0,2

Column 0 is A000698, the number of shellings of an n-cube, divided by 2^n n!. Column 1 is A115974, the number of Feynman diagrams of the proper self-energy at perturbative order n.

Other recurrences exist, as shown by:

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

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

Illustrate the recurrence:

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

at column k=1:

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

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

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;

at column k=2:

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

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

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.

Triangle begins:

1;

2, 2;

10, 6, 3;

74, 42, 12, 4;

706, 414, 108, 20, 5;

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

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

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

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

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

Columns: A000698, A115974, A127059; row sums: A127060; A001147 ((2n-1)!!).

Sequence in context: A163937 A083457 A163808 this_sequence A094359 A129898 A135996

Adjacent sequences: A127055 A127056 A127057 this_sequence A127059 A127060 A127061

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 04 2007