A309775 -id:A309775 - cp's OEIS frontend

A335977 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*(1 - exp(x)) + x).

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -1, -1, 1, 1, -3, 1, 3, 2, 1, 1, -4, 5, 7, 7, 9, 1, 1, -5, 11, 5, -8, -13, 9, 1, 1, -6, 19, -9, -43, -65, -89, -50, 1, 1, -7, 29, -41, -74, -27, 37, -45, -267, 1, 1, -8, 41, -97, -53, 221, 597, 1024, 1191, -413, 1, 1, -9, 55, -183, 92, 679, 961, 805, 1351, 4723, 2180, 1

Offset: 0

Seiichi Manyama, Jul 03 2020

			Square array begins:
  1,  1,   1,   1,   1,   1,    1, ...
  1,  0,  -1,  -2,  -3,  -4,   -5, ...
  1, -1,  -1,   1,   5,  11,   19, ...
  1, -1,   3,   7,   5,  -9,  -41, ...
  1,  2,   7,  -8, -43, -74,  -53, ...
  1,  9, -13, -65, -27, 221,  679, ...
  1,  9, -89,  37, 597, 961, -341, ...

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

Columns k=0-4 give: A000012, A293037, A309775, A320432, A320433.
Main diagonal gives A334241.
Cf. A292861, A335975.

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, 11}, {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!.

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)

A341586 E.g.f.: (exp(1 - exp(x)) - 1)^2 / 2.

Original entry on oeis.org

1, 0, -4, -5, 22, 98, -5, -1458, -5136, 9053, 161328, 549822, -1954067, -30099188, -114161728, 500200027, 8875931202, 42311243830, -149028931789, -3816065804086, -24704581255020, 33033659868037, 2184285021783940, 20047242475274290, 30117550563701293

Offset: 2

Ilya Gutkovskiy, Feb 15 2021

sign

Cf. A000225, A000558, A000587, A008277, A213170, A309775.

nmax = 26; CoefficientList[Series[(Exp[1 - Exp[x]] - 1)^2/2, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
Table[Sum[(-1)^k StirlingS2[n, k] StirlingS2[k, 2], {k, 2, n}], {n, 2, 26}]

a(n) = Sum_{k=2..n} (-1)^k * Stirling2(n, k) * Stirling2(k, 2).
a(n) = Sum_{k=2..n} (-1)^k * Stirling2(n, k) * (2^(k-1) - 1).
a(n) = Sum_{k=1..n-1} binomial(n-1, k) * A000587(k) * A000587(n-k).

cp's OEIS Frontend

A335977 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*(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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A341586 E.g.f.: (exp(1 - exp(x)) - 1)^2 / 2.

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cp's OEIS Frontend

A335977 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*(1 - exp(x)) + x).

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Programs

Mathematica

Formula

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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Maple

Formula

A341586 E.g.f.: (exp(1 - exp(x)) - 1)^2 / 2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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).