A217389
Partial sums of the ordered Bell numbers (number of preferential arrangements) A000670.
Original entry on oeis.org
1, 2, 5, 18, 93, 634, 5317, 52610, 598445, 7685706, 109933269, 1732565842, 29824133437, 556682481818, 11198025452261, 241481216430114, 5557135898411469, 135927902927547370, 3521462566184392693, 96323049885512803826, 2774010846129897006941, 83898835844633970888762
Offset: 0
-
A000670:=func;
[&+[A000670(k): k in [0..n]]: n in [0..19]]; // Bruno Berselli, Oct 03 2012
-
b:= proc(n, k) option remember;
`if`(n=0, k!, k*b(n-1, k)+b(n-1, k+1))
end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n, 0)) end:
seq(a(n), n=0..23); # Alois P. Heinz, Feb 20 2025
-
t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[t[k], {k, 0, n}], {n, 0, 100}]
(* second program: *)
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}]; Fubini[0, 1] = 1; Table[Fubini[n, 1], {n, 0, 20}] // Accumulate (* Jean-François Alcover, Mar 31 2016 *)
-
t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum(t(k),k,0,n),n,0,40);
-
for(n=0,30, print1(sum(k=0,n, sum(j=0,k, j!*stirling(k,j,2))), ", ")) \\ G. C. Greubel, Feb 07 2018
A006957
Self-convolution of numbers of preferential arrangements.
Original entry on oeis.org
1, 2, 7, 32, 185, 1310, 11067, 109148, 1234045, 15752858, 224169407, 3518636504, 60381131265, 1124390692886, 22577494959427, 486212633129300, 11177317486573445, 273173247028616594, 7072436847620016327, 193351544314753174736, 5565941751233499986185
Offset: 0
-
f:= proc(n) option remember; `if`(n<=1, 1,
add(binomial(n, k) *f(n-k), k=1..n))
end:
a:= n-> add(f(k)*f(n-k), k=0..n):
seq(a(n), n=0..25); # Alois P. Heinz, Feb 02 2009
-
t[n_] := Sum[StirlingS2[n, k]*k!, {k, 0, n}]; Table[Sum[t[k]*t[n-k], {k, 0, n}], {n, 0, 20}] (* Jean-François Alcover, Apr 09 2014, after Emanuele Munarini *)
-
t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum(t(k)*t(n-k),k,0,n),n,0,20); /* Emanuele Munarini, Oct 02 2012 */
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a006957(n)=my(x='x+O('x^(n+2))); Vec((x-2*log(2-exp(x)))/(4-exp(x)))[n+1]*(n+1)! \\ Hugo Pfoertner, Dec 27 2024
A217391
Partial sums of the squares of the ordered Bell numbers (number of preferential arrangements) A000670.
Original entry on oeis.org
1, 2, 11, 180, 5805, 298486, 22228975, 2258856824, 300194704049, 50529463186170, 10505093602625139, 2643441560563225468, 791779611505017309493, 278371498870260182630654, 113516551713466910954246903, 53143864598655784249290736512, 28309328562668956145157858372537
Offset: 0
-
A000670:=func;
[&+[A000670(k)^2: k in [0..n]]: n in [0..14]]; // Bruno Berselli, Oct 03 2012
-
t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[t[k]^2, {k, 0, n}], {n, 0, 100}]
-
t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum(t(k)^2,k,0,n),n,0,40);
-
for(n=0,30, print1(sum(k=0,n, (sum(j=0,k, j!*stirling(k,j,2)))^2), ", ")) \\ G. C. Greubel, Feb 07 2018
A217392
Alternating sums of the squares of the ordered Bell numbers (number of preferential arrangements) A000670.
Original entry on oeis.org
1, 0, 9, 160, 5465, 287216, 21643273, 2214984576, 295720862649, 49933547619472, 10404630591819497, 2622531836368780832, 786513638108085303193, 276793205620647080017968, 112961387008976003691598281, 52917386659933341334644891328, 28203267311410367019573922744697
Offset: 0
-
A000670:=func;
[&+[(-1)^(n-k)*A000670(k)^2: k in [0..n]]: n in [0..14]]; // Bruno Berselli, Oct 03 2012
-
t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[(-1)^(n-k)t[k]^2, {k, 0, n}], {n, 0, 100}]
-
t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum((-1)^(n-k)*t(k)^2,k,0,n),n,0,40);
-
for(n=0,30, print1(sum(k=0,n, (-1)^(n-k)*(sum(j=0,k, j!*stirling(k,j,2)))^2), ", ")) \\ G. C. Greubel, Feb 07 2018
Showing 1-4 of 4 results.