A321189 -id:A321189 - cp's OEIS frontend

A094420 Generalized ordered Bell numbers Bo(n,n).

1, 1, 10, 219, 8676, 544505, 49729758, 6232661239, 1026912225160, 215270320769109, 55954905981282210, 17662898483917308083, 6655958151527584785900, 2951503248457748982755953, 1521436331153097968932487206, 902143190212525713006814917615, 609729139653483641913607434550800

Offset: 0

Ralf Stephan, May 02 2004

nonn

Main diagonal of array A094416.

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A094416, A321189.
The coefficients of the Fubini polynomials are A131689.
Central column of A344499.

A094420:= func< n | (&+[Factorial(k)*n^k*StirlingSecond(n,k): k in [0..n]]) >;
[A094420(n): n in [0..25]]; // G. C. Greubel, Jan 12 2024

F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n-k)*x, k=1..n)) end:
a := n -> subs(x = n, F(n)):
seq(a(n), n = 0..16); # Peter Luschny, May 21 2021

Table[Sum[k!*n^k*StirlingS2[n, k], {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jul 23 2018 *)

{a(n) = sum(k=0, n, k!*n^k*stirling(n, k, 2))} \\ Seiichi Manyama, Jun 12 2020

def aList(len):
    R. = PowerSeriesRing(QQ)
    f = lambda n: R(1/(1 + n * (1 - exp(x))))
    return [factorial(n)*f(n).list()[n] for n in (0..len-1)]
print(aList(17)) # Peter Luschny, May 21 2021

a(n) ~ sqrt(2*Pi) * n^(2*n + 5/2) / exp(n - 3/2). - Vaclav Kotesovec, Jul 23 2018
a(n) = Sum_{k=0..n} k!*n^k*Stirling2(n, k). - Seiichi Manyama, Jun 12 2020
From Peter Luschny, May 21 2021: (Start)
a(n) = F_{n}(n), the Fubini polynomial F_{n}(x) evaluated at x = n.
a(n) = n! * [x^n] (1 / (1 + n * (1 - exp(x)))).  (End)

More terms from Seiichi Manyama, Jun 12 2020

A335529 a(n) = n! * [x^n] (1 - (n-1)log(1 + x))/(1 - nlog(1 + x)).

Original entry on oeis.org

1, 1, 3, 38, 1042, 49774, 3661128, 383653080, 54275300112, 9964363066848, 2303245150868640, 654457584668128416, 224205104879416320768, 91129285853151907958544, 43356207229026959513863680, 23868203329368882698589532800, 15053662436260897659550535387136

Offset: 0

Seiichi Manyama, Jun 12 2020

nonn

Main diagonal of A334369.

Cf. A317172, A321189.

a[0] = 1; a[n_] := Sum[k! * n^(k - 1) * StirlingS1[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Jun 12 2020 *)

{a(n) = if(n==0, 1, sum(k=0, n, k!*n^(k-1)*stirling(n, k, 1)))}

a(n) = A317172(n)/n = Sum_{k=0..n} k!*n^(k-1)*Stirling1(n,k) for n > 1.

a(n) ~ sqrt(2*Pi) * n^(2*n - 1/2) / exp(n + 1/2). - Vaclav Kotesovec, Jun 12 2020

A369435 Square array A(n, k) = n! * [t^n] (exp(t)/(1+k-k*exp(t))) for n >= 0 and k >= 0, read by antidiagonals upwards.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 26, 15, 4, 1, 1, 150, 111, 28, 5, 1, 1, 1082, 1095, 292, 45, 6, 1, 1, 9366, 13503, 4060, 605, 66, 7, 1, 1, 94586, 199815, 70564, 10845, 1086, 91, 8, 1, 1, 1091670, 3449631, 1471708, 243005, 23826, 1771, 120, 9, 1, 1, 14174522, 68062695, 35810212, 6534045, 653406, 45955, 2696, 153, 10, 1

Offset: 0

Werner Schulte, Jan 23 2024

The following formulae are conjectures:
  (1) det(A(0..n, k..k+n)) = (Product_{i=1..n} i!)^2 for k >= 0 and n >= 0.
  (2) A(n, k) = 1 + k * (Sum_{i=0..n-1} binomial(n, i) * A(i, k)) for k >= 0 and
      n > 0 with initial values A(0, k) = 1 for k >= 0.
  (3) A(n, k) = (k+1)^n + k * (Sum_{i=0..n-2} binomial(n, i) * A(i, k) *
      ((k+1)^(n-i) - (k+1) * k^(n-1-i))) for k >= 0 and n > 1 with initial values
      A(n, k) = (k+1)^n for k >= 0 and n < 2.
  (4) Let B(n, k) = (k!) * (Sum_{i=k..n} (i!) * S2(i, k) * S2(n+1, i+1)) for 0 <=
      k <= n where S2(i, j) = A048993(i, j). Then holds:
      (a) B(n, k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * A(n, i) for 0 <= k
          <= n;
      (b) E.g.f. of row n >= 0: exp(x) * (Sum_{k=0..n} B(n, k) * x^k / (k!)).

			Array A(n, k) starts:
n\k : 0        1        2        3       4        5        6         7         8
================================================================================
 0  : 1        1        1        1       1        1        1         1         1
 1  : 1        2        3        4       5        6        7         8         9
 2  : 1        6       15       28      45       66       91       120       153
 3  : 1       26      111      292     605     1086     1771      2696      3897
 4  : 1      150     1095     4060   10845    23826    45955     80760    132345
 5  : 1     1082    13503    70564  243005   653406  1490587   3024008   5618169
 6  : 1     9366   199815  1471708 6534045 21502866 58018051 135878520 286195833
 7  : 1    94586  3449631 35810212
 8  : 1  1091670 68062695
 9  : 1 14174522
.
Triangle T(n, k) starts:
[0] 1;
[1] 1,       1;
[2] 1,       2,       1;
[3] 1,       6,       3,       1;
[4] 1,      26,      15,       4,      1;
[5] 1,     150,     111,      28,      5,     1;
[6] 1,    1082,    1095,     292,     45,     6,   1;
[7] 1,    9366,   13503,    4060,    605,    66,   7,    1;
[8] 1,   94586,  199815,   70564,  10845,  1086,   91,   8, 1;
[9] 1, 1091670, 3449631, 1471708, 243005, 23826, 1771, 120, 9, 1;

Cf. A000012 (col 0 and row 0), A000629 (col 1), A201339 (col 2), A201354 (col 3), A201365 (col 4), A000027 (row 1), A000384 (row 2), A163626, A028246.

Cf. A372312.

egf := exp(t) / (1 + x*(1 - exp(t))): sert := series(egf, t, 12):
col := k -> local j; seq(subs(x=k, j!*coeff(sert, t, j)), j=0..9):
T := (n, k) -> col(k)[n - k + 1]:  # Triangle
for n from 0 to 9 do seq(T(n, k), k=0..n) od;  # Peter Luschny, Jan 24 2024
with(combinat): # WP Worpitzky polynomials, WC coefficients of WP.
WC := (n, k) -> local j; add(eulerian1(n, j)*binomial(n-j, n-k), j=0..n):
WP := n -> local j; add(WC(n, j) * x^j, j=0..n):
A369435row := (n, k) -> subs(x = k, WP(n)):
seq(lprint(seq(A369435row(n, k), k = 0..7)), n = 0..7);
# Peter Luschny, Apr 26 2024

{A(n, k) = n! * polcoeff(exp(x+x*O(x^n)) / (1+k-k*exp(x+x*O(x^n))), n)}

A(n, k) = Sum_{i=0..n} A163626(n, i) * (-k)^i for n >= 0 and k >= 0.
A(n, k) = Sum_{i=0..n} A028246(n+1, i+1) * k^i for n >= 0 and k >= 0.
E.g.f. of column k >= 0: exp(t) / (1 + k - k * exp(t)).
A(n, n) = Sum_{i=0..n} A163626(n, i) * (-n)^i = Sum_{i=0..n} A028246(n+1, i+1) * n^i for n >= 0.
Conjecture: A(n, n) = (n + 1) * A321189(n) for n >= 0. [This is true. - Peter Luschny, Apr 26 2024]
A(n, n) = A372312(n). - Peter Luschny, Apr 26 2024

A331345 a(n) = (1/n^2) * Sum_{k>=1} k^n * (1 - 1/n)^(k - 1).

Original entry on oeis.org

1, 3, 37, 1015, 48601, 3583811, 376372333, 53343571695, 9808511445361, 2270198126932219, 645790373135121061, 221449391959470686375, 90084675298978081317961, 42890688646618728144279987, 23627228721958495690763944861, 14910259060767841554203065990111

Offset: 1

Ilya Gutkovskiy, Jun 08 2020

nonn

Index entries for sequences related to rooted trees

Cf. A000670, A050351, A050352, A050353, A094420, A257565, A321189, A334240.

Join[{1}, Table[1/n^2 Sum[k^n (1 - 1/n)^(k - 1), {k, 1, Infinity}], {n, 2, 16}]]
Table[n! SeriesCoefficient[(Exp[x] - 1)/(Exp[x] - n (Exp[x] - 1)), {x, 0, n}], {n, 1, 16}]

a(n) = n! * [x^n] (exp(x) - 1) / (exp(x) - n * (exp(x) - 1)).
a(n) = Sum_{k=1..n} Stirling2(n,k) * (n - 1)^(k - 1) * k!.
a(n) ~ sqrt(2*Pi) * n^(2*n - 1/2) / exp(n + 1/2). - Vaclav Kotesovec, Jun 08 2020

cp's OEIS Frontend

A094420 Generalized ordered Bell numbers Bo(n,n).

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A335529 a(n) = n! * [x^n] (1 - (n-1)*log(1 + x))/(1 - n*log(1 + x)).

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A369435 Square array A(n, k) = n! * [t^n] (exp(t)/(1+k-k*exp(t))) for n >= 0 and k >= 0, read by antidiagonals upwards.

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A331345 a(n) = (1/n^2) * Sum_{k>=1} k^n * (1 - 1/n)^(k - 1).

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A335529 a(n) = n! * [x^n] (1 - (n-1)log(1 + x))/(1 - nlog(1 + x)).