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%I #28 Sep 08 2022 08:46:01
%S 1,4,1,20,8,1,112,56,12,1,672,384,108,16,1,4224,2640,880,176,20,1,
%T 27456,18304,6864,1664,260,24,1,183040,128128,52416,14560,2800,360,28,
%U 1,1244672,905216,396032,121856,27200,4352,476,32,1,8599552,6449664,2976768
%N Triangle read by rows, T(n,k) for 0<=k<=n, generalizes the Motzkin lattice paths with weights of A003645.
%H Muniru A Asiru, <a href="/A201639/b201639.txt">Table of n, a(n) for n = 0..9044</a>
%F Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n>=1 T(n,k) = T(n-1,k-1)+4*T(n-1,k)+4*T(n-1,k+1).
%F G.f.: -(4*x+sqrt(1-8*x)-1)/((4*x^2-x)*y+sqrt(1-8*x)*x*y+8*x^2). - _Vladimir Kruchinin_, Apr 06 2018
%F T(n,k) = (k+1)*2^(n-k)*C(2*(n+1),n-k)/(n+1). - _Vladimir Kruchinin_, Apr 06 2018
%e [0] [1]
%e [1] [4, 1]
%e [2] [20, 8, 1]
%e [3] [112, 56, 12, 1]
%e [4] [672, 384, 108, 16, 1]
%e [5] [4224, 2640, 880, 176, 20, 1]
%e [6] [27456, 18304, 6864, 1664, 260, 24, 1]
%e [7] [183040, 128128, 52416, 14560, 2800, 360, 28, 1]
%t Flatten[Table[(k + 1) 2^(n - k) Binomial[2 (n + 1), n - k] / (n + 1), {n, 0, 11}, {k, 0, n}]] (* _Vincenzo Librandi_, Apr 07 2018 *)
%o (Sage)
%o def A201639_triangle(dim):
%o T = matrix(ZZ,dim,dim)
%o for n in range(dim): T[n,n] = 1
%o for n in (1..dim-1):
%o for k in (0..n-1):
%o T[n,k] = T[n-1,k-1]+4*T[n-1,k]+4*T[n-1,k+1]
%o return T
%o A201639_triangle(9)
%o (PARI) T(n,k) = (k+1)*2^(n-k)*binomial(2*(n+1),n-k)/(n+1);
%o tabl(nn) = for(n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Apr 07 2018
%o (Magma) /* As triangle */ [[(k+1)*2^(n-k)*Binomial(2*(n+1),n-k)/(n+1): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Apr 07 2018
%o (GAP) Flat(List([0..10], n->List([0..n],k->(k+1)*2^(n-k)*Binomial(2*(n+1),n-k)/(n+1)))); # _Muniru A Asiru_, Apr 07 2018
%Y Sum of row n is A194723(n+1).
%Y Cf. A003645.
%K nonn,tabl
%O 0,2
%A _Peter Luschny_, Sep 20 2012
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