A173075 T(n,k) = binomial(n, k) - 1 + q^(floor(n/2))*binomial(n-2, k-1) for 0 < k < n with T(n,0) = T(n,n) = 1 and q = 1. Triangle read by rows.

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

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

Cf. A132044 (q=0), this sequence (q=1), A173076 (q=2), A173077 (q=3).

Cf. A132044 (p=0), this sequence (p=1), A173046 (p=2), A173047 (p=3).

Cf. A167763.

Sequence in context: A050447 A248601 A167172 * A166293 A094525 A130671

Adjacent sequences: A173072 A173073 A173074 * A173076 A173077 A173078

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 09 2010

Status

approved