A292767 Square array read by antidiagonals downwards: T(k,n) = sum of the site-perimeters of words of length n >= 1 over an alphabet of size k >= 1.

4, 6, 10, 8, 28, 18, 10, 72, 74, 28, 12, 176, 281, 152, 40, 14, 416, 1020, 762, 270, 54, 16, 960, 3591, 3664, 1680, 436, 70, 18, 2176, 12366, 17120, 10050, 3238, 658, 88, 20, 4864, 41877, 78336, 58500, 23160, 5677, 944, 108, 22, 10752, 139968, 352768, 333750, 161352, 47236, 9276, 1302, 130

Offset

1,1

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, and Toufik Mansour, The Site-Perimeter of Words, Transactions on Combinatorics, Vol. 6 No. 2 (2017), pp. 37-48. ISSN (print): 2251-8657, ISSN (on-line): 2251-8665.

Formula

G.f. of row k: k*x*(36 + 12*k + (8 - 24*k - 8*k^2)*x + (2 - 5*k + 4*k^2 - k^3)*x^2)/(12*(1 - k*x)^2). - Andrew Howroyd, Oct 27 2018

Example

Array begins (rows are indexed by k = 1,2,3,4,...; columns by n = 1,2,3,4,...):
   4,   6,    8,    10,     12,      14,       16, ...
  10,  28,   72,   176,    416,     960,     2176, ...
  18,  74,  281,  1020,   3591,   12366,    41877, ...
  28, 152,  762,  3664,  17120,   78336,   352768, ...
  40, 270, 1680, 10050,  58500,  333750,  1875000, ...
  54, 436, 3238, 23160, 161352, 1102464,  7420896, ...
  70, 658, 5677, 47236, 383131, 3049270, 23916361, ...
  ...

Mathematica

RowGf[k_] := k x (36 + 12k + (8 - 24k - 8k^2) x + (2 - 5k + 4k^2 - k^3) x^2)/(12(1 - k x)^2);
T[k_, n_] := SeriesCoefficient[RowGf[k], {x, 0, n}];
Table[T[k - n + 1, n], {k, 1, 10}, {n, k, 1, -1}] // Flatten (* Jean-François Alcover, Aug 27 2019, from PARI *)

Prog

(PARI)
RowGf(k) = {k*x*(36 + 12*k + (8 - 24*k - 8*k^2)*x + (2 - 5*k + 4*k^2 - k^3)*x^2)/(12*(1 - k*x)^2)}
M(k, n)={Mat(vectorv(k, k, Vec(RowGf(k) + O(x*x^n))))}
{ M(10, 8) } \\ Andrew Howroyd, Oct 27 2018

Crossrefs

Row k=2 is A128135.

Sequence in context: A263483 A363700 A249982 * A117622 A188673 A365448

Adjacent sequences: A292764 A292765 A292766 * A292768 A292769 A292770

Keyword

nonn,tabl

Author

N. J. A. Sloane, Sep 27 2017

Extensions

Terms a(16) and beyond from Andrew Howroyd, Oct 27 2018

Status

approved