A309084 -id:A309084 - cp's OEIS frontend

A292861 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*(1 - exp(x))).

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 2, 1, 0, 1, -4, 6, 2, 1, 0, 1, -5, 12, -3, -6, -2, 0, 1, -6, 20, -20, -21, -14, -9, 0, 1, -7, 30, -55, -20, 24, 26, -9, 0, 1, -8, 42, -114, 45, 172, 195, 178, 50, 0, 1, -9, 56, -203, 246, 370, 108, -111, 90, 267, 0, 1, -10, 72, -328, 679, 318, -1105, -2388, -3072, -2382, 413, 0

Offset: 0

Seiichi Manyama, Sep 25 2017

			Square array begins:
   1,  1,   1,   1,   1,     1,     1, ...
   0, -1,  -2,  -3,  -4,    -5,    -6, ...
   0,  0,   2,   6,  12,    20,    30, ...
   0,  1,   2,  -3, -20,   -55,  -114, ...
   0,  1,  -6, -21, -20,    45,   246, ...
   0, -2, -14,  24, 172,   370,   318, ...
   0, -9,  26, 195, 108, -1105, -4074, ...

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

Columns k=0..4 give A000007, A000587, A213170, A309084, A309085.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292866.
Cf. A292860, A309386.

A:= proc(n, k) option remember; `if`(n=0, 1,
      -(1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017

A[n_, k_] := Sum[(-k)^j StirlingS2[n, j], {j, 0, n}];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 10 2021 *)
A292861[n_, k_] := BellB[k, k - n];
Table[A292861[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 23 2021 *)

A(0,k) = 1 and A(n,k) = -k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.
A(n,k) = Sum_{j=0..n} (-k)^j * Stirling2(n,j). - Seiichi Manyama, Jul 27 2019
A(n,k) = BellPolynomial(n, -k). - Peter Luschny, Dec 23 2021

A309085 a(n) = exp(4) * Sum_{k>=0} (-4)^k*k^n/k!.

Original entry on oeis.org

1, -4, 12, -20, -20, 172, 108, -2388, -3220, 47532, 161900, -1062740, -8532628, 13623212, 431041132, 1206169260, -17833021588, -169685043796, 180187176044, 13462762665132, 79377664422252, -553096696140884, -11670986989785492, -44371854928405844, 829755609457185644

Offset: 0

Ilya Gutkovskiy, Jul 11 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..564

Column k = 4 of A292861.

Cf. A000587, A078944, A213170, A309084, A318179.

[1] cat [(&+[((-4)^k*StirlingSecond(m,k)):k in [0..m]]):m in [1..24]]; // Marius A. Burtea, Jul 11 2019

Table[Exp[4] Sum[(-4)^k k^n/k!, {k, 0, Infinity}], {n, 0, 24}]
Table[BellB[n, -4], {n, 0, 24}]
nmax = 24; CoefficientList[Series[Sum[(-4)^j x^j/Product[(1 - k x), {k, 1, j}] , {j, 0, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[4 (1 - Exp[x])], {x, 0, nmax}], x] Range[0, nmax]!

a(n) = sum(k=0, n, (-4)^k * stirling(n,k,2)); \\ Michel Marcus, Jul 12 2019

G.f.: Sum_{j>=0} (-4)^j*x^j / Product_{k=1..j} (1 - k*x).
E.g.f.: exp(4*(1 - exp(x))).
a(n) = Sum_{k=0..n} (-4)^k * Stirling2(n,k).

A335981 Expansion of e.g.f. exp(3 * (1 - exp(-x)) + x).

Original entry on oeis.org

1, 4, 13, 31, 40, -23, -95, 490, 823, -8393, 3766, 174775, -658787, -2751404, 34033297, -55552037, -1170734432, 9362348365, 3277050925, -562286419646, 3848880970147, 8815342530739, -356804325202730, 2389771436686339, 8677476137729929, -302470260552857660

Offset: 0

Ilya Gutkovskiy, Jul 03 2020

sign

Cf. A027710, A078940, A109747, A309084, A335868, A335980, A335982.

nmax = 25; CoefficientList[Series[Exp[3 (1 - Exp[-x]) + x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = a[n - 1] + 3 Sum[(-1)^(n - k - 1) Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]

a(n) = exp(3) * (-1)^n * Sum_{k>=0} (-3)^k * (k - 1)^n / k!.

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

cp's OEIS Frontend

A292861 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*(1 - exp(x))).

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A309085 a(n) = exp(4) * Sum_{k>=0} (-4)^k*k^n/k!.

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A335981 Expansion of e.g.f. exp(3 * (1 - exp(-x)) + x).

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