%I #24 Feb 12 2021 18:04:47

%S 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,

%T 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,

%U 10,52,147,265,321,265,147,52,10,1

%N 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.

%C Rows two through six appear in the table on p. 8 of Getzler. Cf. also A167763. - _Tom Copeland_, Jan 22 2020

%C 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

%H G. C. Greubel, <a href="/A173075/b173075.txt">Rows n = 0..100 of the triangle, flattened</a>

%H E. Getzler, <a href="http://arxiv.org/abs/alg-geom/9612005">The semi-classical approximation for modular operads</a>, arXiv:alg-geom/9612005, 1996.

%F T(n, k) = binomial(n, k) - 1 + binomial(n-2, k-1) for 0 < k < n.

%F T(n, 0) = T(n, n) = 1.

%F From _G. C. Greubel_, Feb 12 2021: (Start)

%F 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.

%F Sum_{k=0..n} T(n, k, 1) = 2^(n-2) + 2^n - (n-1) - (5/4)*[n=0] -(1/2)*[n=1]. (End)

%e Triangle begins:

%e 1,

%e 1, 1;

%e 1, 2, 1;

%e 1, 3, 3, 1;

%e 1, 4, 7, 4, 1;

%e 1, 5, 12, 12, 5, 1;

%e 1, 6, 18, 25, 18, 6, 1;

%e 1, 7, 25, 44, 44, 25, 7, 1;

%e 1, 8, 33, 70, 89, 70, 33, 8, 1;

%e 1, 9, 42, 104, 160, 160, 104, 42, 9, 1;

%e 1, 10, 52, 147, 265, 321, 265, 147, 52, 10, 1;

%e ...

%e Row sums: {1, 2, 4, 8, 17, 36, 75, 154, 313, 632, 1271, ...}.

%t T[n_, m_]:= If[m==0 || m==n, 1, Binomial[n, m] - 1 + Binomial[n-2, m-1]];

%t Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten

%o (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

%o (Sage)

%o 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

%o flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 12 2021

%o (Magma)

%o 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 >;

%o [T(n,k,1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 12 2021

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

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

%Y Cf. A167763.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 09 2010