cp's OEIS Frontend

a(n) is also the constant term in product 1 <= i,j <= n, i different from j (1 - x_i/x_j)^4. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 16 2002

From Tilman Piesk, Oct 28 2014: (Start)

Number of permutations of a multiset that contains m different elements n times. These multisets have the signatures A249543(m,n-1) for m>=1 and n>=2.

In an m-dimensional Pascal tensor (the generalization of a symmetric Pascal matrix) P(x1,...,xn) = (x1+...+xn)!/(x1!*...*xn!), so the main diagonal of an m-dimensional Pascal tensor is D(n) = (m*n)!/(n!^m). These diagonals are the rows of this array (with m>0), which begins like this:

m\n:0 1 2 3 4 5

0: 1 1 1 1 1 1 ... A000012;

1: 1 1 1 1 1 1 ... A000012;

2: 1 2 6 20 70 252 ... A000984;

3: 1 6 90 1680 34650 756756 ... A006480;

4: 1 24 2520 369600 63063000 11732745024 ... A008977;

5: 1 120 113400 168168000 305540235000 623360743125120 ... A008978;

6: 1 720 7484400 137225088000 3246670537110000 88832646059788350720 ... A008979;

with columns: A000142 (n=1), A000680 (n=2), A014606 (n=3), A014608 (n=4), A014609 (n=5).

A089759 is the transpose of this matrix. A034841 is its diagonal. A141906 is its lower triangle. A120666 is the upper triangle of this matrix with indices starting from 1. A248827 are the diagonal sums (or the row sums of the triangle).

(End)

T(n,k) is the number of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1. - Alois P. Heinz, May 06 2013

From Yahia Kahloune, Jan 30 2014: (Start)

In general, let b(k,e,p) = Sum_{i=0..k} (-1)^i*binomial(e*p+1,i)*binomial(k+e-i,e)^p. Then T(n,k) = b(k,3,n).

With these coefficients we can calculate: Sum_{i=1..n} binomial(i+e-1,e)^p = Sum_{i=0..e*(p-1)} b(i,e,p)*binomial(n+e+i,e*p+1).

For example, A086020(n) = Sum_{i=1..n} binomial(2+i, 3)^2 = T(2,0)*binomial(n+3, 7) + T(2,1)*binomial(n+4,7) + T(2,2)*binomial(n+5,7) + T(2,3)*binomial(n+6,7) = (1/5040)*(20*n^7 + 210*n^6 + 854*n^5 + 1680*n^4 + 1610*n^3 + 630*n^2 + 36*n). (End)

T(n,k) is the number of permutations of 3 indistinguishable copies of 1..n with exactly k descents. A descent is a pair of adjacent elements with the second element less than the first. - Andrew Howroyd, May 06 2020

a(n) is also the constant term in product 1 <= i,j <= n, i different from j (1 - x_i/x_j)^5. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 18 2002

An alignment of n strings of various lengths is a way of inserting blank characters into the n strings so that the resulting strings all have the same length. We don't allow insertion of a blank character into the same position in each of the n strings.

In this case, let s_1,...,s_n be n strings each of length 3 over an alphabet A. Let - be a gap symbol not in A and let A' = union of A and {-}. An alignment of the n strings is an n-tuple (s_1',...,s_n') of strings each of length >= 3 over the alphabet A' such that

(a) the strings s_i', 1 <= i <= n, have the same length. This common length is called the length of the alignment.

(b) deleting the gap symbols from s_i' yields the string s_i for 1 <= i <= n

(c) there is no value j such that all the strings s_i', 1 <= i <= n have a gap symbol at position j.

By writing the strings s_i' one under another we can consider an alignment of n strings as an n X L matrix, where L, the length of the alignment, ranges from a minimum value of 3 to a maximum value of 3*n. Each row of the matrix has 3 characters from the alphabet A and (L - 3) gap characters.

For example,

s_1' = ABC------

s_2' = ---DEF---

s_3' = ------GHI

is an alignment (of maximum length L = 9) of three strings s_1 = ABC, s_2 = DEF and s_3 = GHI each of length 3.

For the number of alignments of length k of n strings of length 1 (resp. 2) see A131689 (resp. A122193).

The generalized Eulerian polynomials F_{m}(x) are defined F_{m; 0}(x) = 1 for all m >= 0 and for n > 0:

F_{0; n}(x) = Sum_{k=0..n} A097805(n, k)*(x-1)^(n-k) with coeffs. in A129186.

F_{1; n}(x) = Sum_{k=0..n} A131689(n, k)*(x-1)^(n-k) with coeffs. in A173018.

F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) with coeffs. in A292604.

F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) with coeffs. in A292605.

F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) with coeffs. in A292606.

The case m = 1 are the Eulerian polynomials whose coefficients are the Eulerian numbers which are displayed in Euler's triangle A173018.

Evaluated at x in {-1, 1, 0} these families of polynomials give for the first few m:

F_{m} : F_{0} F_{1} F_{2} F_{3} F_{4}

x = -1: A165326 A155585 A002105 A292609 A292607

x = 1: A000012 A000142 A000680 A014606 A014608 ... (m*n)!/m!^n

x = 0: -- A000012 A000364 A002115 A211212 ... m-alternating permutations of length m*n.

Note that the constant terms of the polynomials are the generalized Euler numbers as defined in A181985. In this sense generalized Euler numbers are also generalized Eulerian numbers.