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
Examples
[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;
Links
- 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.
Crossrefs
Programs
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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 *) -
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
Formula
F(n,r) = Sum_{k=0..n} {n over k}_r *k!.