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A259691 Triangle read by rows: T(n,k) number of arrangements of non-attacking rooks on an n X n right triangular board where the top rook is in row k (n >= 0, 1 <= k <= n+1). 4
1, 1, 1, 2, 2, 1, 5, 6, 3, 1, 15, 20, 12, 4, 1, 52, 74, 51, 20, 5, 1, 203, 302, 231, 104, 30, 6, 1, 877, 1348, 1116, 564, 185, 42, 7, 1, 4140, 6526, 5745, 3196, 1175, 300, 56, 8, 1, 21147, 34014, 31443, 18944, 7700, 2190, 455, 72, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Another version of A056857.

See Becker (1948/49) for precise definition.

The case of n=k+1 corresponds to the empty board where there is no top rook. - Andrew Howroyd, Jun 13 2017

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1274

H. W. Becker, Rooks and rhymes, Math. Mag., 22 (1948/49), 23-26. See Table II.

H. W. Becker, Rooks and rhymes, Math. Mag., 22 (1948/49), 23-26. [Annotated scanned copy]

FORMULA

T(n,n+1) = 1, T(n,k) = k*Sum_{i=0..n-k} binomial(n-k,i) * k^i * Bell(n-k-i) for k<=n. - Andrew Howroyd, Jun 13 2017

EXAMPLE

Triangle begins:

1,

1, 1,

2, 2, 1,

5, 6, 3, 1,

15, 20, 12, 4, 1,

52, 74, 51, 20, 5, 1,

203, 302, 231, 104, 30, 6, 1,

...

From Andrew Howroyd, Jun 13 2017: (Start)

For n=3 the 5 solutions with the top rook in row 1 are:

  x      x      x      x      x

  . .    . .    . .    . x    . x

  . . .  . . x  . x .  . . .  . . x

For n=3 the 6 solutions with the top rook in row 2 are:

  .      .      .      .      .      .

  x .    x .    x .    . x    . x    . x

  . . .  . x .  . . x  . . .  x . .  . . x

(End)

MATHEMATICA

T[n_, k_] := If[k>n, 1, k*Sum[Binomial[n-k, i]*k^i*BellB[n-k-i], {i, 0, n - k}]];

Table[T[n, k], {n, 0, 10}, {k, 1, n+1}] // Flatten (* Jean-Fran├žois Alcover, Jul 03 2018, after Andrew Howroyd *)

PROG

(PARI)

bell(n) = sum(k=0, n, stirling(n, k, 2));

T(n, k) = if(k>n, 1, k*sum(i=0, n-k, binomial(n-k, i) * k^i * bell(n-k-i)));

for(n=0, 6, for(k=1, n+1, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, Jun 13 2017

CROSSREFS

First column is A000110.

Row sums are A000110(n+1).

Cf. A056857, A259697, A108087.

Sequence in context: A118806 A171670 A124644 * A056857 A175579 A129100

Adjacent sequences:  A259688 A259689 A259690 * A259692 A259693 A259694

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jul 05 2015

EXTENSIONS

Name edited and terms a(28) and beyond from Andrew Howroyd, Jun 13 2017

STATUS

approved

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Last modified February 19 08:57 EST 2019. Contains 320309 sequences. (Running on oeis4.)