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
Examples
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;
References
- O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 46.
Links
- Seiichi Manyama, Rows n = 0..139, flattened
- Wikipedia, Rook polynomial
Crossrefs
Programs
-
Magma
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
-
Maple
A176120 := proc(i,j) add(binomial(i,k)*binomial(j,k)*k!,k=0..j) ; end proc: # R. J. Mathar, Jul 28 2016
-
Mathematica
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 -
SageMath
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
Formula
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)
Comments