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A142459 Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).

%I A142459 #22 Mar 12 2020 03:01:52
%S A142459 1,1,1,1,10,1,1,59,59,1,1,308,1062,308,1,1,1557,13562,13562,1557,1,1,
%T A142459 7806,148527,352612,148527,7806,1,1,39055,1500669,7108915,7108915,
%U A142459 1500669,39055,1,1,195304,14482396,123929944,241703110,123929944,14482396,195304,1
%N A142459 Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).
%C A142459 Row sums are A001813.
%C A142459 This is the case m=4 of a group of triangles defined by the recursion T(n,k,m) = (m*n-m*k+1) *T(n-1,k-1) + (m*k-m+1)* T(n - 1, k).
%H A142459 Michael De Vlieger, <a href="/A142459/b142459.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150, flattened).
%H A142459 Nick Early, <a href="https://arxiv.org/abs/1810.03246">Honeycomb tessellations and canonical bases for permutohedral blades</a>, arXiv:1810.03246 [math.CO], 2018.
%H A142459 G. Strasser, <a href="http://dx.doi.org/10.1017/S0305004110000538">Generalisation of the Euler adic</a>, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_4(n,k).
%F A142459 From _Peter Bala_, Feb 22 2011: (Start)
%F A142459 E.g.f: sqrt[u^2*(1-u)*exp(2*(u+1)*t)/(exp(4*u*t)-u*exp(4*t))] = Sum_{n >= 1} R(n,u)*t^n/n! = u + (u+u^2)*t + (u+10*u^2+u^3)*t^3/3! + ....
%F A142459 The row polynomials R(n,u) are related to the row polynomials P(n,u) of A186492 via R(n+1,u) = (-i)^n *(1-u)^n *P(n,i*(1+u)/(1-u)), where i = sqrt(-1). (End)
%e A142459 Triangle begins as:
%e A142459   1;
%e A142459   1,      1;
%e A142459   1,     10,        1;
%e A142459   1,     59,       59,         1;
%e A142459   1,    308,     1062,       308,         1;
%e A142459   1,   1557,    13562,     13562,      1557,         1;
%e A142459   1,   7806,   148527,    352612,    148527,      7806,        1;
%e A142459   1,  39055,  1500669,   7108915,   7108915,   1500669,    39055,      1;
%e A142459   1, 195304, 14482396, 123929944, 241703110, 123929944, 14482396, 195304, 1;
%p A142459 A142459 := proc(n, k) if n = k then 1; elif k > n or k < 1 then 0 ; else (4*n-4*k+1)*procname(n-1, k-1)+(4*k-3)*procname(n-1, k) ; end if; end proc:
%p A142459 seq(seq(A142459(n, k), k=1..n), n=1..10) ; # _R. J. Mathar_, May 11 2012
%t A142459 T[n_, 1]:= 1; T[n_, n_]:= 1; T[n_, k_]:= (4*n -4*k +1)*T[n-1, k-1] + (4*k - 3)*T[n-1, k]; Table[T[n, k], {n, 10}, {k, n}]//Flatten
%o A142459 (Sage)
%o A142459 @CachedFunction
%o A142459 def T(n, k):
%o A142459     if (k==1 or k==n): return 1
%o A142459     else: return (4*k-3)* T(n-1, k) + (4*(n-k)+1)*T(n-1, k-1)
%o A142459 [[T(n, k) for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, Mar 12 2020
%Y A142459 Cf. A001813, A186492.
%K A142459 nonn,tabl,easy
%O A142459 1,5
%A A142459 _Roger L. Bagula_, Sep 19 2008
%E A142459 Edited by the Assoc. Eds. of the OEIS, Mar 25 2010
%E A142459 Edited by _N. J. A. Sloane_, May 11 2013