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Triangle, read by rows, defined by: T(n,k) = Sum_{j=0..n-k-1} T(j+k,k)*T(n-j,k+1) for n > k >= 0, with T(n,n) = n+1.
3

%I #13 Dec 29 2023 11:03:44

%S 1,2,2,10,6,3,74,42,12,4,706,414,108,20,5,8162,5058,1332,220,30,6,

%T 110410,72486,19908,3260,390,42,7,1708394,1182762,342252,57700,6750,

%U 630,56,8,29752066,21573054,6583788,1159700,138150,12474,952,72,9,576037442

%N Triangle, read by rows, defined by: T(n,k) = Sum_{j=0..n-k-1} T(j+k,k)*T(n-j,k+1) for n > k >= 0, with T(n,n) = n+1.

%C Column 0 is A000698, the number of shellings of an n-cube, divided by 2^n n!.

%C Column 1 is A115974, the number of Feynman diagrams of the proper self-energy at perturbative order n.

%H G. C. Greubel, <a href="/A127058/b127058.txt">Rows n = 0..15 of triangle, flattened</a>

%e Other recurrences exist, as shown by:

%e column 0 = A000698: T(n,0) = (2n+1)!! - Sum_{k=1..n} (2k-1)!!*T(n-k,0);

%e column 1 = A115974: T(n,1) = T(n+1,0) - Sum_{k=0..n-1} T(k,1)*T(n-k,0).

%e Illustrate the recurrence:

%e T(n,k) = Sum_{j=0..n-k-1) T(j+k,k)*T(n-j,k+1) (n > k >= 0)

%e at column k=1:

%e T(2,1) = T(1,1)*T(2,2) = 2*3 = 6;

%e T(3,1) = T(1,1)*T(3,2) + T(2,1)*T(2,2) = 2*12 + 6*3 = 42;

%e T(4,1) = T(1,1)*T(4,2) + T(2,1)*T(3,2) + T(3,1)*T(2,2) = 2*108 + 6*12 + 42*3 = 414;

%e at column k=2:

%e T(3,2) = T(2,2)*T(3,3) = 3*4 = 12;

%e T(4,2) = T(2,2)*T(4,3) + T(3,2)*T(3,3) = 3*20 + 12*4 = 108;

%e T(5,2) = T(2,2)*T(5,3) + T(3,2)*T(4,3) + T(4,2)*T(3,3) = 3*220 + 12*20 + 108*4 = 1332.

%e Triangle begins:

%e 1;

%e 2, 2;

%e 10, 6, 3;

%e 74, 42, 12, 4;

%e 706, 414, 108, 20, 5;

%e 8162, 5058, 1332, 220, 30, 6;

%e 110410, 72486, 19908, 3260, 390, 42, 7;

%e 1708394, 1182762, 342252, 57700, 6750, 630, 56, 8;

%e 29752066, 21573054, 6583788, 1159700, 138150, 12474, 952, 72, 9; ...

%t T[n_,k_]:= If[k==n, n+1, Sum[T[j+k,k]*T[n-j,k+1], {j,0,n-k-1}]];

%t Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 03 2019 *)

%o (PARI) {T(n,k)=if(n==k,n+1,sum(j=0,n-k-1,T(j+k,k)*T(n-j,k+1)))}

%o (Sage)

%o def T(n, k):

%o if (k==n): return n+1

%o else: return sum(T(j+k,k)*T(n-j,k+1) for j in (0..n-k-1))

%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jun 03 2019

%Y Columns: A000698, A115974, A127059.

%Y Row sums: A127060.

%Y Cf. A001147 ((2n-1)!!).

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Jan 04 2007