cp's OEIS Frontend

A110197 Number triangle of sums of squared binomial coefficients.

%I A110197 #20 Mar 21 2025 09:41:00
%S A110197 1,2,1,3,5,1,4,14,10,1,5,30,46,17,1,6,55,146,117,26,1,7,91,371,517,
%T A110197 251,37,1,8,140,812,1742,1476,478,50,1,9,204,1596,4878,6376,3614,834,
%U A110197 65,1,10,285,2892,11934,22252,19490,7890,1361,82,1,11,385,4917,26334,66352,82994,51990,15761,2107,101,1
%N A110197 Number triangle of sums of squared binomial coefficients.
%C A110197 Alternatively, number square T(n,k) = Sum_{i=0..n} binomial(i+k,k)^2 read by antidiagonals.
%F A110197 T(n,k) = Sum_{i=0..n-k} binomial(i+k,k)^2.
%F A110197 G.f.: 1/((1-x)*sqrt(x^2*y^2-2*x^2*y-2*x*y+x^2-2*x+1)). - _Vladimir Kruchinin_, Mar 20 2025
%e A110197 Rows start:
%e A110197   1;
%e A110197   2,   1;
%e A110197   3,   5,   1;
%e A110197   4,  14,  10,   1;
%e A110197   5,  30,  46,  17,   1;
%e A110197   6,  55, 146, 117,  26,   1;
%e A110197   ...
%o A110197 (PARI) T(n,k) = sum(i=0, n-k, binomial(i+k,k)^2);
%o A110197 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print();); \\ _Michel Marcus_, Dec 03 2016
%Y A110197 Columns include A000027, A000330, A024166, A086020, A086023, A086025.
%Y A110197 Row sums are A006134.
%Y A110197 Antidiagonal sums are A110198.
%Y A110197 T(2n,n) gives A112029.
%K A110197 easy,nonn,tabl
%O A110197 0,2
%A A110197 _Paul Barry_, Jul 15 2005