A058877 -id:A058877 - cp's OEIS frontend

A361718 Triangular array read by rows. T(n,k) is the number of labeled directed acyclic graphs on [n] with exactly k nodes of indegree 0.

1, 0, 1, 0, 2, 1, 0, 15, 9, 1, 0, 316, 198, 28, 1, 0, 16885, 10710, 1610, 75, 1, 0, 2174586, 1384335, 211820, 10575, 186, 1, 0, 654313415, 416990763, 64144675, 3268125, 61845, 441, 1, 0, 450179768312, 286992935964, 44218682312, 2266772550, 43832264, 336924, 1016, 1

Offset: 0

Geoffrey Critzer, Apr 02 2023

Also the number of sets of n nonempty subsets of {1..n}, k of which are singletons, such that there is only one way to choose a different element from each. For example, row n = 3 counts the following set-systems:
  {{1},{1,2},{1,3}}    {{1},{2},{1,3}}    {{1},{2},{3}}
  {{1},{1,2},{2,3}}    {{1},{2},{2,3}}
  {{1},{1,3},{2,3}}    {{1},{3},{1,2}}
  {{2},{1,2},{1,3}}    {{1},{3},{2,3}}
  {{2},{1,2},{2,3}}    {{2},{3},{1,2}}
  {{2},{1,3},{2,3}}    {{2},{3},{1,3}}
  {{3},{1,2},{1,3}}    {{1},{2},{1,2,3}}
  {{3},{1,2},{2,3}}    {{1},{3},{1,2,3}}
  {{3},{1,3},{2,3}}    {{2},{3},{1,2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{1,3},{1,2,3}}
  {{2},{1,2},{1,2,3}}
  {{2},{2,3},{1,2,3}}
  {{3},{1,3},{1,2,3}}
  {{3},{2,3},{1,2,3}}

			Triangle begins:
  1;
  0,     1;
  0,     2,     1;
  0,    15,     9,    1;
  0,   316,   198,   28,  1;
  0, 16885, 10710, 1610, 75, 1;
  ...

E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.
R. W. Robinson, Counting digraphs with restrictions on the strong components, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.

Cf. A058876 (mirror), A361579, A224069.
Row-sums are A003024, unlabeled A003087.
Column k = 1 is A003025(n) = |n*A134531(n)|.
Column k = n-1 is A058877.
For fixed sinks we get A368602.
A058891 counts set-systems, unlabeled A000612.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A000169, A059201, A082402, A088957, A133686, A334282, A350415, A367904, A367908, A368600, A368601.

nn = 8; B[n_] := n! 2^Binomial[n, 2] ;ggf[egf_] := Normal[Series[egf, {z, 0, nn}]] /. Table[z^i -> z^i/2^Binomial[i, 2], {i, 0, nn}];Table[Take[(Table[B[n], {n, 0, nn}] CoefficientList[ Series[ggf[Exp[(u - 1) z]]/ggf[Exp[-z]], {z, 0, nn}], {z, u}])[[i]], i], {i, 1, nn + 1}] // Grid
nv=4;Table[Length[Select[Subsets[Subsets[Range[n]],{n}], Count[#,{_}]==k&&Length[Select[Tuples[#], UnsameQ@@#&]]==1&]],{n,0,nv},{k,0,n}]

T(n,k) = A368602(n,k) * binomial(n,k). - Gus Wiseman, Jan 03 2024

A368602 Triangle read by rows where T(n,k) is the number of labeled acyclic digraphs on {1..n} with sinks {1..k}.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 3, 1, 0, 79, 33, 7, 1, 0, 3377, 1071, 161, 15, 1, 0, 362431, 92289, 10591, 705, 31, 1, 0, 93473345, 19856703, 1832705, 93375, 2945, 63, 1, 0, 56272471039, 10249747713, 789619327, 32382465, 782719, 12033, 127, 1

Offset: 0

Gus Wiseman, Jan 02 2024

Also the number of set-systems with n vertices and n edges such that {i} is a singleton edge iff i <= k, and such that there is only one way to choose a different vertex from each edge.

			Triangle begins:
    1
    0    1
    0    1    1
    0    5    3    1
    0   79   33    7    1
    0 3377 1071  161   15    1
    ...
Row n = 3 counts the following set-systems:
  {{1},{1,2},{1,3}}    {{1},{2},{1,3}}    {{1},{2},{3}}
  {{1},{1,2},{2,3}}    {{1},{2},{2,3}}
  {{1},{1,3},{2,3}}    {{1},{2},{1,2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{1,3},{1,2,3}}

Column k = n-1 is A000225 = A058877(n)/n.
Column k = 1 is A134531 (up to sign) or A003025(n)/n, non-fixed A350415.
For any choice of k sinks we get A361718.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A000169, A003024, A003087, A082402, A088957, A334282, A367862, A367904, A367908, A368600, A368601.

Table[Length[Select[Subsets[Subsets[Range[n]],{n}], Union@@Cases[#,{_}]==Range[k] && Length[Select[Tuples[#],UnsameQ@@#&]]==1&]], {n,0,3},{k,0,n}]

T(n,k) = A361718(n,k)/binomial(n,k).

More terms from Alois P. Heinz, Jan 04 2024

A091913 Triangle read by rows: a(n,k) = C(n,k)*(2^(n-k) - 1) for k= n, where k=0..max(n-1,0).

Original entry on oeis.org

0, 1, 3, 2, 7, 9, 3, 15, 28, 18, 4, 31, 75, 70, 30, 5, 63, 186, 225, 140, 45, 6, 127, 441, 651, 525, 245, 63, 7, 255, 1016, 1764, 1736, 1050, 392, 84, 8, 511, 2295, 4572, 5292, 3906, 1890, 588, 108, 9, 1023, 5110, 11475, 15240, 13230, 7812, 3150, 840, 135, 10, 2047

Offset: 0

Ross La Haye, Mar 10 2004

Row lengths are 1,1,2,3,4,... = A028310. - M. F. Hasler, Jul 21 2012
Rows: Sum of the n-th row = A001047(n); Sum of the n-th row excluding column 0 = A028243(n+1). Columns: a(n,0) = A000225(n); a(n,1) = A058877(n). Diagonals: a(n,n-2) = A045943(n-1). Also note that the sums of the antidiagonals = A006684.
As an infinite lower triangular matrix * the Bernoulli numbers as a vector (Cf. A027641) = the natural numbers: [1, 2, 3, ...]. The same matrix * the Bernoulli number version starting [1, 1/2, 1/6, ...] = A001787: (1, 4, 12, 32, ...). - Gary W. Adamson, Mar 13 2012

			Triangle begins
   0;
   1;
   3,   2;
   7,   9,   3;
  15,  28,  18,   4;
  31,  75,  70,  30,   5;
  63, 186, 225, 140,  45,   6;
  ...
a(5,3) = 30 because C(5,3) = 10, 2^(5 - 3) - 1 = 3 and 10 * 3 = 30.

Cf. A007318, A038207.

Cf. A001787, A027641, A027642.

For k>=n, a(n, k) = 0; for k < n, a(n, k) = C(n, k) * (2^(n-k) - 1) = Sum [C(n,k) * C(n-k, m), {m=1 to n-k}]. [Formula corrected Aug 22 2006]
The triangle (1; 3,2; 7,9,3; ...) = A007318^2 - A007318, then delete the right border of zeros. - Gary W. Adamson, Nov 16 2007
O.g.f.: 1/( (1 - (1 + x)*t)*(1 - (2 + x)*t) ) = 1 + (3 + 2*x)*t + (7 + 9*x + 3*x^2)*t^2 + .... - Peter Bala, Jul 16 2013

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A126136 Binomial transform of A107430.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 8, 7, 9, 3, 16, 15, 28, 16, 6, 32, 31, 75, 55, 40, 10, 64, 63, 186, 156, 165, 75, 20, 128, 127, 441, 399, 546, 336, 175, 35, 256, 255, 1016, 960, 1596, 1176, 896, 336, 70, 512, 511, 2295, 2223, 4320, 3564, 3528, 1848, 756, 126, 1024, 1023, 5110, 5020, 11115, 9855, 11880, 7680, 4620, 1470, 252

Offset: 0

Gary W. Adamson, Dec 18 2006

Row sums = powers of 3.

			First few rows of the triangle are:
1;
2, 1;
4, 3, 2;
8, 7, 9, 3;
16, 15, 28, 16, 6;
32, 31, 75, 55, 40, 10;
...

Cf. A107430.

Columns : A000079, A000225, A058877, A027540

tabl(nn) = {p = matrix(nn+1, nn+1, n, k, binomial(n-1, k-1)); m = matrix(nn+1, nn+1, n, k, if (k<=n, binomial(n-1, (k-1)\2), 0)); r = p*m; for (n=0, nn, for (k=0, n, print1(r[n+1,k+1], ", ");); print(););} \\ Michel Marcus, Jul 03 2017

Given M = A107430 as an infinite lower triangular matrix and P = Pascal's triangle, A126136 = P*M.

More terms from Philippe Deléham, Jul 02 2017

A130265 Triangle read by rows: matrix product A007318 * A051340.

Original entry on oeis.org

1, 2, 2, 4, 5, 3, 8, 10, 10, 4, 16, 19, 23, 17, 5, 32, 36, 46, 46, 26, 6, 64, 69, 87, 102, 82, 37, 7, 128, 134, 162, 204, 204, 134, 50, 8, 256, 263, 303, 387, 443, 373, 205, 65, 9, 512, 520, 574, 718, 886, 886, 634, 298, 82, 10

Offset: 0

Gary W. Adamson, May 18 2007

			First few rows of the triangle are:
   1;
   2,  2;
   4,  5,  3;
   8, 10, 10,   4;
  16, 19, 23,  17,  5;
  32, 36, 46,  46, 26,  6;
  64, 69, 87, 102, 82, 37,  7;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001787 (row sums), A007318, A051340, A058877, A084633, A258431.

A130265:= func< n,k | k eq n select n+1 else (k+1)*Binomial(n,k) + (&+[Binomial(n, j+k): j in [1..n-k]]) >;
[A130265(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2023

A051340 := proc(n,k)
    if k = n then
        n+1 ;
    elif k <= n then
        1;
    else
        0;
    end if;
end proc:
A130265 := proc(n,k)
    add( binomial(n,j)*A051340(j,k),j=k..n) ;
end proc:
seq(seq(A130265(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Aug 06 2016

T[n_, k_]:= (k+1)*Binomial[n,k] + Binomial[n,k+1]*Hypergeometric2F1[1, k-n+1, k+2, -1];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2023 *)

def A130265(n,k): return (k+1)*binomial(n,k) + sum(binomial(n, j+k) for j in range(1,n-k+1))
flatten([[A130265(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 18 2023

Binomial transform of A051340.
From G. C. Greubel, Mar 18 2023: (Start)
T(n, k) = (k+1)*binomial(n,k) + Sum_{j=1..n-k} binomial(n, j+k).
T(n, k) = (k+1)*binomial(n,k) + binomial(n,k+1)*Hypergeometric2F1([1, k-n+1], [k+2], -1).
T(2*n, n) = (1/2)*T(2*n+1, n) = A258431(n+1).
Sum_{k=0..n} T(n, k) = A001787(n+1).
Sum_{k=0..n-1} T(n, k) = A058877(n+1), for n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A084633(n). (End)

Missing term inserted by R. J. Mathar, Aug 06 2016

A178759 Expansion of e.g.f. 3xexp(x)*(exp(x)-1)^2.

Original entry on oeis.org

0, 0, 0, 18, 144, 750, 3240, 12642, 46368, 163350, 559800, 1881066, 6229872, 20406750, 66273480, 213759090, 685601856, 2188698150, 6959413080, 22053083514, 69672773520, 219535296750, 690106487400, 2164714299138, 6777100916064, 21179698653750, 66083277045240, 205880260458762

Offset: 0

Geoffrey Critzer, Dec 26 2010

a(n) is the sum of the digits in ternary sequences of length n, in which each element of the alphabet, {0,1,2} appears at least once in the sequence.
Generally, the e.g.f. for such sum of n-ary sequences (taken on an alphabet of {0,1,2,...,n-1}) is binomial(n,2)*x*exp(x)*(exp(x)-1)^(n-1).
Cf. A058877 which is the sum of the digits in such binary sequences.

			a(3)=18 because there are six length 3 sequences on {0,1,2} that contain at least one 0, at least one 1 and at least one 2: (0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1),(2,1,0).  The digits sum to 18.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-58,144,-193,132,-36).

Cf. A058877, A178756.

List([0..30], n -> (3^n - 3*2^n + 3)*n); # G. C. Greubel, Jan 24 2019

[(3^n - 3*2^n + 3)*n: n in [0..30]]; // G. C. Greubel, Jan 24 2019

Range[0,20]! CoefficientList[Series[3x Exp[x](Exp[x]-1)^2,{x,0,20}],x]
Table[(3^n -3*2^n +3)*n, {n,0,30}] (* G. C. Greubel, Jan 24 2019 *)

my(x='x+O('x^30)); concat([0,0,0],Vec(serlaplace(3*x*exp(x)*(exp(x)-1)^2))) \\ Joerg Arndt, May 13 2013

concat([0,0,0], Vec(6*x^3*(11*x^2-12*x+3)/((x-1)^2*(2*x-1)^2*(3*x-1)^2) + O(x^100))) \\ Colin Barker, Nov 30 2014

vector(30, n, n--; (3^n - 3*2^n + 3)*n) \\ G. C. Greubel, Jan 24 2019

[(3^n - 3*2^n + 3)*n for n in (0..30)] # G. C. Greubel, Jan 24 2019

E.g.f.: 3*x*exp(x)*(exp(x)-1)^2.
a(n) = (3^n - 3*2^n + 3)*n. - Mark van Hoeij, May 13 2013
G.f.: 6*x^3*(11*x^2-12*x+3) / ((x-1)^2*(2*x-1)^2*(3*x-1)^2). - Colin Barker, Nov 30 2014

A276659 Accumulation of the upper left triangle used in binomial transform of nonnegative integers.

Original entry on oeis.org

0, 2, 11, 39, 114, 300, 741, 1757, 4052, 9162, 20415, 44979, 98214, 212888, 458633, 982905, 2097000, 4456278, 9436995, 19922735, 41942810, 88080132, 184549101, 385875669, 805306044, 1677721250, 3489660551, 7247756907, 15032385102, 31138512432, 64424508945

Offset: 0

Jean-François Alcover and Francois Alcover, Sep 11 2016

After 0, is this the second column of A108284? [Bruno Berselli, Sep 13 2016 - this comment may be removed if the property is confirmed.]

			Starting from the triangle:
   0,  1,  2,  3,  4,  5, ...
   1,  3,  5,  7,  9, ...
   4,  8, 12, 16, ...
  12, 20, 28, ...
  32, 48, ...
  80, ...
  ...
the first terms are:
a(0) = 0;
a(1) = a(0) + 1 + 1 = 2;
a(2) = a(1) + 4 + 3 + 2 = 11;
a(3) = a(2) + 12 + 8 + 5 + 3 = 39, etc.
First column is A001787: n*2^(n-1).

Index entries for linear recurrences with constant coefficients, signature (7,-19,25,-16,4).

Cf. A001787, A058877, A062111, A152920.

[(2^(n+2)-n-3)*n/2: n in [0..40]]; // Vincenzo Librandi, Sep 13 2016

A276659:=n->n*(2^(n+2) - n - 3)/2: seq(A276659(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

t[0, k_] := k; t[n_, k_] := t[n, k] = t[n - 1, k] + t[n - 1, k + 1]; a[n_] := Sum[t[m, k], {m, 0, n}, {k, 0, n - m}]; Table[a[n], {n, 0, 30}]
Table[(2^(n + 2) - n - 3) n / 2, {n, 0, 30}] (* Vincenzo Librandi, Sep 13 2016 *)

x='x+O('x^99); concat(0, Vec(x*(2-3*x)/((1-x)^3*(1-2*x)^2))) \\ Altug Alkan, Sep 14 2017

O.g.f.: x*(2 - 3*x)/((1 - x)^3*(1 - 2*x)^2).
E.g.f.: x*exp(x)*(8*exp(x) - x - 4)/2.
a(n) = n*(2^(n+2) - n - 3)/2.
a(n) = 7*a(n-1) - 19*a(n-2) + 25*a(n-3) - 16*a(n-4) + 4*a(n-5) for n > 4.
a(n) = a(n-1) + A058877(n+1). - R. J. Mathar, Sep 14 2016
a(n) = Sum_{k=2..n+3} Sum_{i=2..n+3} k * C(n-i+3,k). - Wesley Ivan Hurt, Sep 20 2017

Edited and extended by Bruno Berselli, Sep 13 2016

A369919 Triangular array read by rows. T(n,k) is the number of labeled posets on [n] of rank at most one with exactly k elements of positive indegree, n >= 0, 0 <= k <= max{0,n-1}.

Original entry on oeis.org

1, 1, 1, 2, 1, 9, 3, 1, 28, 54, 4, 1, 75, 490, 270, 5, 1, 186, 3375, 6860, 1215, 6, 1, 441, 20181, 118125, 84035, 5103, 7, 1, 1016, 111132, 1668296, 3543750, 941192, 20412, 8, 1, 2295, 580644, 21003948, 116363646, 95681250, 9882516, 78732, 9

Offset: 0

Geoffrey Critzer, Feb 05 2024

The rank of a poset is the number of cover relations in a maximal chain.

Equivalently, T(n,k) is the number of labeled posets P on [n] of rank at most one such that |image(P)| = k.

			Triangle begins
  1;
  1;
  1,   2;
  1,   9,    3;
  1,  28,   54,    4;
  1,  75,  490,  270,    5;
  1, 186, 3375, 6860, 1215, 6;
  ...

Cf. A001831 (row sums), A058877, A263859, A369921.

nn = 9; Map[Select[#, # > 0 &] &,Table[n!, {n, 0, nn}] CoefficientList[Series[ Sum[ Exp[y  x]^(2^n - 1)  x^n/n!, {n, 0, nn}], {x, 0, nn}], {x, y}]] // Grid

E.g.f.: Sum_{n>=0} x^n/n!*exp(y*x)^(2^n-1).

T(n,1) = A058877(n).

A381888 Triangle read by rows: T(n, k) = (n + 1) * Sum_{j=k..n} binomial(n, j) * Eulerian1(j, j - k).

Original entry on oeis.org

1, 2, 2, 3, 9, 3, 4, 28, 28, 4, 5, 75, 165, 75, 5, 6, 186, 786, 786, 186, 6, 7, 441, 3311, 6181, 3311, 441, 7, 8, 1016, 12888, 40888, 40888, 12888, 1016, 8, 9, 2295, 47529, 241191, 404361, 241191, 47529, 2295, 9, 10, 5110, 168670, 1312750, 3445510, 3445510, 1312750, 168670, 5110, 10

Offset: 0

Peter Luschny, Mar 11 2025

Consider A381706, the number of permutations of k chosen numbers in [n] with i-1 descents, as a sequence of squares of size 1x1, 2x2, 3x3, ..., as displayed in the example section of A381706. Conjecture: T(n, k) is the sum of column k+1 of the (n+1)th square; in other words: T(n, k) = Sum_{j=0..n} b(n+1, j+1, k+1).

			Triangle starts:
  [0] 1;
  [1] 2,    2;
  [2] 3,    9,     3;
  [3] 4,   28,    28,      4;
  [4] 5,   75,   165,     75,      5;
  [5] 6,  186,   786,    786,    186,      6;
  [6] 7,  441,  3311,   6181,   3311,    441,     7;
  [7] 8, 1016, 12888,  40888,  40888,  12888,  1016,    8;
  [8] 9, 2295, 47529, 241191, 404361, 241191, 47529, 2295, 9;

Cf. A046802, A173018 (Eulerian1), A122045 (Euler), A058877 (column 1), A007526 (row sums), A381706 (generalized Eulerian).

T := (n, k) -> (n + 1)*add(binomial(n, j)*combinat:-eulerian1(j, j - k), j = k .. n):
for n from 0 to 8 do seq(T(n, k), k=0..n) od;
# Using the e.g.f.:
egf := ((y - 1)*(y*exp(x*y)*(x*y + 1) - (x + 1)*exp(x*(2*y - 1))))/(exp(x*(y - 1)) - y)^2:
ser := simplify(series(egf, x, 10)):
seq(seq(n!*coeff(coeff(ser, x, n), y, k), k = 0..n), n = 0..9);

# Using function eulerian1 from A173018.
def T(n: int, k: int) -> int:
    return (n + 1) * sum(binomial(n, j) * eulerian1(j, j-k) for j in (k..n))
def Trow(n) -> list[int]: return [T(n, k) for k in (0..n)]
for n in (0..8): print(f"{n}: ", Trow(n))

T(n, k) = n! * [y^k] [x^n] ((y - 1)*(y*exp(x*y)*(x*y + 1) - (x + 1)*exp(x*(2*y - 1)))) / (exp(x*(y - 1)) - y)^2.
Sum_{k=0..n} (-1)^k * T(n, k) = (-1)^n * (n + 1) * Euler(n).
T(n, k) = (n + 1) * A046802(n, k).

A171150 Triangle related to T(x,2x).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 9, 7, 1, 6, 20, 28, 15, 1, 10, 50, 85, 75, 31, 1, 20, 105, 255, 294, 186, 63, 1, 35, 245, 651, 1029, 903, 441, 127, 1, 70, 504, 1736, 3108, 3612, 2568, 1016, 255, 1, 126, 1134, 4116, 9324, 12636, 11556, 6921, 2295, 511, 1, 252, 2310, 10290, 25080, 42120, 46035, 34605, 17930, 5110, 1023, 1

Offset: 0

Philippe Deléham, Dec 04 2009

Let the triangle T_(x,y)=T defined by T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1.
This triangle gives the coefficients of Sum_{k=0..n} T(n,k) where y=2x.
T_(0,0) = A053121, T_(1,2) = A039599, T_(2,4) = A124575.
First column of T_(x,2x) is given by A126222.

			Triangle begins:
   1;
   1,  1;
   2,  3,  1;
   3,  9,  7,  1;
   6, 20, 28, 15,  1;
  10, 50, 85, 75, 31,  1;
  ...

M. Barnabei, F. Bonetti, and M. Silimbani, The Eulerian numbers on restricted centrosymmetric permutations, PU. M. A. Vol. 21 (2010), No. 2, pp. 99-118 (see Table p. 118, with additional zeros); see also, arXiv:0910.2376 [math.CO], 2009.

Cf. A000012, A000225, A058877, A126222.

Row sums give A000984.

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001405(n), A000984(n), A133158(n) for x = -1, 0, 1, 2 respectively.

More terms from Alois P. Heinz, Jan 31 2023

cp's OEIS Frontend

A361718 Triangular array read by rows. T(n,k) is the number of labeled directed acyclic graphs on [n] with exactly k nodes of indegree 0.

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A368602 Triangle read by rows where T(n,k) is the number of labeled acyclic digraphs on {1..n} with sinks {1..k}.

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A091913 Triangle read by rows: a(n,k) = C(n,k)*(2^(n-k) - 1) for k= n, where k=0..max(n-1,0).

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A126136 Binomial transform of A107430.

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A130265 Triangle read by rows: matrix product A007318 * A051340.

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A178759 Expansion of e.g.f. 3*x*exp(x)*(exp(x)-1)^2.

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A276659 Accumulation of the upper left triangle used in binomial transform of nonnegative integers.

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A178759 Expansion of e.g.f. 3xexp(x)*(exp(x)-1)^2.