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
KEYWORD
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