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

Author

Roger L. Bagula, Feb 09 2010

Keywords

• nonn
• tabl

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

Examples

			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, ...}.
		

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.

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.

Programs

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

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

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

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)