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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). 22
 1, 1, 1, 1, 10, 1, 1, 59, 59, 1, 1, 308, 1062, 308, 1, 1, 1557, 13562, 13562, 1557, 1, 1, 7806, 148527, 352612, 148527, 7806, 1, 1, 39055, 1500669, 7108915, 7108915, 1500669, 39055, 1, 1, 195304, 14482396, 123929944, 241703110, 123929944, 14482396, 195304, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums are A001813. 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). LINKS Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened). Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018. G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_4(n,k). FORMULA From Peter Bala, Feb 22 2011: (Start) 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! + .... 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) EXAMPLE Triangle begins as:   1;   1,      1;   1,     10,        1;   1,     59,       59,         1;   1,    308,     1062,       308,         1;   1,   1557,    13562,     13562,      1557,         1;   1,   7806,   148527,    352612,    148527,      7806,        1;   1,  39055,  1500669,   7108915,   7108915,   1500669,    39055,      1;   1, 195304, 14482396, 123929944, 241703110, 123929944, 14482396, 195304, 1; MAPLE 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: seq(seq(A142459(n, k), k=1..n), n=1..10) ; # R. J. Mathar, May 11 2012 MATHEMATICA 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 PROG (Sage) @CachedFunction def T(n, k):     if (k==1 or k==n): return 1     else: return (4*k-3)* T(n-1, k) + (4*(n-k)+1)*T(n-1, k-1) [[T(n, k) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Mar 12 2020 CROSSREFS Cf. A001813, A186492. Sequence in context: A174109 A171692 A152971 * A157641 A129274 A176021 Adjacent sequences:  A142456 A142457 A142458 * A142460 A142461 A142462 KEYWORD nonn,tabl,easy AUTHOR Roger L. Bagula, Sep 19 2008 EXTENSIONS Edited by the Assoc. Eds. of the OEIS, Mar 25 2010 Edited by N. J. A. Sloane, May 11 2013 STATUS approved

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Last modified April 8 11:56 EDT 2020. Contains 333314 sequences. (Running on oeis4.)