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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.)