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 A127058 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
 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, 576037442 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. LINKS G. C. Greubel, Rows n = 0..15 of triangle, flattened EXAMPLE 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; ... MATHEMATICA T[n_, k_]:= If[k==n, n+1, Sum[T[j+k, k]*T[n-j, k+1], {j, 0, n-k-1}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 03 2019 *) PROG (PARI) {T(n, k)=if(n==k, n+1, sum(j=0, n-k-1, T(j+k, k)*T(n-j, k+1)))} (Sage) def T(n, k): if (k==n): return n+1 else: return sum(T(j+k, k)*T(n-j, k+1) for j in (0..n-k-1)) [[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 03 2019 CROSSREFS Columns: A000698, A115974, A127059. Row sums: A127060. Cf. A001147 ((2n-1)!!). Sequence in context: A083457 A163808 A223126 * A242002 A094359 A293060 Adjacent sequences: A127055 A127056 A127057 * A127059 A127060 A127061 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Jan 04 2007 STATUS approved

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Last modified January 29 11:19 EST 2023. Contains 359922 sequences. (Running on oeis4.)