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 A073387 #54 Jan 10 2025 02:40:16
%S A073387 1,2,1,6,4,1,16,16,6,1,44,56,30,8,1,120,188,128,48,10,1,328,608,504,
%T A073387 240,70,12,1,896,1920,1872,1080,400,96,14,1,2448,5952,6672,4512,2020,
%U A073387 616,126,16,1,6688,18192,23040,17856,9352,3444,896,160,18,1
%N A073387 Convolution triangle of A002605(n) (generalized (2,2)-Fibonacci), n>=0.
%C A073387 The g.f. for the row polynomials P(n,x) = Sum_{m=0..n} T(n,m)*x^m is 1/(1-(2+x+2*z)*z). See Shapiro et al. reference and comment under A053121 for such convolution triangles.
%C A073387 T(n, k) is the number of words of length n over {0,1,2,3} having k letters 3 and avoiding runs of odd length for the letters 0,1. - _Milan Janjic_, Jan 14 2017
%H A073387 G. C. Greubel, <a href="/A073387/b073387.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A073387 Wolfdieter Lang, <a href="/A073387/a073387.txt">First 10 rows</a>.
%F A073387 T(n, k) = 2*(p(k-1, n-k)*(n-k+1)*T(n-k+1) + q(k-1, n-k)*(n-k+2)*T(n-k))/(k!*12^k), n >= k >= 1, with T(n) = T(n, k=0) = A002605(n), else 0; p(m, n) = Sum_{j=0..m} A(m, j)*n^(m-j) and q(m, n) = Sum_{j=0..m} B(m, j)*n^(m-j) with the number triangles A(k, m) = A073403(k, m) and B(k, m) = A073404(k, m).
%F A073387 T(n, k) = Sum_{j=0..floor((n-k)/2)} 2^(n-k-j)*binomial(n-j, k)*binomial(n-k-j, j) if n > k, else 0.
%F A073387 T(n, k) = ((n-k+1)*T(n, k-1) + 2*(n+k)*T(n-1, k-1))/(6*k), n >= k >= 1, T(n, 0) = A002605(n+1), else 0.
%F A073387 Sum_{k=0..n} T(n, k) = A007482(n).
%F A073387 G.f. for column m (without leading zeros): 1/(1-2*x*(1+x))^(m+1), m>=0.
%F A073387 T(n,k) = 2^(n-k)*binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -2) for n>=1. - _Peter Luschny_, Apr 25 2016
%F A073387 From _G. C. Greubel_, Oct 03 2022: (Start)
%F A073387 T(n, n-1) = A005843(n), n >= 1.
%F A073387 T(n, n-2) = 2*A005563(n-1), n >= 2.
%F A073387 T(n, n-3) = 4*A159920(n-1), n >= 2.
%F A073387 Sum_{k=0..n} (-1)^k*T(n, k) = A001045(n+1).
%F A073387 Sum_{k=0..floor(n/2)} T(n-k, k) = A015518(n+1). (End)
%e A073387 Lower triangular matrix, T(n,k), n >= k >= 0, else 0:
%e A073387 1;
%e A073387 2, 1;
%e A073387 6, 4, 1;
%e A073387 16, 16, 6, 1;
%e A073387 44, 56, 30, 8, 1;
%e A073387 120, 188, 128, 48, 10, 1;
%e A073387 328, 608, 504, 240, 70, 12, 1;
%e A073387 896, 1920, 1872, 1080, 400, 96, 14, 1;
%p A073387 T := (n,k) -> `if`(n=0,1,2^(n-k)*binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -2)): seq(seq(simplify(T(n,k)),k=0..n),n=0..10); # _Peter Luschny_, Apr 25 2016
%t A073387 T[n_, k_]:=T[n,k]=Sum[2^(n-k-j)*Binomial[n-j,k]*Binomial[n-k-j,j], {j,0,(n-k)/2}];
%t A073387 Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* _Jean-François Alcover_, Jun 04 2019 *)
%o A073387 (Magma)
%o A073387 A073387:= func< n,k | (&+[2^(n-k-j)*Binomial(n-j,k)*Binomial(n-k-j,j): j in [0..Floor((n-k)/2)]]) >;
%o A073387 [A073387(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 03 2022
%o A073387 (SageMath)
%o A073387 def A073387(n,k): return sum(2^(n-k-j)*binomial(n-j,k)*binomial(n-k-j,j) for j in range(((n-k+2)//2)))
%o A073387 flatten([[A073387(n,k) for k in range(n+1)] for n in range(12)]) # _G. C. Greubel_, Oct 03 2022
%Y A073387 Cf. A002605, A007482 (row sums), A053121, A073403, A073404.
%Y A073387 Cf. A001045, A005563, A015518, A159920.
%Y A073387 Columns: A002605 (k=0), A073388 (k=1), A073389 (k=2), A073390 (k=3), A073391 (k=4), A073392 (k=5), A073393 (k=6), A073394 (k=7), A073397 (k=8), A073398 (k=9).
%K A073387 nonn,easy,tabl
%O A073387 0,2
%A A073387 _Wolfdieter Lang_, Aug 02 2002