A218142
a(n) = Stirling2(n^2+n, n).
Original entry on oeis.org
1, 1, 31, 86526, 45232115901, 7713000216608565075, 666480349285726891499539272955, 41929298560838945526242744414099901692285884, 2610516895723221966171633379256064857587637240616032299710417
Offset: 0
O.g.f.: A(x) = 1 + x + 31*x^2 + 86526*x^3 + 45232115901*x^4 +...
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Table[StirlingS2[n^2+n, n],{n,0,10}] (* Vaclav Kotesovec, May 11 2014 *)
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makelist(stirling2(n^2+n,n),n,0,30 ); /* Martin Ettl, Oct 21 2012 */
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{a(n)=polcoeff(sum(k=0,n,(k^(n+1))^k*exp(-k^(n+1)*x +x*O(x^n))*x^k/k!),n)}
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{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(n^2))), n^2)}
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{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(n^2+n, n)}
for(n=0, 10, print1(a(n), ", "))
A218143
a(n) = Stirling2(n*(n+1)/2, n).
Original entry on oeis.org
1, 1, 3, 90, 34105, 210766920, 26585679462804, 82892803728383735268, 7529580759157036060608585183, 22982258052528294182955639980819773510, 2672446997421818663856559987803834697952486978300, 13239043631590111512460321918828937597837325561187113535696980
Offset: 0
O.g.f.: A(x) = 1 + x + 3*x^2 + 90*x^3 + 34105*x^4 + 210766920*x^5 + 26585679462804*x^6 +...
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Table[StirlingS2[n*(n+1)/2, n],{n,0,10}] (* Vaclav Kotesovec, May 11 2014 *)
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makelist(stirling2(n*(n+1)/2,n),n,0,30 ); /* Martin Ettl, Oct 21 2012 */
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{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(n*(n-1)/2))), n*(n-1)/2)}
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{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(n*(n+1)/2, n)}
for(n=0, 15, print1(a(n), ", "))
A344532
Number of cycle-up-down permutations of [n^2] having n cycles.
Original entry on oeis.org
1, 1, 7, 14698, 51629528080, 914192102910317528125, 199979553262025879510473132453855232, 1131253316618666789979709230473744963049785439771172168, 309491168658231587025767619097898747214052900521443034546657433273562730332160
Offset: 0
a(2) = 7: (1)(243), (143)(2), (142)(3), (132)(4), (12)(34), (13)(24), (14)(23).
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b:= proc(u, o) option remember; `if`(u+o=0, 1,
add(b(o-1+j, u-j), j=1..u))
end:
g:= proc(n) option remember; expand(`if`(n=0, 1,
add(g(n-j)*binomial(n-1, j-1)*x*b(j-1, 0), j=1..n)))
end:
a:= n-> coeff(g(n^2), x, n):
seq(a(n), n=0..9);
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b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]];
g[n_] := g[n] = Expand[If[n == 0, 1,
Sum[g[n-j]*Binomial[n-1, j-1]*x*b[j-1, 0], {j, 1, n}]]];
a[n_] := Coefficient[g[n^2], x, n];
a /@ Range[0, 9] (* Jean-François Alcover, Jun 10 2021, after Alois P. Heinz *)
A384504
a(n) = Stirling1(n^2, n).
Original entry on oeis.org
1, 1, 11, 118124, 5056995703824, 2677503356427960382362624, 43103055200236892507668550744976954163200, 44206966751754314698168885550132827351582613259130314424320000
Offset: 0
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Table[StirlingS1[n^2, n], {n, 0, 10}]
Showing 1-4 of 4 results.
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