A219374 Triangle of F(n,r) of r-geometric numbers, 1 <= r <= n.

1, 3, 2, 13, 10, 6, 75, 62, 42, 24, 541, 466, 342, 216, 120, 4683, 4142, 3210, 2184, 1320, 720, 47293, 42610, 34326, 24696, 15960, 9360, 5040, 545835, 498542, 413322, 310344, 211560, 131760, 75600, 40320, 7087261, 6541426, 5544342, 4304376, 3063000, 2005200, 1214640, 685440, 362880, 102247563, 95160302

Offset

1,2

Links

Table of n, a(n) for n=1..47.

S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, Unit interval parking functions and the r-Fubini numbers, arXiv:2401.06937 [math.CO], 2024. See Table 1 at page 9.

A. Dil and V. Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series, II, Appl. An. Disc. Math. 5 (2011) 212-229, section 3.2.

Formula

F(n,r) = Sum_{k=0..n} {n over k}_r *k!.

Example

[1] 1;
[2] 3, 2;
[3] 13, 10, 6;
[4] 75, 62, 42, 24;
[5] 541, 466, 342, 216, 120;
[6] 4683, 4142, 3210, 2184, 1320, 720;

Maple

Stirr := proc(n, k, r)
    option remember;
    if n < r then
        0;
    elif n = r then
        if k = r then
            1 ;
        else
            0 ;
        end if;
    else
        procname(n-1, k-1, r) + k*procname(n-1, k, r) ;
    end if;
end proc:
A := proc(n, r)
    add( k!*Stirr(n, k, r), k=0..n) ;
end proc:
seq(seq( A(n, r), r=1..n), n=1..12) ;

Mathematica

Stirr[n_, k_, r_] := Stirr[n, k, r] = Which[n < r, 0, n == r, If[k == r, 1, 0], True, Stirr[n-1, k-1, r] + k*Stirr[n-1, k, r]]; a[n_, r_] := Sum[ k!*Stirr[n, k, r], {k, 0, n}]; Table[Table[a[n, r], {r, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jan 10 2014, translated from Maple *)
Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Table[Fubini[n, r], {n, 1, 10}, {r, 1, n}] // Flatten (* Jean-François Alcover, Mar 30 2016 *)

Prog

(Sage)
@CachedFunction
def stirling_number2r(n, k, r) :
    if n < r: return 0
    if n == r: return 1 if k == r else 0
    return stirling_number2r(n-1, k-1, r)+ k*stirling_number2r(n-1, k, r)
def A219374(n, r):
    return add(factorial(k)*stirling_number2r(n, k, r) for k in (0..n))
for n in (1..6):
    print([A219374(n, r) for r in (1..n)]) # Peter Luschny, Nov 19 2012

Crossrefs

Cf. A008277 {n over k}_1, A143494 {n over k}_2, A143495 {n over k}_3, A000670 (first column).

Sequence in context: A191705 A252669 A131050 * A084416 A210604 A005352

Adjacent sequences: A219371 A219372 A219373 * A219375 A219376 A219377

Keyword

tabl,nonn,easy

Author

R. J. Mathar, Nov 19 2012

Status

approved