A131689 -id:A131689 - cp's OEIS frontend

A278073 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.

1, 0, 1, 0, 1, 20, 0, 1, 168, 1680, 0, 1, 1364, 55440, 369600, 0, 1, 10920, 1561560, 33633600, 168168000, 0, 1, 87380, 42771456, 2385102720, 34306272000, 137225088000, 0, 1, 699048, 1160164320, 158411809920, 5105916816000, 54752810112000, 182509367040000

Offset: 0

Peter Luschny, Jan 22 2017

			Triangle begins:
[1]
[0, 1]
[0, 1,    20]
[0, 1,   168,    1680]
[0, 1,  1364,   55440,   369600]
[0, 1, 10920, 1561560, 33633600, 168168000]

Cf. A014606 (diagonal), A243664 (row sums), A002115 (alternating row sums), A281479 (central coefficients), A327023 (refinement).

Cf. A097805 (m=0), A131689 (m=1), A241171 (m=2), A278074 (m=4).

P := proc(m, n) option remember; if n = 0 then 1 else
add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:
for n from 0 to 6 do PolynomialTools:-CoefficientList(P(3,n), x) od;
# Alternatively:
A278073_row := proc(n)
1/(1-t*((1/3)*exp(x)+(2/3)*exp(-(1/2)*x)*cos((1/2)*x*sqrt(3))-1));
expand(series(%,x,3*n+1)); (3*n)!*coeff(%,x,3*n);
PolynomialTools:-CoefficientList(%,t) end:
for n from 0 to 6 do A278073_row(n) od;

With[{m = 3}, Table[Expand[j!*SeriesCoefficient[1/(1 - t*(MittagLefflerE[m, x^m] - 1)), {x, 0, j}]], {j, 0, 21, m}]];
Function[arg, CoefficientList[arg, t]] /@ % // Flatten

R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(30)
@cached_function
def P(m, n):
    if n == 0: return R(1)
    return expand(sum(binomial(m*n, m*k)*P(m, n-k)*x for k in (1..n)))
def A278073_row(n): return list(P(3, n))
for n in (0..6): print(A278073_row(n)) # Peter Luschny, Mar 24 2020

E.g.f.: 1/(1-t*((1/3)*exp(x)+(2/3)*exp(-(1/2)*x)*cos((1/2)*x*sqrt(3))-1)), nonzero terms.

A038719 Triangle T(n,k) (0 <= k <= n) giving number of chains of length k in partially ordered set formed from subsets of n-set by inclusion.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 8, 19, 18, 6, 16, 65, 110, 84, 24, 32, 211, 570, 750, 480, 120, 64, 665, 2702, 5460, 5880, 3240, 720, 128, 2059, 12138, 35406, 57120, 52080, 25200, 5040, 256, 6305, 52670, 213444, 484344, 650160, 514080, 221760, 40320, 512, 19171

Offset: 0

N. J. A. Sloane, May 02 2000

The relation of this triangle to A143494 given in the Formula section leads to the following combinatorial interpretation: T(n,k) gives the number of partitions of the set {1,2,...,n+2} into k + 2 blocks where 1 and 2 belong to two distinct blocks and the remaining k blocks are labeled from a fixed set of k labels. - Peter Bala, Jul 10 2014

Also, the number of distinct k-level fuzzy subsets of a set consisting of n elements ordered by set inclusion. - Rajesh Kumar Mohapatra, Mar 16 2020

			Triangle begins
   1;
   2,   1;
   4,   5,   2;
   8,  19,  18,   6;
  16,  65, 110,  84,  24;
  ...
From _Peter Bala_, Feb 02 2022: (Start)
Table of successive differences of k^2 starting at k = 2
4   9   16
  5   7
    2
gives [4, 5, 2] as row 2 of this triangle.
Table of successive differences of k^3 starting at k = 2
8   27   64   125
  19   37   61
     18   24
        6
gives [8, 19, 8, 6] as row 3 of this triangle. (End)

Alois P. Heinz, Rows n = 0..140, flattened
Peter Bala, Deformations of the Hadamard product of power series
L. Bartlomiejczyk and J. Drewniak, A characterization of sets and operations invariant under bijections, Aequationes Mathematicae 68 (2004), pp. 1-9.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.
Index entries for sequences related to posets

Row sums give A007047. Columns give A000079, A001047, A038721. Next-to-last diagonal gives A038720.
Diagonal gives A000142. - Rajesh Kumar Mohapatra, Mar 16 2020
Cf. A028246. A019538, A143494.

a038719 n k = a038719_tabl !! n !! k
a038719_row n = a038719_tabl !! n
a038719_tabl = iterate f [1] where
   f row = zipWith (+) (zipWith (*) [0..] $ [0] ++ row)
                       (zipWith (*) [2..] $ row ++ [0])
-- Reinhard Zumkeller, Jul 08 2012

T:= proc(n, k) option remember;
      `if` (n=0, `if`(k=0, 1, 0), k*T(n-1, k-1) +(k+2)*T(n-1, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Aug 02 2011

t[n_, k_] := Sum[ (-1)^(k-i)*Binomial[k, i]*(2+i)^n, {i, 0, k}]; Flatten[ Table[ t[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, after Pari *)

T(n,k)=sum(i=0,k,(-1)^(k-i)*binomial(k,i)*(2+i)^n)

T(n, k) = Sum_{j=0..k} (-1)^j*C(k, j)*(k+2-j)^n.
T(n+1, k) = k*T(n, k-1) + (k+2)*T(n, k), T(0,0) = 1, T(0,k) = 0 for k>0.
E.g.f.: exp(2*x)/(1+y*(1-exp(x))). - Vladeta Jovovic, Jul 21 2003
A038719 as a lower triangular matrix is the binomial transform of A028246. - Gary W. Adamson, May 15 2005
Binomial transform of n-th row = 2^n + 3^n + 4^n + ...; e.g., binomial transform of [8, 19, 18, 6] = 2^3 + 3^3 + 4^3 + 5^3 + ... = 8, 27, 64, 125, ... - Gary W. Adamson, May 15 2005
From Peter Bala, Jul 09 2014: (Start)
T(n,k) = k!*( Stirling2(n+2,k+2) - Stirling2(n+1,k+2) ).
T(n,k) = k!*A143494(n+2,k+2).
n-th row polynomial = 1/(1 + x)*( sum {k >= 0} (k + 2)^n*(x/(1 + x))^k ). Cf. A028246. (End)
The row polynomials have the form (2 + x) o (2 + x) o ... o (2 + x), where o denotes the black diamond multiplication operator of Dukes and White. See example E12 in the Bala link. - Peter Bala, Jan 18 2018
Z(P,m) = Sum_{k=0..n} T(n,k)Binomial(m-2,k) = m^n, the zeta polynomial of the poset B_n. Each length m multichain from 0 to 1 in B_n corresponds to a function from [n] into [m]. - Geoffrey Critzer, Dec 25 2020
The entries in row n are the first terms in a table of the successive differences of the sequence [2^n, 3^n, 4^n, ...]. Examples are given below. - Peter Bala, Feb 02 2022

More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000

A285824 Number T(n,k) of ordered set partitions of [n] into k blocks such that equal-sized blocks are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 6, 1, 0, 1, 11, 18, 1, 0, 1, 30, 75, 40, 1, 0, 1, 52, 420, 350, 75, 1, 0, 1, 126, 1218, 3080, 1225, 126, 1, 0, 1, 219, 4242, 17129, 15750, 3486, 196, 1, 0, 1, 510, 14563, 82488, 152355, 63756, 8526, 288, 1, 0, 1, 896, 42930, 464650, 1049895, 954387, 217560, 18600, 405, 1

Offset: 0

Alois P. Heinz, Apr 27 2017

			T(3,1) = 1: 123.
T(3,2) = 6: 1|23, 23|1, 2|13, 13|2, 3|12, 12|3.
T(3,3) = 1: 1|2|3.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   6,    1;
  0, 1,  11,   18,     1;
  0, 1,  30,   75,    40,     1;
  0, 1,  52,  420,   350,    75,    1;
  0, 1, 126, 1218,  3080,  1225,  126,   1;
  0, 1, 219, 4242, 17129, 15750, 3486, 196, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000007, A057427, A285917, A285918, A285919, A285920, A285921, A285922, A285923, A285924, A285925.
Main diagonal and first lower diagonal give: A000012, A002411.
Row sums give A120774.
T(2n,n) gives A285926.
Cf. A048993, A131689, A226874, A285849.

b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*x^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..12);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n-i*j, i-1, p+j]*x^j*multinomial[n, Join[{n-i*j}, Table[i, j]]]/ j!^2, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 28 2018, after Alois P. Heinz *)

A278074 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 4.

Original entry on oeis.org

1, 0, 1, 0, 1, 70, 0, 1, 990, 34650, 0, 1, 16510, 2702700, 63063000, 0, 1, 261630, 213519150, 17459442000, 305540235000, 0, 1, 4196350, 17651304000, 4350310965000, 231905038365000, 3246670537110000

Offset: 0

Peter Luschny, Jan 22 2017

			Triangle starts:
[1]
[0, 1]
[0, 1,     70]
[0, 1,    990,     34650]
[0, 1,  16510,   2702700,    63063000]
[0, 1, 261630, 213519150, 17459442000, 305540235000]

Robert Israel, Table of n, a(n) for n = 0..8510

Cf. A014608 (diagonal), A243665 (row sums), A211212 (alternating row sums), A281480 (central coefficients).
Cf. A097805 (m=0), A131689 (m=1), A241171 (m=2), A278073 (m=3).
Cf. A327024 (refinement).

P := proc(m,n) option remember; if n = 0 then 1 else
add(binomial(m*n,m*k)* P(m,n-k)*x, k=1..n) fi end:
for n from 0 to 6 do PolynomialTools:-CoefficientList(P(4,n), x) od;
# Alternatively:
A278074_row := proc(n) 1/(1-t*((cosh(x)+cos(x))/2-1)); expand(series(%,x,4*n+1));
(4*n)!*coeff(%,x,4*n); PolynomialTools:-CoefficientList(%,t) end:
for n from 0 to 5 do A278074_row(n) od;

With[{m = 4}, Table[Expand[j!*SeriesCoefficient[1/(1 - t*(MittagLefflerE[m, x^m] - 1)), {x, 0, j}]], {j, 0, 24, m}]];
Function[arg, CoefficientList[arg, t]] /@ % // Flatten

# uses [P from A278073]
def A278074_row(n): return list(P(4, n))
for n in (0..6): print(A278074_row(n)) # Peter Luschny, Mar 24 2020

E.g.f.: 1/(1-t*((cosh(x)+cos(x))/2-1)), nonzero terms.

A094420 Generalized ordered Bell numbers Bo(n,n).

Original entry on oeis.org

1, 1, 10, 219, 8676, 544505, 49729758, 6232661239, 1026912225160, 215270320769109, 55954905981282210, 17662898483917308083, 6655958151527584785900, 2951503248457748982755953, 1521436331153097968932487206, 902143190212525713006814917615, 609729139653483641913607434550800

Offset: 0

Ralf Stephan, May 02 2004

nonn

Main diagonal of array A094416.

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A094416, A321189.
The coefficients of the Fubini polynomials are A131689.
Central column of A344499.

A094420:= func< n | (&+[Factorial(k)*n^k*StirlingSecond(n,k): k in [0..n]]) >;
[A094420(n): n in [0..25]]; // G. C. Greubel, Jan 12 2024

F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n-k)*x, k=1..n)) end:
a := n -> subs(x = n, F(n)):
seq(a(n), n = 0..16); # Peter Luschny, May 21 2021

Table[Sum[k!*n^k*StirlingS2[n, k], {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jul 23 2018 *)

{a(n) = sum(k=0, n, k!*n^k*stirling(n, k, 2))} \\ Seiichi Manyama, Jun 12 2020

def aList(len):
    R. = PowerSeriesRing(QQ)
    f = lambda n: R(1/(1 + n * (1 - exp(x))))
    return [factorial(n)*f(n).list()[n] for n in (0..len-1)]
print(aList(17)) # Peter Luschny, May 21 2021

a(n) ~ sqrt(2*Pi) * n^(2*n + 5/2) / exp(n - 3/2). - Vaclav Kotesovec, Jul 23 2018
a(n) = Sum_{k=0..n} k!*n^k*Stirling2(n, k). - Seiichi Manyama, Jun 12 2020
From Peter Luschny, May 21 2021: (Start)
a(n) = F_{n}(n), the Fubini polynomial F_{n}(x) evaluated at x = n.
a(n) = n! * [x^n] (1 / (1 + n * (1 - exp(x)))).  (End)

More terms from Seiichi Manyama, Jun 12 2020

A122193 Triangle T(n,k) of number of loopless multigraphs with n labeled edges and k labeled vertices and without isolated vertices, n >= 1; 2 <= k <= 2*n.

Original entry on oeis.org

1, 1, 6, 6, 1, 24, 114, 180, 90, 1, 78, 978, 4320, 8460, 7560, 2520, 1, 240, 6810, 63540, 271170, 604800, 730800, 453600, 113400, 1, 726, 43746, 774000, 6075900, 25424280, 61923960, 90720000, 78813000, 37422000, 7484400

Offset: 1

Vladeta Jovovic, Aug 24 2006

T(n,k) equals the number of arrangements on a line of n (nondegenerate) finite closed intervals having k distinct endpoints. See the 'IBM Ponder This' link. An example is given below. - Peter Bala, Jan 28 2018

T(n,k) equals the number of alignments of length k of n strings each of length 2. See Slowinski. Cf. A131689 (alignments of strings of length 1) and A299041 (alignments of strings of length 3). - Peter Bala, Feb 04 2018

			Triangle begins:
  1;
  1,  6,   6;
  1, 24, 114,  180,   90;
  1, 78, 978, 4320, 8460, 7560, 2520;
  ...
From _Francisco Santos_, Nov 17 2017: (Start)
For n=3 edges and k=4 vertices there are three loopless multigraphs without isolated vertices: a path, a Y-graph, and the multigraph {12, 34, 34}. The number of labelings in each is 3!4!/a, where a is the number of automorphisms. This gives respectively 3!4!/2 = 72, 3!4!/6 = 24 and 3!4!/8 = 18, adding up to 72 + 24 + 18 = 114. (End)
From _Peter Bala_, Jan 28 2018: (Start)
T(2,3) = 6: Consider 2 (nondegenerate) finite closed intervals [a, b] and [c, d]. There are 6 arrangements of these two intervals with 3 distinct endpoints:
  ...a--b--d....  a < b = c < d
  ...a...c--b...  a < c < b = d
  ...a--d...b...  a = c < d < b
  ...a--b...d...  a = c < b < d
  ...c...a--d...  c < a < b = d
  ...c--a--b....  c < a = d < b
T(2,4) = 6: There are 6 arrangements of the two intervals with 4 distinct endpoints:
  ...a--b...c--d.....  no intersection a < b < c < d
  ...a...c...b...d...  a < c < b < d
  ...a...c--d...b....  [c,d] is a proper subset of [a,b]
  ...c...a...d...b...  c < a < d < b
  ...c...a--b...d... [a,b] is a proper subset of [c,d]
  ...c--d...a--b.....  no intersection c < d < a < b.
Sums of powers of triangular numbers:
Row 2: Sum_{i = 2..n-1} C(i,2)^2 = C(n,3) + 6*C(n,4) + 6*C(n,5);
Row 3: Sum_{i = 2..n-1} C(i,2)^3 = C(n,3) + 24*C(n,4) + 114*C(n,5) + 180*C(n,6) + 90*C(n,7). See A024166 and A085438.
exp( Sum_{n >= 1} R(n,2)*x^n/n ) = (1 + x + 19*x^2 + 1147*x^3 + 145606*x^4 + 31784062*x^5 + ... )^4
exp( Sum_{n >= 1} R(n,3)*x^n/n ) = (1 + x + 37*x^2 + 4453*x^3 + 1126375*x^4 + 489185863*x^5 + ... )^9
exp( Sum_{n >= 1} R(n,4)*x^n/n ) = (1 + x + 61*x^2 + 12221*x^3 + 5144411*x^4 + 3715840571*x^5 + ... )^16 (End)
From _Peter Bala_, Feb 04 2018: (Start)
T(3,3) = 24 alignments of length 3 of 3 strings each of length 2. Examples include
  (i) A B -    (ii) A - B
      - C D         - C D
      - E F         E F -
There are 18 alignments of type (i) with two gap characters in one of the columns (3 ways of putting 2 gap characters in a column x 2 ways to place the other letter in the row which doesn't yet have a gap character x 3 columns: there are 6 alignments of type (ii) with a single gap character in each column (3 ways to put a single gap character in the first column x 2 ways to then place a single gap character in the second column). (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (rows 1 <= n <= 100).
Peter Bala, Deformations of the Hadamard product of power series
Peter Bala, Notes on A122193
S. Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
J. Engbers and C. Stocker, Two Combinatorial Proofs of Identities Involving Sums of Powers of Binomial Coefficients, Integers 16 (2016), #A58.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
IBM Ponder This, Jan 01 2001
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522

Row sums give A055203.
Cf. A078739, A005799, A019538, A131689, A154283, A299041, A087127.
For Sum_{i = 2..n} C(i,2)^k see A024166 (k = 2), A085438 - A085442 ( k = 3 thru 7).

# Note that the function implements the full triangle because it can be
# much better reused and referenced in this form.
A122193 := (n,k) -> A078739(n,k)*k!/2^n:
# Displays the truncated triangle from the definition:
seq(print(seq(A122193(n,k),k=2..2*n)),n=1..6); # Peter Luschny, Mar 25 2011

t[n_, k_] := Sum[(-1)^(n - r) Binomial[n, r] StirlingS2[n + r, k], {r, 0, n}]; Table[t[n, k] k!/2^n, {n, 6}, {k, 2, 2 n}] // Flatten (* Michael De Vlieger, Nov 18 2017, after Jean-François Alcover at A078739 *)

Double e.g.f.: exp(-x)*Sum_{n>=0} exp(binomial(n,2)*y)*x^n/n!.
T(n,k) = S_{2,2}(n,k)*k!/2^n; S_{2,2} the generalized Stirling numbers A078739. - Peter Luschny, Mar 25 2011
From Peter Bala, Jan 28 2018: (Start)
T(n,k) = Sum_{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i*(i-1)/2)^n.
T(n,k) = k*(k-1)/2*( T(n-1,k) + 2*T(n-1,k-1) + T(n-1,k-2) ) for 2 < k <= 2*n with boundary conditions T(n,2) = 1 for n >= 1 and T(n,k) = 0 if (k < 2) or (k > 2*n).
n-th row polynomial R(n,x) = Sum_{i >= 2} (i*(i-1)/2)^n * x^i/(1+x)^(i+1) for n >= 1.
1/(1-x)*R(n,x/(1-x)) = Sum_{i >= 2} (i*(i-1)/2)^n*x^i for n >= 1.
R(n,x) = 1/2^n*Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*F(n+k,x), where F(n,x) = Sum_{k = 0..n} k!*Stirling2(n,k)*x^k is the n-th Fubini polynomial, the n-th row polynomial of A131689.
R(n,x) = x/(1+x)*1/2^n*Sum_{k = 0..n} binomial(n,k)*F(n+k,x) for n >= 1.
The polynomials Sum_{k = 2..2*n} T(n,k)*x^(k-2)*(1-x)^(2*n-k) are the row polynomials of A154283.
A154283 * A007318 equals the row reverse of this array.
Sum_{k = 2..2*n} T(n,k)*binomial(x,k) = ( binomial(x,2) )^n. Equivalently, Sum_{k = 2..2*n} (-1)^k*T(n,k)*binomial(x+k,k) = ( binomial(x+2,2) )^n. Cf. the Worpitzky-type identity Sum_{k = 1..n} A019538(n,k)*binomial(x,k) = x^n.
Sum_{i = 2..n-1} (i*(i-1)/2)^m = Sum_{k = 2..2*m} T(m,k) * binomial(n,k+1) for m >= 1. See Examples below.
R(n,x) = x^2 o x^2 o ... o x^2 (n factors), where o is the black diamond product of power series defined in Dukes and White. Note the polynomial x o x o ... o x (n factors) is the n-th row polynomial of A019538.
x^2*R(n,-1-x) = (1+x)^2*R(n,x) for n >= 1.
R(n+1,x) = 1/2*x^2*(d/dx)^2 ((1+x)^2*R(n,x)).
The zeros of R(n,x) belong to the interval [-1, 0].
Alternating row sums equal 1, that is R(n,-1) = 1.
R(n,-2) = 4*R(n,1) = 4*A055203(n).
4^n*Sum_{k = 2..2*n} T(n,k)*(-1/2)^k appears to equal (-1)^(n+1)*A005799(n) for n >= 1.
For k a nonzero integer, the power series A(k,x) := exp( Sum_{n >= 1} 1/k^2*R(n,k)*x^n/n ) appear to have integer coefficients. See the Example section.
Sum_{k = 2..2*n} T(n,k)*binomial(x,k-2) = binomial(x,2)^n - 2*binomial(x+1,2)^n + binomial(x+2,2)^n. These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane (the corresponding property also holds for the row polynomials of A019538 with a factor of x removed). (End)
From Peter Bala, Mar 08 2018: (Start)
n-th row polynomial R(n,x) = coefficient of (z_1)^2 * ... * (z_n)^2 in the expansion of the rational function 1/(1 + x - x*(1 + z_1)*...*(1 + z_n)).
The n-th row of the table is given by the matrix product P^(-1)*v_n, where P denotes Pascal's triangle A007318 and v_n is the sequence (0, 0, 1, 3^n, 6^n, 10^n, ...) regarded as an infinite column vector, where 1, 3, 6, 10, ... is the sequence of triangular numbers A000217. Cf. A087127. (End)

Definition corrected by Francisco Santos, Nov 17 2017

A037960 a(n) = n(3n+1)*(n+2)!/24.

Original entry on oeis.org

0, 1, 14, 150, 1560, 16800, 191520, 2328480, 30240000, 419126400, 6187104000, 97037740800, 1612798387200, 28332944640000, 524813313024000, 10226013557760000, 209144207720448000, 4480594531725312000, 100357207837286400000, 2345925761384325120000, 57136703662028390400000

Offset: 0

N. J. A. Sloane

For n>=1, a(n) is equal to the number of surjections from {1,2,..,n+2} onto {1,2,...,n}. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007

Identity (1.18) in H. W. Gould, Combinatorial Identities, Morgantown, 1972; page 3.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
H. W. Gould, ed. J. Quaintance, Combinatorial Identities, May 2010 (identity 10.3, p.45)

Cf. A000142, A001286, A001296, A019538, A037960, A131689.

Cf. A037959, A037961, A037962, A037963.

[Factorial(n+2)*n*(3*n+1)/24: n in [0..25]]; // Vincenzo Librandi, Feb 20 2017

Table[(n+2)!*n*(3n+1)/24,{n,0,20}] (* Harvey P. Dale, Oct 16 2014 *)

n*(3*n+1)*(n+2)!/24 \\ Charles R Greathouse IV, Nov 02 2011

[factorial(n)*stirling_number2(n+2, n) for n in (0..30)] # G. C. Greubel, Jun 20 2022

a(n) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+2). - Vladimir Kruchinin, Jun 01 2013
(3*n-2)*(n-1)*a(n) - n*(n+2)*(3*n+1)*a(n-1) = 0. - R. J. Mathar, Jul 26 2015
E.g.f.: x*(1 + 2*x)/(1 - x)^5. - Ilya Gutkovskiy, Feb 20 2017
From G. C. Greubel, Jun 20 2022: (Start)
a(n) = n!*StirlingS2(n+2, n).
a(n) = A131689(n+2, n).
a(n) = A019538(n+2, n). (End)

More terms from Vincenzo Librandi, Feb 20 2017

A210029 Number of sequences over the alphabet of n symbols of length 2n which have n distinct symbols. Also number of placements of 2n balls into n cells where no cell is empty.

Original entry on oeis.org

1, 14, 540, 40824, 5103000, 953029440, 248619571200, 86355926616960, 38528927611574400, 21473732319740064000, 14620825330739032204800, 11941607887300551753216000, 11523529003703200697461248000, 12970646659082235068963297280000

Offset: 1

Washington Bomfim, Mar 16 2012

nonn

			a(2) = 14 because the 2^4 sequences on 2 symbols of length 4 can be represented by 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100,1110, and 1111. Only two of them do not have n distinct symbols.
a(10)= 21473732319740064000 since all digits appear in 21473732319740064000 nonnegative integers with 20 digits.
O.g.f.: A(x) = 1 + 14*x + 540*x^2 + 40824*x^3 + 5103000*x^4 + ... where
A(x) = x/(1+x)^2 + 2^4*x^2/(1+4*x)^3 + 3^6*x^3/(1+9*x)^4 + 4^8*x^4/(1+16*x)^5 + 5^10*x^5/(1+25*x)^6 +... - _Paul D. Hanna_, Feb 24 2013
E.g.f.: E(x) = 1 + 14*x + 540*x^2/2! + 40824*x^3/3! + 5103000*x^4/4! + ... where
E(x) = exp(-x)*x + 2^4*exp(-4*x)*x^2/2! + 3^6*exp(-9*x)*x^3/3! + 4^8*exp(-16*x)*x^4/4! + 5^10*exp(-25*x)*x^5/5! +... - _Paul D. Hanna_, Feb 24 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A131689, A210024, A209899, A209900, A007820, A008277.

P := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n,k)*P(n-k)*x, k=1..n)) end:
a := n -> coeff(P(2*n), x, n); # Peter Luschny, Sep 11 2019

Table[Sum[((-1)^v*Binomial[n, v]*(n - v)^(2 n)), {v, 0, n - 1}], {n, 20}] (* T. D. Noe, Mar 16 2012 *)

{a(n)=n!*polcoeff(sum(m=1, n, (m^2)^m*exp(-m^2*x+x*O(x^n))*x^m/m!), n)} \\ Paul D. Hanna, Oct 26 2012

{a(n)=polcoeff(sum(k=1, n, (k^2)^k*x^k/(1+k^2*x +x*O(x^n))^(k+1)), n)} for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 24 2013

a(n) = Sum_{v=0..n-1}( (-1)^v * binomial(n,v) * (n-v)^(2n) ).
a(n) = n! * S2(2*n,n), where S2(n,k) = A008277(n,k) are the Stirling numbers of the second kind. - Paul D. Hanna, Oct 26 2012 [Also the central column of A131689 (suggesting a(0) = 1). - Peter Luschny, Sep 11 2019]
E.g.f.: Sum_{n>=1} (n^2)^n * exp(-n^2*x) * x^n/n! = Sum_{n>=1} a(n)*x^n/n!. - Paul D. Hanna, Oct 26 2012
a(n) ~ n^(2*n)*(2/(exp(c)*(2-c)))^n / sqrt(1-c), where c = -LambertW(-2/exp(2)) = 0.406375739959959907676958... - Vaclav Kotesovec, Jan 02 2013
O.g.f.: Sum_{n>=1} n^(2*n) * x^n / (1 + n^2*x)^(n+1). - Paul D. Hanna, Feb 24 2013
a(n) = [x^n] P(2*n) where P(n) = Sum_{k=1..n} binomial(n, k)*P(n-k)*x based in P(0) = 1. - Peter Luschny, Sep 11 2019

A278075 Coefficients of the signed Fubini polynomials in ascending order, F_n(x) = Sum_{k=0..n} (-1)^nStirling2(n,k)k!*(-x)^k.

Original entry on oeis.org

1, 0, 1, 0, -1, 2, 0, 1, -6, 6, 0, -1, 14, -36, 24, 0, 1, -30, 150, -240, 120, 0, -1, 62, -540, 1560, -1800, 720, 0, 1, -126, 1806, -8400, 16800, -15120, 5040, 0, -1, 254, -5796, 40824, -126000, 191520, -141120, 40320, 0, 1, -510, 18150, -186480, 834120, -1905120, 2328480, -1451520, 362880

Offset: 0

Peter Luschny, Jan 09 2017

sign

Signed version of A131689.

Integral_{x=0..1} F_n(x) = B_n(1) where B_n(x) are the Bernoulli polynomials.

			Triangle of coefficients starts:
[1]
[0,  1]
[0, -1,    2]
[0,  1,   -6,    6]
[0, -1,   14,  -36,    24]
[0,  1,  -30,  150,  -240,   120]
[0, -1,   62, -540,  1560, -1800,    720]
[0,  1, -126, 1806, -8400, 16800, -15120, 5040]

Peter Luschny, Illustration of the polynomials.
Peter Luschny, The Bernoulli Manifesto.
Grzegorz Rządkowski, Bernoulli numbers and solitons - revisited, Journal of Nonlinear Mathematical Physics, (2010) 17:1, 121-126.
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

Row sums are A000012, diagonal is A000142.
Cf. A131689 (unsigned), A019538 (n>0, k>0), A090582.
Let F(n, x) = Sum_{k=0..n} T(n,k)*x^k then, apart from possible differences in the sign or the offset, we have: F(n, -5) = A094418(n), F(n, -4) = A094417(n), F(n, -3) = A032033(n), F(n, -2) = A004123(n), F(n, -1) = A000670(n), F(n, 0) = A000007(n), F(n, 1) = A000012(n), F(n, 2) = A000629(n), F(n, 3) = A201339(n), F(n, 4) = A201354(n), F(n, 5) = A201365(n).
Cf. A163626, A173018.

function T(n, k)
    if k < 0 || k > n return 0 end
    if n == 0 && k == 0 return 1 end
    k*(T(n-1, k-1) - T(n-1, k))
end
for n in 0:7
    println([T(n,k) for k in 0:n])
end
# Peter Luschny, Mar 26 2020

F := (n,x) -> add((-1)^n*Stirling2(n,k)*k!*(-x)^k, k=0..n):
for n from 0 to 10 do PolynomialTools:-CoefficientList(F(n,x), x) od;

T[ n_, k_] := If[ n < 0 || k < 0, 0, (-1)^(n - k) k! StirlingS2[n, k]]; (* Michael Somos, Jul 08 2018 *)

{T(n, k) = if( n<0, 0, sum(i=0, k, (-1)^(n + i) * binomial(k, i) * i^n))};
/* Michael Somos, Jul 08 2018 */

T(n, k) = (-1)^(n-k) * Stirling2(n, k) * k!.
E.g.f.: 1/(1-x*(1-exp(-t))) = Sum_{n>=0} F_n(x) t^n/n!.
T(n, k) = k*(T(n-1, k-1) - T(n-1, k)) for 0 <= k <= n, T(0, 0) = 1, otherwise 0.
Bernoulli numbers are given by B(n) = Sum_{k = 0..n} T(n, k) / (k+1) with B(1) = 1/2. - Michael Somos, Jul 08 2018
Let F_n(x) be the row polynomials of this sequence and W_n(x) the row polynomials of A163626. Then F_n(1 - x) = W_n(x) and Integral_{x=0..1} U(n, x) = Bernoulli(n, 1) for U in {W, F}. - Peter Luschny, Aug 10 2021
T(n, k) = [z^k] Sum_{k=0..n} Eulerian(n, k)*z^(k+1)*(z-1)^(n-k-1) for n >= 1, where Eulerian(n, k) = A173018(n, k). - Peter Luschny, Aug 15 2022

A371292 Numbers whose binary indices have prime indices covering an initial interval of positive integers.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 22, 23, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 86, 87, 92, 93, 94, 95, 112, 113, 114, 115, 116, 117, 118, 119

Offset: 0

Gus Wiseman, Mar 27 2024

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their prime indices of binary indices begin:
   0: {}
   1: {{}}
   2: {{1}}
   3: {{},{1}}
   6: {{1},{2}}
   7: {{},{1},{2}}
   8: {{1,1}}
   9: {{},{1,1}}
  10: {{1},{1,1}}
  11: {{},{1},{1,1}}
  12: {{2},{1,1}}
  13: {{},{2},{1,1}}
  14: {{1},{2},{1,1}}
  15: {{},{1},{2},{1,1}}
  22: {{1},{2},{3}}
  23: {{},{1},{2},{3}}
  28: {{2},{1,1},{3}}
  29: {{},{2},{1,1},{3}}
  30: {{1},{2},{1,1},{3}}
  31: {{},{1},{2},{1,1},{3}}
  32: {{1,2}}

John Tyler Rascoe, Table of n, a(n) for n = 0..10000

The case with squarefree product of prime indices is A371293.
For binary indices of each prime index we have A371447, A371448.
The connected components of this multiset system are counted by A371452.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts patterns, ranked by A333217.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A001222, A055887, A255906, A326782, A368109, A371291, A371294.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],normQ[Join@@prix/@bpe[#]]&]

from itertools import count, islice
from sympy import sieve, factorint
def a_gen():
    for n in count(0):
        s = set()
        b = [(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1']
        for i in b:
            p = factorint(i)
            for j in p:
                s.add(sieve.search(j)[0])
        x = sorted(s)
        y = len(x)
        if sum(x) == (y*(y+1))//2:
            yield n
A371292_list = list(islice(a_gen(), 65)) # John Tyler Rascoe, May 21 2024

cp's OEIS Frontend

A278073 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(m*n, m*k)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.

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A038719 Triangle T(n,k) (0 <= k <= n) giving number of chains of length k in partially ordered set formed from subsets of n-set by inclusion.

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A285824 Number T(n,k) of ordered set partitions of [n] into k blocks such that equal-sized blocks are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A278074 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(m*n, m*k)* P(m, n-k)*z with P(m, 0) = 1 and m = 4.

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A094420 Generalized ordered Bell numbers Bo(n,n).

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A122193 Triangle T(n,k) of number of loopless multigraphs with n labeled edges and k labeled vertices and without isolated vertices, n >= 1; 2 <= k <= 2*n.

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A037960 a(n) = n*(3*n+1)*(n+2)!/24.

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A278073 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.

A278074 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 4.

A037960 a(n) = n(3n+1)*(n+2)!/24.

A278075 Coefficients of the signed Fubini polynomials in ascending order, F_n(x) = Sum_{k=0..n} (-1)^nStirling2(n,k)k!*(-x)^k.