OFFSET
0,5
COMMENTS
Rows two through six appear in the table on p. 8 of Getzler. Cf. also A167763. - Tom Copeland, Jan 22 2020
The triangle sequences having the form T(n,k,p) = binomial(n, k) + p^n*binomial(n-2, k-1) - 1 have the row sums Sum_{k=0..n} T(n,k,p) = 2^(n-2)*p^n + 2^n - (n-1) - (5/4)*[n=0] -(p/2)*[n=1]. - G. C. Greubel, Feb 12 2021
LINKS
G. C. Greubel, Rows n = 0..100 of the triangle, flattened
E. Getzler, The semi-classical approximation for modular operads, arXiv:alg-geom/9612005, 1996.
FORMULA
T(n, k) = binomial(n, k) - 1 + binomial(n-2, k-1) for 0 < k < n.
T(n, 0) = T(n, n) = 1.
From G. C. Greubel, Feb 12 2021: (Start)
T(n, k, p) = binomial(n, k) + p^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and p = 1.
Sum_{k=0..n} T(n, k, 1) = 2^(n-2) + 2^n - (n-1) - (5/4)*[n=0] -(1/2)*[n=1]. (End)
EXAMPLE
Triangle begins:
1,
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 7, 4, 1;
1, 5, 12, 12, 5, 1;
1, 6, 18, 25, 18, 6, 1;
1, 7, 25, 44, 44, 25, 7, 1;
1, 8, 33, 70, 89, 70, 33, 8, 1;
1, 9, 42, 104, 160, 160, 104, 42, 9, 1;
1, 10, 52, 147, 265, 321, 265, 147, 52, 10, 1;
...
Row sums: {1, 2, 4, 8, 17, 36, 75, 154, 313, 632, 1271, ...}.
MATHEMATICA
T[n_, m_]:= If[m==0 || m==n, 1, Binomial[n, m] - 1 + Binomial[n-2, m-1]];
Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten
PROG
(PARI) T(n, k)={if(k<=0||k>=n, k==0||k==n, binomial(n, k) - 1 + binomial(n-2, k-1))} \\ Andrew Howroyd, Jan 22 2020
(Sage)
def T(n, k, p): return 1 if (k==0 or k==n) else binomial(n, k) + p^n*binomial(n-2, k-1) -1
flatten([[T(n, k, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 12 2021
(Magma)
T:= func< n, k, p | k eq 0 or k eq n select 1 else Binomial(n, k) + p^n*Binomial(n-2, k-1) -1 >;
[T(n, k, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 12 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 09 2010
STATUS
approved