A145453 -id:A145453 - cp's OEIS frontend

A145460 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where sequence a_k of column k is the exponential transform of C(n,k).

1, 1, 1, 1, 1, 2, 1, 0, 3, 5, 1, 0, 1, 10, 15, 1, 0, 0, 3, 41, 52, 1, 0, 0, 1, 9, 196, 203, 1, 0, 0, 0, 4, 40, 1057, 877, 1, 0, 0, 0, 1, 10, 210, 6322, 4140, 1, 0, 0, 0, 0, 5, 30, 1176, 41393, 21147, 1, 0, 0, 0, 0, 1, 15, 175, 7273, 293608, 115975, 1, 0, 0, 0, 0, 0, 6, 35, 1176, 49932, 2237921, 678570

Offset: 0

Alois P. Heinz, Oct 10 2008

A(n,k) is also the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box k balls are seen at the top. E.g. A(3,1)=10:
  |1.| |2.| |3.| |1|2| |1|2| |1|3| |1|3| |2|3| |2|3| |1|2|3|
  |23| |13| |12| |3|.| |.|3| |2|.| |.|2| |1|.| |.|1| |.|.|.|
  +--+ +--+ +--+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+-+

			Square array A(n,k) begins:
   1,   1,  1,  1,  1,  1,  ...
   1,   1,  0,  0,  0,  0,  ...
   2,   3,  1,  0,  0,  0,  ...
   5,  10,  3,  1,  0,  0,  ...
  15,  41,  9,  4,  1,  0,  ...
  52, 196, 40, 10,  5,  1,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
N. J. A. Sloane, Transforms

Columns k=0-9 give: A000110, A000248, A133189, A145453, A145454, A145455, A145456, A145457, A145458, A145459.
A(2n,n) gives A029651.
Cf.: A007318, A143398, A292948.

exptr:= proc(p) local g; g:=
          proc(n) option remember; `if`(n=0, 1,
             add(binomial(n-1, j-1) *p(j) *g(n-j), j=1..n))
        end: end:
A:= (n,k)-> exptr(i-> binomial(i, k))(n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Exptr[p_] := Module[{g}, g[n_] := g[n] = If[n == 0, 1, Sum[Binomial[n-1, j-1] *p[j]*g[n-j], {j, 1, n}]]; g]; A[n_, k_] := Exptr[Function[i, Binomial[i, k]]][n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A145460(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A145460(20) # Seiichi Manyama, Sep 28 2017

A(0,k) = 1 and A(n,k) = Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0. - Seiichi Manyama, Sep 28 2017

A143398 Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains k labels, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 10, 3, 1, 0, 41, 9, 4, 1, 0, 196, 40, 10, 5, 1, 0, 1057, 210, 30, 15, 6, 1, 0, 6322, 1176, 175, 35, 21, 7, 1, 0, 41393, 7273, 1176, 105, 56, 28, 8, 1, 0, 293608, 49932, 7084, 756, 126, 84, 36, 9, 1, 0, 2237921, 372060, 42120, 6510, 378, 210, 120, 45, 10, 1

Offset: 0

Alois P. Heinz, Aug 12 2008

			T(4,2) = 9: 3->{1,2}<-4, 2->{1,3}<-4, 2->{1,4}<-3, 1->{2,3}<-4, 1->{2,4}<-3, 1->{3,4}<-2, {1,2}{3,4}, {1,3}{2,4}, {1,4}{2,3}.
Triangle begins:
  1;
  0,   1;
  0,   3,  1;
  0,  10,  3,  1;
  0,  41,  9,  4,  1;
  0, 196, 40, 10,  5,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Index entries for sequences related to rooted trees

Columns k=0-9 give: A000007, A000248, A133189, A145453, A145454, A145455, A145456, A145457, A145458, A145459.
Main diagonal gives A000012.
Row sums give A143406.
T(2n,n) gives A029651.
Cf. A000142, A145460, A351703.

u:= (n, k)-> `if`(k=0, 0, floor(n/k)):
T:= (n, k)-> n! *add(i^(n-k*i)/ ((n-k*i)! *i! *k!^i), i=0..u(n, k)):
seq(seq(T(n, k), k=0..n), n=0..12);

t[n_, n_] = 1; t[, 0] = 0; t[n, k_] := n!*Sum[i^(n-k*i)/((n-k*i)!*i!*k!^i), {i, 0, n/k}]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 19 2013 *)

u(n,k) = if(k==0, 0, n\k);
T(n, k) = n!*sum(j=0, u(n, k), j^(n-k*j)/(k!^j*j!*(n-k*j)!)); \\ Seiichi Manyama, May 13 2022

T(n,k) = n! * Sum_{i=0..u(n,k)} i^(n-k*i)/((n-k*i)!*i!*k!^i) with u(n,k) = 0 if k=0 and u(n,k) = floor(n/k) else.

A292889 Expansion of e.g.f. exp(x^3 * exp(x)).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 480, 5250, 40656, 302904, 3024720, 35260830, 400499880, 4619610996, 58758862344, 813678468330, 11686918766880, 173172973038960, 2695921067029536, 44219170187746614, 755363769931590840, 13351503268881437100, 244634794020399856920

Offset: 0

Seiichi Manyama, Sep 26 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..498

Column k=3 of A292978.

Cf. A145453, A292891, A292908.

x='x+O('x^66); Vec(serlaplace(exp(x^3*exp(x))))

a(n) = n!*sum(k=0, n\3, k^(n-3*k)/(k!*(n-3*k)!)); \\ Seiichi Manyama, Jul 10 2022

a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-3*k)/(k! * (n-3*k)!). - Seiichi Manyama, Jul 10 2022

A346889 Expansion of e.g.f. 1 / (1 - x^3 * exp(x) / 3!).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 40, 315, 2296, 15204, 117720, 1127445, 11531740, 120909646, 1370809804, 17111895255, 227853866800, 3182209445640, 47003318806896, 737325061500009, 12187616610231540, 210930852047426770, 3821604062633503300, 72479758506840597451

Offset: 0

Ilya Gutkovskiy, Aug 06 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..464

Column k=3 of A351703.
Cf. A006153, A346888, A346890, A346893.
Cf. A000292, A145453, A353999.

nmax = 23; CoefficientList[Series[1/(1 - x^3 Exp[x]/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 3] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^3*exp(x)/3!))) \\ Michel Marcus, Aug 06 2021

a(n) = n!*sum(k=0, n\3, k^(n-3*k)/(6^k*(n-3*k)!)); \\ Seiichi Manyama, May 13 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * binomial(k,3) * a(n-k).
a(n) ~ n! / ((1 + LambertW(2^(1/3)/3^(2/3))) * 3^(n+1) * LambertW(2^(1/3)/3^(2/3))^n). - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-3*k)/(6^k * (n-3*k)!). - Seiichi Manyama, May 13 2022

A292910 E.g.f.: exp(1/3! * x^3 * exp(-x)).

Original entry on oeis.org

1, 0, 0, 1, -4, 10, -10, -105, 1064, -6356, 25080, -9075, -1056660, 13219206, -106106364, 548948855, 139658960, -48411569800, 761039099824, -7815284148711, 52216924707660, -9385130453790, -6650556642220260, 132749143322588331, -1713641693856894824

Offset: 0

Seiichi Manyama, Sep 26 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..536

Column k=3 of A292948.

Cf. A145453, A292908.

x='x+O('x^66); Vec(serlaplace(exp(1/3!*x^3*exp(-x))))

A346751 Expansion of e.g.f. log( 1 + x^3 * exp(x) / 3! ).

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 10, -105, -1064, -6076, -16680, 129525, 2642860, 25431406, 130210444, -639438345, -26431524560, -382074099000, -3083015556624, 5641134587049, 726952330301940, 14940678486798610, 173111303303845060, 258953439321230731, -43858702741534022936

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..471

Cf. A000292, A009306, A145453, A346750, A346752, A346754.

nmax = 24; CoefficientList[Series[Log[1 + x^3 Exp[x]/3!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 3] - (1/n) Sum[Binomial[n, k] Binomial[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 24}]

a(0) = 0; a(n) = binomial(n,3) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,3) * k * a(k).

a(n) = n! * Sum_{k=1..floor(n/3)} (-1)^(k-1) * k^(n-3*k-1)/(6^k * (n-3*k)!). - Seiichi Manyama, Dec 14 2023

A346754 Expansion of e.g.f. -log( 1 - x^3 * exp(x) / 3! ).

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 30, 175, 1176, 7364, 50520, 425205, 4010380, 39433966, 414654604, 4793188855, 59834495280, 789420239560, 11016095913456, 163423065359529, 2565467553034740, 42320595474149650, 732058678770177220, 13275485607004016011

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

nonn

Cf. A000292, A009444, A145453, A292954, A305990, A346751, A346753, A346755.

nmax = 23; CoefficientList[Series[-Log[1 - x^3 Exp[x]/3!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 3] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]

a(0) = 0; a(n) = binomial(n,3) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,3) * k * a(k).
a(n) ~ (n-1)! / (3*LambertW(2^(1/3)/3^(2/3)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/3)} k^(n-3*k-1)/(6^k * (n-3*k)!). - Seiichi Manyama, Dec 14 2023

cp's OEIS Frontend

A145460 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where sequence a_k of column k is the exponential transform of C(n,k).

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A143398 Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains k labels, n>=0, 0<=k<=n.

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A292889 Expansion of e.g.f. exp(x^3 * exp(x)).

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A346889 Expansion of e.g.f. 1 / (1 - x^3 * exp(x) / 3!).

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A292910 E.g.f.: exp(1/3! * x^3 * exp(-x)).

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A346751 Expansion of e.g.f. log( 1 + x^3 * exp(x) / 3! ).

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A346754 Expansion of e.g.f. -log( 1 - x^3 * exp(x) / 3! ).

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