A187646
(Signless) Central Stirling numbers of the first kind s(2n,n).
Original entry on oeis.org
1, 1, 11, 225, 6769, 269325, 13339535, 790943153, 54631129553, 4308105301929, 381922055502195, 37600535086859745, 4070384057007569521, 480544558742733545125, 61445535102359115635655, 8459574446076318147830625, 1247677142707273537964543265, 196258640868140652967646352465
Offset: 0
-
seq(abs(Stirling1(2*n,n)), n=0..20);
-
Table[Abs[StirlingS1[2n, n]], {n, 0, 12}]
N[1 + 1/(2 LambertW[-1, -Exp[-1/2]/2]), 50] (* The constant z in the asymptotic - Vladimir Reshetnikov, Oct 08 2016 *)
-
makelist(abs(stirling1(2*n,n)),n,0,12);
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for(n=0,50, print1(abs(stirling(2*n, n, 1)), ", ")) \\ G. C. Greubel, Nov 09 2017
A285849
Number T(n,k) of permutations of [n] with k ordered cycles such that equal-sized cycles are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 2, 6, 1, 0, 6, 19, 18, 1, 0, 24, 100, 105, 40, 1, 0, 120, 508, 1005, 430, 75, 1, 0, 720, 3528, 6762, 6300, 1400, 126, 1, 0, 5040, 24876, 61572, 62601, 28700, 3822, 196, 1, 0, 40320, 219168, 558548, 706608, 431445, 105336, 9114, 288, 1
Offset: 0
T(3,1) = 2: (123), (132).
T(3,2) = 6: (1)(23), (23)(1), (2)(13), (13)(2), (3)(12), (12)(3).
T(3,3) = 1: (1)(2)(3).
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 2, 6, 1;
0, 6, 19, 18, 1;
0, 24, 100, 105, 40, 1;
0, 120, 508, 1005, 430, 75, 1;
0, 720, 3528, 6762, 6300, 1400, 126, 1;
0, 5040, 24876, 61572, 62601, 28700, 3822, 196, 1;
Columns k=0-10 give:
A000007,
A104150,
A285853,
A285854,
A285855,
A285856,
A285857,
A285858,
A285859,
A285860,
A285861.
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b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
(p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
[multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..12);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 28 2018, after Alois P. Heinz *)
A285926
Number of ordered set partitions of [2n] into n blocks such that equal-sized blocks are ordered with increasing least elements.
Original entry on oeis.org
1, 1, 11, 420, 17129, 1049895, 97141022, 10742461730, 1370094506209, 207877406991111, 36104901766271975, 7033373902938469086, 1531762189401458287506, 368890302956243012167470, 97283928918541409263666020, 27895730515878936009534815250
Offset: 0
-
b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
(p+n)!/n!*x^n, add(x^j*b(n-i*j, i-1, p+j)*combinat
[multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)))
end:
a:= n-> coeff(b(2*n$2, 0), x, n):
seq(a(n), n=0..20);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n! x^n, Sum[b[n - i j, i - 1, p + j] x^j multinomial[n, Join[{n - i j}, Table[i, j]]]/j!^2, {j, 0, n/i}]]];
a[n_] := Coefficient[b[2n, 2n, 0], x, n];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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