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 A128545 #17 Jun 02 2020 01:15:43 %S A128545 1,1,1,1,2,1,1,3,3,1,1,5,8,5,1,1,7,18,18,7,1,1,11,39,58,39,11,1,1,15, %T A128545 75,155,155,75,15,1,1,22,141,383,526,383,141,22,1,1,30,251,867,1555, %U A128545 1555,867,251,30,1,1,42,433,1860,4192,5448,4192,1860,433,42,1 %N A128545 Triangle, read by rows, where T(n,k) is the coefficient of q^(n*k) in the q-binomial coefficient [2*n, n] for n >= k >= 0. %C A128545 Variant of A047812 (Parker's partition triangle). %C A128545 Column 1 equals the number of partitions of n: A000041(n) is the coefficient of q^n in the central q-binomial coefficient [2*n, n] for n > 0. %H A128545 Paul D. Hanna, <a href="/A128545/b128545.txt">Rows n = 0..45, flattened.</a> %F A128545 Row sums equal the row sums of triangle A123610: A123611(n) = 2*A047996(2*n,n) = 2*A003239(n) for n > 0, where A047996 is the triangle of circular binomial coefficients and A003239(n) = number of rooted planar trees with n non-root nodes. %e A128545 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins: %e A128545 1; %e A128545 1, 1; %e A128545 1, 2, 1; %e A128545 1, 3, 3, 1; %e A128545 1, 5, 8, 5, 1; %e A128545 1, 7, 18, 18, 7, 1; %e A128545 1, 11, 39, 58, 39, 11, 1; %e A128545 1, 15, 75, 155, 155, 75, 15, 1; %e A128545 1, 22, 141, 383, 526, 383, 141, 22, 1; %e A128545 1, 30, 251, 867, 1555, 1555, 867, 251, 30, 1; %e A128545 1, 42, 433, 1860, 4192, 5448, 4192, 1860, 433, 42, 1; %e A128545 ... %o A128545 (PARI) T(n,k)=if(n<k || k<0,0,if(n==0,1,polcoeff(prod(j=n+1,2*n,1-q^j)/prod(j=1,n,1-q^j),n*k,q))) %Y A128545 Cf. A003239, A047812 (variant), A047996, A123610, A123611 (row sums). %Y A128545 Cf. A000041 (column 1), A128552 (column 2), A128553 (column 3), A128554 (column 4). %K A128545 nonn,tabl %O A128545 0,5 %A A128545 _Paul D. Hanna_, Mar 10 2007