19,1,128,1696,4424,4942,2945,1014,

%U 203,22,1,256,4096,12896,17584,13073,5823,1603,268,25,1,512,9728

%N Riordan array ((1-x)/(1-2*x), x(1-x)/(1-2*x)^2).

%C Row sums are A052936(n).

%C Diagonal sums are A121449(n).

%C The triangle T'(n,k) = T(n,k)*(-1)^(n+k) is the inverse of the Riordan array in A090285.

%H G. C. Greubel, <a href="/A236471/b236471.txt">Table of n, a(n) for the first 100 rows, flattened</a>

%F T(n,0) = A011782(n), T(n,1) = A049611(n), T(n,n) = A000012(n) = 1, T(n+1,n) = A016777(n), T(n+2,n) = A062708(n+1).

%F G.f.: (2*x^2-3*x+1)/((x^2-x)*y+4*x^2-4*x+1). - _Vladimir Kruchinin_, Apr 21 2015

%F T(n,k) = Sum_{m=0..n} C(m+k,2*k)*C(n-1,n-m). - _Vladimir Kruchinin_, Apr 21 2015

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 4, 1;

%e 4, 13, 7, 1;

%e 8, 38, 33, 10, 1;

%e 16, 104, 129, 62, 13, 1;

%e 32, 272, 450, 304, 100, 16, 1;

%e 64, 688, 1452, 1289, 590, 147, 19, 1;

%t CoefficientList[CoefficientList[Series[(2*x^2-3*x+1)/((x^2-x)*y +4*x^2 - 4*x+1), {x,0,20}, {y,0,20}], x], y]//Flatten (* _G. C. Greubel_, Apr 19 2018 *)

%o (Maxima)

%o T(n,k):=sum(binomial(m+k,2*k)*binomial(n-1,n-m),m,0,n); /* _Vladimir Kruchinin_, Apr 21 2015 */

%o (PARI) for(n=0,20, for(k=0,n, print1(sum(m=0, n, binomial(m+k,2*k)* binomial(n-1,n-m)), ", "))) \\ _G. C. Greubel_, Apr 19 2018

%Y Cf. A011782, A049611, A000012, A016777, A062708.

%K nonn,tabl

%O 0,4

%A _Philippe Deléham_, Jan 26 2014