A292948 -id:A292948 - 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

A292973 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (-1)^(k+1) * k! * Sum_{i=0..n-1} (-1)^i * binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, -1, 1, 1, 2, 1, 0, -1, -5, 1, 0, 2, -2, 15, 1, 0, 0, -6, 9, -52, 1, 0, 0, 6, 24, -4, 203, 1, 0, 0, 0, -24, -140, -95, -877, 1, 0, 0, 0, 24, 60, 870, 414, 4140, 1, 0, 0, 0, 0, -120, 240, -5922, 49, -21147, 1, 0, 0, 0, 0, 120, 360, -4830, 45416, -10088, 115975

Offset: 0

Seiichi Manyama, Sep 27 2017

			Square array begins:
   1,  1,  1,   1,  1, ...
  -1,  1,  0,   0,  0, ...
   2, -1,  2,   0,  0, ...
  -5, -2, -6,   6,  0, ...
  15,  9, 24, -24, 24, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give: A292935, A003725, A292907, A292908, A292969, A292970.
Rows n=0 gives A000012.
Main diagonal gives A000142.
Cf. A292948, A292978.

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
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 << (-1) ** (k % 2 + 1) * f(k) * (0..i - 1).inject(0){|s, j| s + (-1) ** (j % 2) * ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A292973(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 A292973(20)

T(n,k) = n! * Sum_{j=0..floor(n/k)} (-j)^(n-k*j)/(j! * (n-k*j)!) for k > 0. - Seiichi Manyama, Jul 10 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))))

A292935 E.g.f.: exp(exp(-x) - 1).

Original entry on oeis.org

1, -1, 2, -5, 15, -52, 203, -877, 4140, -21147, 115975, -678570, 4213597, -27644437, 190899322, -1382958545, 10480142147, -82864869804, 682076806159, -5832742205057, 51724158235372, -474869816156751, 4506715738447323, -44152005855084346

Offset: 0

Seiichi Manyama, Sep 27 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..576

Column k=0 of A292948.

Cf. A000110.

x='x+O('x^66); Vec(serlaplace(exp(exp(-x)-1)))

a(n) = (-1)^n * A000110(n).

G.f.: Sum_{k>=0} (-x)^k / Product_{j=1..k} (1 + j*x). - Ilya Gutkovskiy, Dec 14 2019

A292909 E.g.f.: exp(1/2! * x^2 * exp(-x)).

Original entry on oeis.org

1, 0, 1, -3, 9, -40, 210, -1176, 7273, -49932, 372060, -2971540, 25359411, -230364498, 2215550428, -22460391240, 239236043985, -2669869110856, 31134833803728, -378485082644400, 4786085290280275, -62838103267148790, 855122923978737876

Offset: 0

Seiichi Manyama, Sep 26 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..530

Column k=2 of A292948.

Cf. A133189, A292907.

x='x+O('x^66); Vec(serlaplace(exp(1/2!*x^2*exp(-x))))

a(n) = (-1)^n * A133189(n).

A292912 E.g.f.: exp(1/4! * x^4 * exp(-x)).

Original entry on oeis.org

1, 0, 0, 0, 1, -5, 15, -35, 105, -756, 6510, -46530, 283470, -1667380, 11457446, -99776040, 969295145, -9298091180, 86154691680, -804769174536, 8052676029420, -88489327173660, 1038440150703340, -12501684521410700, 151866259113256611

Offset: 0

Seiichi Manyama, Sep 26 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..534

Column k=4 of A292948.

Cf. A145454.

With[{nn=30},CoefficientList[Series[Exp[x^4/4! Exp[-x]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Nov 12 2021 *)

x='x+O('x^66); Vec(serlaplace(exp(1/4!*x^4*exp(-x))))

a(n) = (-1)^n * A145454(n).

A292950 E.g.f.: exp(1/5! * x^5 * exp(-x)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, -6, 21, -56, 126, -126, -2310, 32472, -287001, 2016014, -11978967, 58518096, -159272932, -1367358552, 33638536548, -450849366456, 4857267575853, -44901867052350, 349150315551037, -1912043403523176, -2260563011376054

Offset: 0

Seiichi Manyama, Sep 27 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..544

Column k=5 of A292948.

x='x+O('x^66); Vec(serlaplace(exp(1/5!*x^5*exp(-x))))

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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A292973 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (-1)^(k+1) * k! * Sum_{i=0..n-1} (-1)^i * binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.

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

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A292935 E.g.f.: exp(exp(-x) - 1).

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A292909 E.g.f.: exp(1/2! * x^2 * exp(-x)).

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PARI

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A292912 E.g.f.: exp(1/4! * x^4 * exp(-x)).

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Mathematica

PARI

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A292950 E.g.f.: exp(1/5! * x^5 * exp(-x)).

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PARI