A135859 -id:A135859 - cp's OEIS frontend

A176120 Triangle read by rows: Sum_{j=0..k} binomial(n, j)binomial(k, j)j!.

1, 1, 2, 1, 3, 7, 1, 4, 13, 34, 1, 5, 21, 73, 209, 1, 6, 31, 136, 501, 1546, 1, 7, 43, 229, 1045, 4051, 13327, 1, 8, 57, 358, 1961, 9276, 37633, 130922, 1, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 1, 10, 91, 748, 5509, 36046, 207775, 1047376, 4596553, 17572114

Offset: 0

Roger L. Bagula, Apr 09 2010

The number of ways of placing any number k = 0, 1, ..., min(n,m) of non-attacking rooks on an n X m chessboard. - R. J. Mathar, Dec 19 2014

			Triangle begins
  1;
  1,  2;
  1,  3,   7;
  1,  4,  13,   34;
  1,  5,  21,   73,  209;
  1,  6,  31,  136,  501,  1546;
  1,  7,  43,  229, 1045,  4051,  13327;
  1,  8,  57,  358, 1961,  9276,  37633,  130922;
  1,  9,  73,  529, 3393, 19081,  93289,  394353,  1441729;
  1, 10,  91,  748, 5509, 36046, 207775, 1047376,  4596553, 17572114;
  1, 11, 111, 1021, 8501, 63591, 424051, 2501801, 12975561, 58941091, 234662231;

O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 46.

Seiichi Manyama, Rows n = 0..139, flattened
Wikipedia, Rook polynomial

Cf. A086885 (table without column 0), A129833 (row sums).

Cf. A000262, A002061, A002720, A082545, A135859, A343832.

A176120:=func< n,k| (&+[Factorial(j)*Binomial(n,j)*Binomial(k,j): j in [0..k]]) >;
[A176120(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 11 2022

A176120 := proc(i,j)
        add(binomial(i,k)*binomial(j,k)*k!,k=0..j) ;
end proc: # R. J. Mathar, Jul 28 2016

T[n_, m_]:= T[n,m]= Sum[Binomial[n, k]*Binomial[m, k]*k!, {k, 0, m}];
Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten

def A176120(n,k): return sum(factorial(j)*binomial(n,j)*binomial(k,j) for j in (0..k))
flatten([[A176120(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 11 2022

Sum_{k=0..n} T(n, k) = A129833(n).
T(n,m) = A088699(n, m). - Peter Bala, Aug 26 2013
T(n,m) = A086885(n, m). - R. J. Mathar, Dec 19 2014
From G. C. Greubel, Aug 11 2022: (Start)
T(n, k) = Hypergeometric2F1([-n, -k], [], 1).
T(2*n, n) = A082545(n).
T(2*n+1, n) = A343832(n).
T(n, n) = A002720(n).
T(n, n-1) = A000262(n), n >= 1.
T(n, 1) = A000027(n+1).
T(n, 2) = A002061(n+1).
T(n, 3) = A135859(n+1). (End)

A293985 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-x))/(1-x)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 13, 1, 4, 13, 34, 73, 1, 5, 21, 73, 209, 501, 1, 6, 31, 136, 501, 1546, 4051, 1, 7, 43, 229, 1045, 4051, 13327, 37633, 1, 8, 57, 358, 1961, 9276, 37633, 130922, 394353, 1, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 4596553

Offset: 0

Seiichi Manyama, Oct 21 2017

			Square array begins:
    1,    1,    1,    1,     1, ... A000012;
    1,    2,    3,    4,     5, ... A000027;
    3,    7,   13,   21,    31, ... A002061;
   13,   34,   73,  136,   229, ... A135859;
   73,  209,  501, 1045,  1961, ...
  501, 1546, 4051, 9276, 19081, ...
Antidiagonal rows begin as:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  7, 13;
  1, 4, 13, 34,  73;
  1, 5, 21, 73, 209, 501; - _G. C. Greubel_, Mar 09 2021

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Columns k=0..6 give: A000262, A002720, A000262(n+1), A052852(n+1), A062147, A062266, A062192.
Main diagonal gives A152059.
Similar table: A086885, A088699, A176120.

function t(n,k)
  if n eq 0 then return 1;
  else return Factorial(n-1)*(&+[(j+k)*t(n-j,k)/Factorial(n-j): j in [1..n]]);
  end if; return t;
end function;
[t(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021

t[n_, k_]:= t[n, k]= If[n==0, 1, (n-1)!*Sum[(j+k)*t[n-j,k]/(n-j)!, {j,n}]];
T[n_,k_]:= t[k,n-k]; Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)

@CachedFunction
def t(n,k): return 1 if n==0 else factorial(n-1)*sum( (j+k)*t(n-j,k)/factorial(n-j) for j in (1..n) )
def T(n,k): return t(k,n-k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} (j+k)*A(n-j,k)/(n-j)! for n > 0.
A(0,k) = 1, A(1,k) = k+1 and A(n,k) = (2*n-1+k)*A(n-1,k) - (n-1)*(n-2+k)*A(n-2,k) for n > 1.
From Seiichi Manyama, Jan 25 2025: (Start)
A(n,k) = n! * Sum_{j=0..n} binomial(n+k-1,j)/(n-j)!.
A(n,k) = n! * LaguerreL(n, k-1, -1). (End)

A093190 Array t read by antidiagonals: number of {112,212}-avoiding words.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 9, 4, 1, 8, 21, 16, 5, 1, 10, 39, 52, 25, 6, 1, 12, 63, 136, 105, 36, 7, 1, 14, 93, 292, 365, 186, 49, 8, 1, 16, 129, 544, 1045, 816, 301, 64, 9, 1, 18, 171, 916, 2505, 3006, 1603, 456, 81, 10, 1, 20, 219, 1432, 5225, 9276, 7315, 2864, 657, 100, 11

Offset: 1

Ralf Stephan, Apr 20 2004

t(k,n) = number of n-long k-ary words that simultaneously avoid the patterns 112 and 212.

			Square array begins as:
  1  1   1   1    1    1 ... 1*A000012;
  2  4   6   8   10   12 ... 2*A000027;
  3  9  21  39   63   93 ... 3*A002061;
  4 16  52 136  292  544 ... 4*A135859;
  5 25 105 365 1045 2505 ... ;
Antidiagonal rows begins as:
  1;
  1,  2;
  1,  4,  3;
  1,  6,  9,   4;
  1,  8, 21,  16,   5;
  1, 10, 39,  52,  25,  6;
  1, 12, 63, 136, 105, 36, 7;

G. C. Greubel, Antidiagonal rows n = 1..50, flattened
A. Burstein and T. Mansour, Words restricted by patterns with at most 2 distinct letters, arXiv:math/0110056 [math.CO], 2001.

Main diagonal is A052852.

Antidiagonal sums are in A084261 - 1.

[(&+[Factorial(j)*Binomial(k,j)*Binomial(n-k,j-1): j in [0..n-k+1]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 09 2021

T[n_, k_]:= Sum[j!*Binomial[k, j]*Binomial[n-k, j-1], {j,0,n-k+1}];
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)

t(n,k)=sum(j=0,k,j!*binomial(k,j)*binomial(n-1,j-1))

flatten([[ sum(factorial(j)*binomial(k,j)*binomial(n-k,j-1) for j in (0..n-k+1)) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 09 2021

t(n, k) = Sum{j=0..n} j!*C(n, j)*C(k-1, j-1). (square array)

T(n, k) = Sum_{j=0..n-k+1} j!*binomial(k,j)*binomial(n-k,j-1). (number triangle) - G. C. Greubel, Mar 09 2021

A135858 A128229^2 * A000012.

Original entry on oeis.org

1, 3, 1, 7, 5, 1, 13, 13, 7, 1, 21, 21, 21, 9, 1, 31, 31, 31, 31, 11, 1, 43, 43, 43, 43, 43, 13, 1, 57, 57, 57, 57, 57, 57, 15, 1, 73, 73, 73, 73, 73, 73, 73, 17, 1, 91, 91, 91, 91, 91, 91, 91, 91, 19, 1

Offset: 1

Gary W. Adamson, Dec 01 2007

Row sums = A135859: (1, 4, 13, 34, 73, 136, ...).

			First few rows of the triangle:
   1;
   3,  1;
   7,  5,  1;
  13, 13,  7,  1;
  21, 21, 21,  9,  1;
  31, 31, 31, 31, 11,  1;
  43, 43, 43, 43, 43, 13, 1;
  ...

Cf. A128229, A135859.

A128229^2 * A000012, where A128229 = a bidiagonal matrix with all 1's in the main diagonal and (1, 2, 3, ...) in the subdiagonal.

cp's OEIS Frontend

A176120 Triangle read by rows: Sum_{j=0..k} binomial(n, j)*binomial(k, j)*j!.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A293985 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-x))/(1-x)^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A093190 Array t read by antidiagonals: number of {112,212}-avoiding words.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A135858 A128229^2 * A000012.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A176120 Triangle read by rows: Sum_{j=0..k} binomial(n, j)binomial(k, j)j!.