A122193 - cp's OEIS frontend

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.

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

cp's OEIS Frontend

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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