# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a119304 Showing 1-1 of 1 %I A119304 #20 Jun 22 2024 07:59:07 %S A119304 1,4,1,28,7,1,220,55,10,1,1820,455,91,13,1,15504,3876,816,136,16,1, %T A119304 134596,33649,7315,1330,190,19,1,1184040,296010,65780,12650,2024,253, %U A119304 22,1,10518300,2629575,593775,118755,20475,2925,325,25,1,94143280,23535820 %N A119304 Triangle read by rows: T(n,k) = binomial(4n-k,n-k), 0 <= k <= n. %H A119304 Indranil Ghosh, Rows 0..125, flattened %F A119304 Riordan array (1/(1-4f(x)),f(x)) where f(x)(1-f(x))^3 = x. %F A119304 From _Peter Bala_, Jun 04 2024: (Start) %F A119304 'Horizontal' recurrence equation: T(n, 0) = binomial(4*n,n) and for k >= 1, T(n, k) = Sum_{i = 1..n+1-k} i*(i+1)/2 * T(n-1, k-2+i). %F A119304 T(n, k) = Sum_{j = 0..n} binomial(n+j-1, j)*binomial(3*n-k-j, 2*n). (End) %e A119304 Triangle begins %e A119304 1; %e A119304 4, 1; %e A119304 28, 7, 1; %e A119304 220, 55, 10, 1; %e A119304 1820, 455, 91, 13, 1; %e A119304 15504, 3876, 816, 136, 16, 1; %e A119304 134596, 33649, 7315, 1330, 190, 19, 1; %t A119304 Flatten[Table[Binomial[4n-k,n-k],{n,0,9},{k,0,n}]] (* _Indranil Ghosh_, Feb 26 2017 *) %o A119304 (PARI) tabl(nn) = {for (n=0,nn,for (k=0,n,print1(binomial(4*n-k,n-k),", ");); print(););} \\ _Indranil Ghosh_, Feb 26 2017 %o A119304 (Python) %o A119304 from sympy import binomial %o A119304 i=0 %o A119304 for n in range(12): %o A119304 for k in range(n+1): %o A119304 print(str(i)+" "+str(binomial(4*n-k,n-k))) %o A119304 i+=1 # _Indranil Ghosh_, Feb 26 2017 %Y A119304 Rows sums are A052203. First column is A005810. Inverse of A119305. %Y A119304 Cf. A092392, A119301. %K A119304 easy,nonn,tabl %O A119304 0,2 %A A119304 _Paul Barry_, May 13 2006 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE