A257501 - OEIS

%I #16 Sep 08 2022 08:46:12

%S 1,4,1,14,8,1,48,44,12,1,165,208,90,16,1,572,910,544,152,20,1,2002,

%T 3808,2907,1120,230,24,1,7072,15504,14364,7084,2000,324,28,1,25194,

%U 62016,67298,40480,14625,3248,434,32,1,90440,245157,303600

%N Triangle, read by rows, T(n,k) = 2*k*C(2*(n+k),n-k)/(n+k).

%H Indranil Ghosh, <a href="/A257501/b257501.txt">Rows 1..125, flattened</a>

%F G.f.: 1/(1-(C(x)-1)^2*y/x)-1, where C(x) is g.f. of Catalan numbers (A000108).

%e 1;

%e 4, 1;

%e 14, 8, 1;

%e 48, 44, 12, 1;

%e 165, 208, 90, 16, 1;

%t Flatten@ Table[2 k Binomial[2 (n + k), n - k]/(n + k), {n, 10}, {k, n}] (* _Michael De Vlieger_, Apr 27 2015 *)

%o (Maxima)

%o T(n,k):=(2*k*binomial(2*(n+k),n-k))/(n+k);

%o (Magma) /* Us triangle */ [[(2*k*Binomial(2*(n+k), n-k))/(n+k): k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Apr 27 2015

%o (PARI)

%o tabl(nn) = {for (n=1, nn, for(k=1, n, print1(2*k*binomial(2*(n+k),n-k)/(n+k),", ");); print(););};

%o tabl(10); \\ _Indranil Ghosh_, Mar 04 2017

%o (Python)

%o import math

%o f=math.factorial

%o def C(n,r): return f(n)/f(r)/f(n-r)

%o i=1

%o for n in range(1,126):

%o ....for k in range(1,n+1):

%o ........print str(i)+" "+str(2*k*C(2*(n+k),n-k)/(n+k))

%o ........i+=1 # _Indranil Ghosh_, Mar 04 2017

%Y Cf. A000108.

%K nonn,tabl

%O 1,2

%A _Vladimir Kruchinin_, Apr 27 2015