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%I A173075 #24 Feb 12 2021 18:04:47
%S A173075 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 A173075 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 A173075 10,52,147,265,321,265,147,52,10,1
%N 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.
%C A173075 Rows two through six appear in the table on p. 8 of Getzler. Cf. also A167763. - _Tom Copeland_, Jan 22 2020
%C A173075 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 A173075 G. C. Greubel, <a href="/A173075/b173075.txt">Rows n = 0..100 of the triangle, flattened</a>
%H A173075 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 A173075 T(n, k) = binomial(n, k) - 1 + binomial(n-2, k-1) for 0 < k < n.
%F A173075 T(n, 0) = T(n, n) = 1.
%F A173075 From _G. C. Greubel_, Feb 12 2021: (Start)
%F A173075 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 A173075 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 A173075 Triangle begins:
%e A173075   1,
%e A173075   1,  1;
%e A173075   1,  2,  1;
%e A173075   1,  3,  3,   1;
%e A173075   1,  4,  7,   4,   1;
%e A173075   1,  5, 12,  12,   5,   1;
%e A173075   1,  6, 18,  25,  18,   6,   1;
%e A173075   1,  7, 25,  44,  44,  25,   7,   1;
%e A173075   1,  8, 33,  70,  89,  70,  33,   8,  1;
%e A173075   1,  9, 42, 104, 160, 160, 104,  42,  9,  1;
%e A173075   1, 10, 52, 147, 265, 321, 265, 147, 52, 10, 1;
%e A173075   ...
%e A173075 Row sums: {1, 2, 4, 8, 17, 36, 75, 154, 313, 632, 1271, ...}.
%t A173075 T[n_, m_]:= If[m==0 || m==n, 1, Binomial[n, m] - 1 + Binomial[n-2, m-1]];
%t A173075 Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten
%o A173075 (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 A173075 (Sage)
%o A173075 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 A173075 flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 12 2021
%o A173075 (Magma)
%o A173075 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 A173075 [T(n,k,1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 12 2021
%Y A173075 Cf. A132044 (q=0), this sequence (q=1), A173076 (q=2), A173077 (q=3).
%Y A173075 Cf. A132044 (p=0), this sequence (p=1), A173046 (p=2), A173047 (p=3).
%Y A173075 Cf. A167763.
%K A173075 nonn,tabl
%O A173075 0,5
%A A173075 _Roger L. Bagula_, Feb 09 2010

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