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%I #26 Feb 26 2024 11:01:22
%S 1,1,3,1,9,12,1,18,66,55,1,30,210,455,273,1,45,510,2040,3060,1428,1,
%T 63,1050,6650,17955,20349,7752,1,84,1932,17710,74382,148764,134596,
%U 43263,1,108,3276,40950,245700,753480,1184040,888030,246675,1,135,5220,85260,690606,2992626,7125300,9161100,5852925,1430715
%N Triangle of Generalized Runyon numbers R_{n,k}^(3) read by rows.
%C The Runyon numbers R_{n,k}^(1) are A001263, R_{n,k}^(2) are A108767.
%C Row sums are in A002293.
%D Chunwei Song, The Generalized Schroeder Theory, El. J. Combin. 12 (2005) #R53
%H G. C. Greubel, <a href="/A173020/b173020.txt">Rows n = 1..100 of the triangle, flattened</a>
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020. See Fig. 6.
%H Tad White, <a href="https://arxiv.org/abs/2401.01462">Quota Trees</a>, arXiv:2401.01462 [math.CO], 2024. See p. 20.
%F T(n, k) = R(n,k,3) with R(n,k,m)= binomial(n,k)*binomial(m*n,k-1)/n, 1<=k<=n.
%F T(n, n) = A001764(n).
%F T(n, n-1) = A003408(n-2).
%F T(n, 2) = A045943(n-1).
%F T(n, 3) = n*(n-1)*(n-2)*(3*n-1)/4 = 3*A052149(n-1).
%F O.g.f. is series reversion with respect to x of x/((1+x)*(1+x*u)^3). - _Peter Bala_, Sep 12 2012
%F Sum_{k=1..n} T(n, k, 3) = binomial(4*n, n)/(3*n+1) = A002293(n). - _G. C. Greubel_, Feb 20 2021
%F n-th row polynomial = x * hypergeom([1 - n, -3*n], [2], x). - _Peter Bala_, Aug 30 2023
%e The triangle starts in row n=1 as
%e 1;
%e 1, 3;
%e 1, 9, 12;
%e 1, 18, 66, 55;
%e 1, 30, 210, 455, 273;
%e 1, 45, 510, 2040, 3060, 1428;
%e 1, 63, 1050, 6650, 17955, 20349, 7752;
%e 1, 84, 1932, 17710, 74382, 148764, 134596, 43263;
%t T[n_, k_, m_]:= Binomial[n, k]*Binomial[m*n, k-1]/n;
%t Table[T[n, k, 3], {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Feb 20 2021 *)
%o (Sage)
%o def A173020(n,k,m): return binomial(n,k)*binomial(m*n,k-1)/n
%o flatten([[A173020(n,k,3) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 20 2021
%o (Magma)
%o A173020:= func< n,k,m | Binomial(n,k)*Binomial(m*n,k-1)/n >;
%o [A173020(n,k,3): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Feb 20 2021
%Y Cf. A010054 (m=0), A001263 (m=1), A108767 (m=2), this sequence (m=3).
%Y Cf. A001764, A002293, A003408, A045943, A052149.
%K easy,nonn,tabl
%O 1,3
%A _R. J. Mathar_, Nov 08 2010
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