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 A097710 #23 Feb 22 2024 09:09:05
%S A097710 1,1,1,2,3,1,7,13,7,1,41,88,61,15,1,397,951,781,257,31,1,6377,16691,
%T A097710 15566,6231,1041,63,1,171886,484490,500057,231721,48303,4161,127,1,
%U A097710 7892642,23701698,26604323,13843968,3406505,374127,16577,255,1
%N A097710 Lower triangular matrix T, read by rows, such that row (n) is formed from the sums of adjacent terms in row (n-1) of the matrix square T^2, with T(0,0)=1.
%C A097710 Column 0 is equal to sequence A008934, which is the number of tournament sequences.
%C A097710 This triangle has the same row sums and first column terms as in rows 2^n, for n>=0, of triangle A093654.
%H A097710 Paul D. Hanna, <a href="/A097710/b097710.txt">Table of n, a(n) for n = 0..495, of rows 0..30 of triangle in flattened form.</a>
%F A097710 T(n, k) = T^2(n-1, k-1) + T^2(n-1, k) for n>=1 and k>1, with T(n, 1) = T^2(n-1, 1) and T(n,n) = 1 for n>=0, where T^2 is the matrix square of this triangle T.
%F A097710 T(n, k) = Sum_{j=0..n-1} T(n-1, j)*(T(j, k-1) + T(j,k)), with T(n, 0) = Sum_{j=0..n-1} T(n-1,j)*T(j,0), and T(n, n) = 1.
%F A097710 T(n, 0) = A008934(n).
%F A097710 T(n, 1) = A097711(n).
%F A097710 Sum_{k=0..n} T(n, k) = A093657(n+1) (row sums).
%F A097710 From _G. C. Greubel_, Feb 21 2024: (Start)
%F A097710 T(n, n-1) = A000225(n).
%F A097710 Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)
%e A097710 Rows of this triangle T begin:
%e A097710 1;
%e A097710 1, 1;
%e A097710 2, 3, 1;
%e A097710 7, 13, 7, 1;
%e A097710 41, 88, 61, 15, 1;
%e A097710 397, 951, 781, 257, 31, 1;
%e A097710 6377, 16691, 15566, 6231, 1041, 63, 1;
%e A097710 171886, 484490, 500057, 231721, 48303, 4161, 127, 1;
%e A097710 Rows of T^2 begin:
%e A097710 1;
%e A097710 2, 1;
%e A097710 7, 6, 1;
%e A097710 41, 47, 14, 1;
%e A097710 397, 554, 227, 30, 1;
%e A097710 6377, 10314, 5252, 979, 62, 1;
%e A097710 171886, 312604, 187453, 44268, 4035, 126, 1;
%e A097710 7892642, 15809056, 10795267, 3048701, 357804, 16323, 254, 1;
%e A097710 The sums of adjacent terms in row (n) of T^2 forms row (n+1) of T:
%e A097710 T(5,0) = T^2(4,0) = 397;
%e A097710 T(5,1) = T^2(4,0) + T^2(4,1) = 397 + 554 = 951;
%e A097710 T(5,2) = T^2(4,1) + T^2(4,2) = 554 + 227 = 781.
%e A097710 Rows of matrix inverse T^(-1) begins:
%e A097710 1;
%e A097710 -1, 1;
%e A097710 1, -3, 1;
%e A097710 -1, 8, -7, 1;
%e A097710 1, -25, 44, -15, 1;
%e A097710 -1, 111, -346, 208, -31, 1;
%e A097710 1, -809, 4045, -3720, 912, -63, 1;
%e A097710 -1, 10360, -77351, 99776, -35136, 3840, -127, 1; ...
%e A097710 which is a signed version of A097712.
%t A097710 T[n_, k_] := T[n, k] = Which[n<0 || k>n, 0, n == k, 1, k == 0, Sum[T[n-1, j]*T[j, 0], {j, 0, n-1}], True, Sum[T[n-1, j]*T[j, k-1], {j, 0, n-1}] + Sum[T[n-1, j]*T[j, k], {j, 0, n-1}]]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 23 2016, adapted from PARI *)
%o A097710 (PARI) /* Using Recurrence relation: */
%o A097710 {T(n,k) = if(n<0||k>n, 0, if(n==k,1, if(k==0, sum(j=0,n-1, T(n-1,j)*T(j,0)), sum(j=0,n-1, T(n-1,j)*T(j,k-1)) + sum(j=0,n-1, T(n-1,j)*T(j,k));)))}
%o A097710 for(n=0,8, for(k=0,n, print1(T(n,k),", "));print(""))
%o A097710 (PARI) /* Faster: using Matrix generating method: */
%o A097710 {T(n,k) = my(M=matrix(2,2,r,c,if(r>=c,1))); for(i=1,n,
%o A097710 N=matrix(#M+1,#M+1,r,c, if(r>=c, if(r<=#M,M[r,c], if(c>1,(M^2)[r-1,c-1]) + if(c<=#M,(M^2)[r-1,c])) ));
%o A097710 M=N;); M[n+1,k+1]}
%o A097710 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print("")) \\ _Paul D. Hanna_, Nov 27 2016
%o A097710 (SageMath)
%o A097710 @CachedFunction
%o A097710 def T(n, k): # T = A097710
%o A097710 if n< 0 or k<0 or k>n: return 0
%o A097710 elif k==n: return 1
%o A097710 elif k==0: return sum(T(n-1,j)*T(j,0) for j in range(n))
%o A097710 else: return sum(T(n-1, j)*(T(j, k-1)+T(j,k)) for j in range(n))
%o A097710 flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Feb 21 2024
%Y A097710 Cf. A000007, A000225, A093654, A097712.
%Y A097710 Cf. A008934 (column k=0), A093657 (row sums), A097711 (column k=1).
%K A097710 nonn,tabl
%O A097710 0,4
%A A097710 _Paul D. Hanna_, Aug 22 2004