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 A382629 #20 Apr 02 2025 04:17:30
%S A382629 1,3,0,7,4,0,15,35,5,0,31,203,115,6,0,63,994,1428,315,7,0,127,4470,
%T A382629 13421,7450,783,8,0,255,19185,108156,121314,32865,1839,9,0,511,80161,
%U A382629 793704,1593902,870191,130665,4171,10,0,1023,329648,5483093,18269658,17591035,5383906,485166,9251,11,0
%N A382629 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = (n-k)*T(n-1,k-1) + 2*(k+1)*T(n-1,k) + A102365(n,k) with T(n,k) = 0 if k < 0 or k > n.
%H A382629 Seiichi Manyama, <a href="/A382629/b382629.txt">Rows n = 0..139, flattened</a>
%F A382629 (2/3)^n * Sum_{k=0..n} T(n,k)/2^k = A098830(n).
%e A382629 Triangle begins:
%e A382629 1;
%e A382629 3, 0;
%e A382629 7, 4, 0;
%e A382629 15, 35, 5, 0;
%e A382629 31, 203, 115, 6, 0;
%e A382629 63, 994, 1428, 315, 7, 0;
%e A382629 127, 4470, 13421, 7450, 783, 8, 0;
%e A382629 255, 19185, 108156, 121314, 32865, 1839, 9, 0;
%e A382629 511, 80161, 793704, 1593902, 870191, 130665, 4171, 10, 0;
%e A382629 ...
%o A382629 (PARI) a102365(n, k) = if(k==0, 1, if(n<k, 0, (n-k)*a102365(n-1, k-1)+(2*k+1)*a102365(n-1, k)));
%o A382629 T(n, k) = if(k<0 || k>n, 0, (n-k)*T(n-1, k-1)+2*(k+1)*T(n-1, k)+a102365(n, k));
%Y A382629 Row sums give A180875.
%Y A382629 Column k=0 gives A126646.
%Y A382629 Cf. A098830, A102365.
%K A382629 nonn,tabl
%O A382629 0,2
%A A382629 _Seiichi Manyama_, Apr 01 2025