A335977 -id:A335977 - cp's OEIS frontend

A293037 E.g.f.: exp(1 + x - exp(x)).

1, 0, -1, -1, 2, 9, 9, -50, -267, -413, 2180, 17731, 50533, -110176, -1966797, -9938669, -8638718, 278475061, 2540956509, 9816860358, -27172288399, -725503033401, -5592543175252, -15823587507881, 168392610536153, 2848115497132448, 20819319685262839

Offset: 0

Seiichi Manyama, Sep 28 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..593

Column k=1 of A293051.
Column k=1 of A335977.
Cf. A000587 (k=0), this sequence (k=1), A293038 (k=2), A293039 (k=3), A293040 (k=4).
Cf. A000296, A014182, A193683, A193684, A196835.

f:= series(exp(1 + x - exp(x)), x= 0, 101): seq(factorial(n) * coeff(f, x, n), n = 0..30); # Muniru A Asiru, Oct 31 2017
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, 1-2*t,
      add(b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n+1, 1):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 01 2021

m = 26; Range[0, m]! * CoefficientList[Series[Exp[1 + x - Exp[x]], {x, 0, m}], x] (* Amiram Eldar, Jul 06 2020 *)
Table[Sum[Binomial[n, k] * BellB[k, -1], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 06 2020 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-exp(x)+1+x)))

a(n) = if(n==0, 1, -sum(k=0, n-2, binomial(n-1, k)*a(k))); \\ Seiichi Manyama, Aug 02 2021

a(n) = exp(1) * Sum_{k>=0} (-1)^k*(k + 1)^n/k!. - Ilya Gutkovskiy, Jun 13 2019
a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -1). - Vaclav Kotesovec, Jul 06 2020
a(0) = 1; a(n) = - Sum_{k=0..n-2} binomial(n-1,k) * a(k). - Seiichi Manyama, Aug 02 2021

A335975 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1) + x).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 15, 1, 1, 5, 19, 47, 52, 1, 1, 6, 29, 103, 227, 203, 1, 1, 7, 41, 189, 622, 1215, 877, 1, 1, 8, 55, 311, 1357, 4117, 7107, 4140, 1, 1, 9, 71, 475, 2576, 10589, 29521, 44959, 21147, 1, 1, 10, 89, 687, 4447, 23031, 88909, 227290, 305091, 115975, 1

Offset: 0

Seiichi Manyama, Jul 03 2020

			Square array begins:
  1,   1,    1,     1,     1,      1,      1, ...
  1,   2,    3,     4,     5,      6,      7, ...
  1,   5,   11,    19,    29,     41,     55, ...
  1,  15,   47,   103,   189,    311,    475, ...
  1,  52,  227,   622,  1357,   2576,   4447, ...
  1, 203, 1215,  4117, 10589,  23031,  44683, ...
  1, 877, 7107, 29521, 88909, 220341, 478207, ...

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

Columns k=0-4 give: A000012, A000110(n+1), A035009(n+1), A078940, A078945.
Main diagonal gives A334240.
Cf. A292860, A335977.

T[0, k_] := 1; T[n_, k_] := T[n - 1, k] + k * Sum[T[j, k] * Binomial[n - 1, j], {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jul 03 2020 *)

T(0,k) = 1 and T(n,k) = T(n-1,k) + k * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

T(n,k) = exp(-k) * Sum_{j>=0} (j + 1)^n * k^j / j!.

A309775 Expansion of e.g.f. exp(2 * (1 - exp(x)) + x).

Original entry on oeis.org

1, -1, -1, 3, 7, -13, -89, -45, 1191, 4723, -6873, -143597, -499289, 1843891, 28132391, 104223059, -508838745, -8597456141, -39770287321, 158845792147, 3788893515687, 23979078221619, -38626203043289, -2200108609291821, -19878849864738137, -27269435066568845

Offset: 0

Seiichi Manyama, Jul 06 2020

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Column k=2 of A335977.

Cf. A335980.

m = 25; Range[0, m]! * CoefficientList[Series[Exp[2 * (1 - Exp[x]) + x], {x, 0, m}], x] (* Amiram Eldar, Jul 06 2020 *)
Table[Sum[Binomial[n, k] * BellB[k, -2], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 06 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(exp(2*(1-exp(x))+x)))

a(0) = 1 and a(n) = a(n-1) - 2 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
a(n) = exp(2) * Sum_{k>=0} (k + 1)^n * (-2)^k / k!.
a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -2). - Vaclav Kotesovec, Jul 06 2020

A341287 Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise).

Original entry on oeis.org

1, 0, 1, -1, 1, 1, -1, -2, 3, 1, 2, -9, 1, 6, 1, 9, -9, -25, 15, 10, 1, 9, 50, -104, -20, 50, 15, 1, -50, 267, -98, -364, 105, 119, 21, 1, -267, 413, 1163, -1610, -539, 574, 238, 28, 1, -413, -2180, 7569, -1511, -6636, 903, 1806, 426, 36, 1, 2180, -17731, 17491, 29580, -32570, -12957, 8757, 4500, 705, 45, 1

Offset: 0

Petros Hadjicostas, Feb 08 2021

To agree with Knopfmacher and Mays (2001), the rows start at n = 0 while the columns start at k = 1.
The row sums equal 1.
"Let G be a labeled graph, with edge set E(G) and vertex set V(G). A composition of G is a partition of V(G) into vertex sets of connected induced subgraphs of G." "We will denote by C(G) the number of distinct compositions that exist for a given graph G." By Theorem 10 in Knofmacher and Mays (2001), C(K_{n,k}) = Sum_{i=1..n+1} T(n,i)*i^k, where K_{n,k} is the bipartite graph with n+k vertices and n*k edges. For values of C(K_{n,k}), see the table on p. 10 of the paper.
We have C(K_{n,k}) = A265417(n,k).
By symmetry, Sum_{i=1..n+1} T(n,i)*i^k = C(K_{n,k}) = C(K_{k,n}) = Sum_{i=1..k+1} T(k,i)*i^n for n, k >= 1.
Denote the bivariate e.g.f.-o.g.f by A(x,y) =  Sum_{n>=0,k>=1} T(n,k)*(x^n/n!)*y^k. Using the definition of T(n,k) and standard manipulations of generating functions, one can prove that A(x,y) = y + int_{w=0..x} A(w,y)*((y-1)*exp(w) + 1) dw. This leads to the initial condition A(0,y) = y and the differential equation dA(x,y)/dx = A(x,y)*((y-1)*exp(x) + 1). Solving this differential equation (for a fixed y), we get A(x,y) = y*exp((1 - y)*(1 - exp(x)) + x). (The bivariate e.g.f.-o.g.f was originally guessed due to the contributions of Seiichi Manyama in A335977.)

			Triangle T(n,k) with rows n >= 0 and columns 1 <= k <= n+1 begins:
      1,
      0,     1,
     -1,     1,    1,
     -1,    -2,    3,     1,
      2,    -9,    1,     6,     1,
      9,    -9,  -25,    15,    10,   1,
      9,    50, -104,   -20,    50,  15,    1,
    -50,   267,  -98,  -364,   105, 119,   21,   1,
   -267,   413, 1163, -1610,  -539, 574,  238,  28,  1,
   -413, -2180, 7569, -1511, -6636, 903, 1806, 426, 36, 1,
   ...

Aminul Huq, Compositions of graphs revisited, The Electronic Journal of Combinatorics, Vol. 14 (2007), Article N15.
A. Knopfmacher and M. E. Mays, Graph compositions I: Basic enumerations, Integers, Vol. 1 (2001), Article A4. (See p. 9 for the table and p. 8 for the recurrence.)

Cf. A000217, A000587, A048993, A265417, A309775, A335977.

egf := k-> (exp(x)-1)^(k-1)/(k-1)!*exp(x-(exp(x)-1)):
A341287 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A341287(n, k), k=1..n+1)), n=0..9); # Mélika Tebni, Apr 20 2022

Sum_{k=1..n+1} (-1)^(k-1)*T(n,k) = A309775(n) for n >= 0.
Sum_{k=1..n+1} (-m)^(k-1)*T(n,k) = A335977(n,m+1) for m >= 1 and n >= 0.
T(n,n+1) = 1 and T(n,n) = A000217(n-1) = n*(n-1)/2 for n >= 1.
T(n,1) = -A000587(n+1) for n >= 0 (complementary Bell numbers).
T(n,2) = -T(n+1,1) for n >= 1.
Bivariate e.g.f.-o.g.f: Sum_{n>=0,k>=1} T(n,k)*(x^n/n!)*y^k = y*exp((1 - y)*(1 -exp(x)) + x).
T(n,k) = Sum_{j=1..n+1} binomial(j - 1, k - 1)*(-1)^(j - k)*Stirling2(n + 1, j) for n >= 0 and 1 <= k <= n+1, where Stirling2(n,k) = A048993(n,k). (This is a modification of a formula in Section 4 of Huq (2007).)
From Mélika Tebni, Apr 20 2022: (Start)
T(n, k) = Sum_{j=0..n} A129334(n, j)*Stirling2(j+1, k) for n >= 0 and 1 <= k <= n+1.
E.g.f. column k: (exp(x) - 1)^(k-1) / (k-1)!*exp(x - (exp(x) - 1)), k >= 1. (End)

A320432 Expansion of e.g.f. exp(3 * (1 - exp(x)) + x).

Original entry on oeis.org

1, -2, 1, 7, -8, -65, 37, 1024, 1351, -19001, -92618, 232513, 4087189, 9953926, -123909155, -1170342533, -676144160, 62840385619, 490129709977, -551829062288, -40624407525941, -305175084654341, 698633855671510, 34571970743398621, 278738497423756153, -663168571756087538

Offset: 0

Seiichi Manyama, Jul 06 2020

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Column k=3 of A335977.

Cf. A078940, A335981.

m = 25; Range[0, m]! * CoefficientList[Series[Exp[3 * (1 - Exp[x]) + x], {x, 0, m}], x] (* Amiram Eldar, Jul 06 2020 *)
Table[Sum[Binomial[n, k] * BellB[k, -3], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 06 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(exp(3*(1-exp(x))+x)))

a(0) = 1 and a(n) = a(n-1) - 3 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
a(n) = exp(3) * Sum_{k>=0} (k + 1)^n * (-3)^k / k!.
a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -3). - Vaclav Kotesovec, Jul 06 2020

A320433 Expansion of e.g.f. exp(4 * (1 - exp(x)) + x).

Original entry on oeis.org

1, -3, 5, 5, -43, -27, 597, 805, -11883, -40475, 265685, 2133157, -3405803, -107760283, -301542315, 4458255397, 42421260949, -45046794011, -3365690666283, -19844416105563, 138274174035221, 2917746747446373, 11092963732101461, -207438902364296411, -3205301465165742187

Offset: 0

Seiichi Manyama, Jul 06 2020

sign

Column k=4 of A335977.

Cf. A078945, A335982.

m = 24; Range[0, m]! * CoefficientList[Series[Exp[4 * (1 - Exp[x]) + x], {x, 0, m}], x] (* Amiram Eldar, Jul 06 2020 *)
Table[Sum[Binomial[n, k] * BellB[k, -4], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 06 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(exp(4*(1-exp(x))+x)))

a(0) = 1 and a(n) = a(n-1) - 4 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
a(n) = exp(4) * Sum_{k>=0} (k + 1)^n * (-4)^k / k!.
a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -4). - Vaclav Kotesovec, Jul 06 2020

cp's OEIS Frontend

A293037 E.g.f.: exp(1 + x - exp(x)).

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A335975 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1) + x).

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A309775 Expansion of e.g.f. exp(2 * (1 - exp(x)) + x).

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A341287 Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise).

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A320432 Expansion of e.g.f. exp(3 * (1 - exp(x)) + x).

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Mathematica

PARI

Formula

A320433 Expansion of e.g.f. exp(4 * (1 - exp(x)) + x).

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A341287 Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise).