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

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

Offset: 1

Author

Roger L. Bagula, Sep 19 2008

Keywords

• nonn
• tabl
• easy

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).

Examples

			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;
		

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).

Crossrefs

Cf. A001813, A186492.

Programs

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

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

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)

Extensions

Edited by the Assoc. Eds. of the OEIS, Mar 25 2010

Edited by N. J. A. Sloane, May 11 2013