A062266 -id:A062266 - cp's OEIS frontend

A062140 Coefficient triangle of generalized Laguerre polynomials n!*L(n,4,x) (rising powers of x).

1, 5, -1, 30, -12, 1, 210, -126, 21, -1, 1680, -1344, 336, -32, 1, 15120, -15120, 5040, -720, 45, -1, 151200, -181440, 75600, -14400, 1350, -60, 1, 1663200, -2328480, 1164240, -277200, 34650, -2310, 77, -1, 19958400, -31933440

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,4,x)= sum(a(n,m)*x^m,m=0..n) have g.f. exp(-z*x/(1-z))/(1-z)^5. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials p(n,x)=sum(|A008297(n,m)|*(-x)^m, m=1..n) and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).

			Triangle begins:
  {1};
  {5,-1};
  {30,-12,1};
  {210,-126,21,-1};
  ...
2!*L(2,4,x)=30-12*x+x^2.

Indranil Ghosh, Rows 0..125
Index entries for sequences related to Laguerre polynomials

For m=0..5 the (unsigned) columns give A001720(n+4), A062199, A062260-A062263. The row sums (signed) give A062265, the row sums (unsigned) give A062266.

Cf. A021009, A062137-A062139, A066667.

Flatten[Table[((-1)^m)*n!*Binomial[n+4,n-m]/m!,{n,0,11},{m,0,n}]] (* Indranil Ghosh, Feb 23 2017 *)

row(n) = Vecrev(n!*pollaguerre(n, 4)); \\ Michel Marcus, Feb 06 2021

import math
f=math.factorial
def C(n,r):
    return f(n)//f(r)//f(n-r)
i=0
for n in range(26):
    for m in range(n+1):
        print(i, (-1)**m*f(n)*C(n+4,n-m)//f(m))
        i+=1 # Indranil Ghosh, Feb 23 2017

T(n, m) = ((-1)^m)*n!*binomial(n+4, n-m)/m!.

E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^5), m >= 0.

A086885 Lower triangular matrix, read by rows: T(i,j) = number of ways i seats can be occupied by any number k (0<=k<=j<=i) of persons.

Original entry on oeis.org

2, 3, 7, 4, 13, 34, 5, 21, 73, 209, 6, 31, 136, 501, 1546, 7, 43, 229, 1045, 4051, 13327, 8, 57, 358, 1961, 9276, 37633, 130922, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 10, 91, 748, 5509, 36046, 207775, 1047376, 4596553, 17572114, 11, 111, 1021, 8501

Offset: 1

Hugo Pfoertner, Aug 22 2003

Compare with A088699. - Peter Bala, Sep 17 2008

T(m, n) gives the number of matchings in the complete bipartite graph K_{m,n}. - Eric W. Weisstein, Apr 25 2017

			One person:
T(1,1)=a(1)=2: 0,1 (seat empty or occupied);
T(2,1)=a(2)=3: 00,10,01 (both seats empty, left seat occupied, right seat occupied).
Two persons:
T(2,2)=a(3)=7: 00,10,01,20,02,12,21;
T(3,2)=a(5)=13: 000,100,010,001,200,020,002,120,102,012,210,201,021.
Triangle starts:
  2;
  3  7;
  4 13  34;
  5 21  73 209;
  6 31 136 501 1546;
  ...

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)
Ed Jones, Number of seatings, discussion in newsgroup sci.math, Aug 9, 2003.
R. J. Mathar, The number of binary nxm matrices with at most k 1's in each row or columns, Table 1.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
Luca Zecchini, Tobias Bleifuß, Giovanni Simonini, Sonia Bergamaschi, and Felix Naumann, Determining the Largest Overlap between Tables, Proc. ACM Manag. Data (SIGMOD 2024) Vol. 2, No. 1, Art. 48. See p. 48:6.
Index entries for sequences related to Laguerre polynomials

Diagonal: A002720, first subdiagonal: A000262, 2nd subdiagonal: A052852, 3rd subdiagonal: A062147, 4th subdiagonal: A062266, 5th subdiagonal: A062192, 2nd row/column: A002061. With column 0: A176120.

[Factorial(k)*Evaluate(LaguerrePolynomial(k, n-k), -1): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 23 2021

A086885 := proc(n,k)
    add( binomial(n,j)*binomial(k,j)*j!,j=0..min(n,k)) ;
end proc: # R. J. Mathar, Dec 19 2014

Table[Table[Sum[k! Binomial[n, k] Binomial[j, k], {k, 0, j}], {j, 1, n}], {n, 1, 10}] // Grid (* Geoffrey Critzer, Jul 09 2015 *)
Table[m! LaguerreL[m, n - m, -1], {n, 10}, {m, n}] // Flatten (* Eric W. Weisstein, Apr 25 2017 *)

T(i, j) = j!*pollaguerre(j, i-j, -1); \\ Michel Marcus, Feb 23 2021

flatten([[factorial(k)*gen_laguerre(k, n-k, -1) for k in [1..n]] for n in (1..10)]) # G. C. Greubel, Feb 23 2021

a(n) = T(i, j) with n=(i*(i-1))/2+j; T(i, 1)=i+1, T(i, j)=T(i, j-1)+i*T(i-1, j-1) for j>1.
The role of seats and persons may be interchanged, so T(i, j)=T(j, i).
T(i, j) = j!*LaguerreL(j, i-j, -1). - Vladeta Jovovic, Aug 25 2003
T(i, j) = Sum_{k=0..j} k!*binomial(i, k)*binomial(j, k). - Vladeta Jovovic, Aug 25 2003

A216294 Triangular array read by rows: T(n,k) is the number of partial permutations of {1,2,...,n} that have exactly k cycles, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 13, 14, 6, 1, 73, 84, 41, 10, 1, 501, 609, 325, 95, 15, 1, 4051, 5155, 2944, 965, 190, 21, 1, 37633, 49790, 30023, 10689, 2415, 343, 28, 1, 394353, 539616, 340402, 129220, 32179, 5348, 574, 36, 1, 4596553, 6478521, 4246842, 1698374, 455511, 84567, 10794, 906, 45, 1

Offset: 0

Geoffrey Critzer, Sep 04 2012

A partial permutation on a set X is a bijection between two subsets of X.
Row sums are A002720.
First column (corresponding to k=0) is A000262.

			1;
1,     1;
3,     3,   1;
13,   14,   6,  1;
73,   84,  41, 10,  1;
501, 609, 325, 95, 15,  1;

Alois P. Heinz, Rows n = 0..50, flattened
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 132.

Cf. A021009. A000262, A002720, A052852, A062147, A062192, A062266, A088699.

gf := exp(x / (1 - x)) / (1 - x)^y:
serx := series(gf, x, 10): poly := n -> simplify(coeff(serx, x, n)):
seq(print(seq(n!*coeff(poly(n), y, k), k = 0..n)), n = 0..9); # Peter Luschny, Feb 23 2023

nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[Series[Exp[ x/(1-x)]/(1-x)^y,{x,0,nn}],{x,y}]]//Flatten

E.g.f.: exp(x/(1-x))/(1-x)^y.
From Peter Bala, Aug 23 2013: (Start)
Exponential Riordan array [exp(x/(1-x)), log(1/(1-x))].
The row polynomials R(n,y), n > = 0, satisfy the 2nd order recurrence equation R(n,y) = (2*n + y - 1)*R(n-1,y) - (n - 1)*(n + y - 2)*R(n-2,y) with R(0,y) = 1 and R(1,y) = 1 + y.
Modulo variations in offset we have: R(n,0) = A000262, R(n,1) = A002720, R(n,2) = A000262, R(n,3) = A052852, R(n,4) = A062147, R(n,5) = A062266 and R(n,6) = A062192. In general, for fixed k, the sequence {R(n,k)}n>=1 gives the entries on a diagonal of the square array A088699. (End)

A293985 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-x))/(1-x)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 13, 1, 4, 13, 34, 73, 1, 5, 21, 73, 209, 501, 1, 6, 31, 136, 501, 1546, 4051, 1, 7, 43, 229, 1045, 4051, 13327, 37633, 1, 8, 57, 358, 1961, 9276, 37633, 130922, 394353, 1, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 4596553

Offset: 0

Seiichi Manyama, Oct 21 2017

			Square array begins:
    1,    1,    1,    1,     1, ... A000012;
    1,    2,    3,    4,     5, ... A000027;
    3,    7,   13,   21,    31, ... A002061;
   13,   34,   73,  136,   229, ... A135859;
   73,  209,  501, 1045,  1961, ...
  501, 1546, 4051, 9276, 19081, ...
Antidiagonal rows begin as:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  7, 13;
  1, 4, 13, 34,  73;
  1, 5, 21, 73, 209, 501; - _G. C. Greubel_, Mar 09 2021

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Columns k=0..6 give: A000262, A002720, A000262(n+1), A052852(n+1), A062147, A062266, A062192.
Main diagonal gives A152059.
Similar table: A086885, A088699, A176120.

function t(n,k)
  if n eq 0 then return 1;
  else return Factorial(n-1)*(&+[(j+k)*t(n-j,k)/Factorial(n-j): j in [1..n]]);
  end if; return t;
end function;
[t(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021

t[n_, k_]:= t[n, k]= If[n==0, 1, (n-1)!*Sum[(j+k)*t[n-j,k]/(n-j)!, {j,n}]];
T[n_,k_]:= t[k,n-k]; Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)

@CachedFunction
def t(n,k): return 1 if n==0 else factorial(n-1)*sum( (j+k)*t(n-j,k)/factorial(n-j) for j in (1..n) )
def T(n,k): return t(k,n-k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} (j+k)*A(n-j,k)/(n-j)! for n > 0.
A(0,k) = 1, A(1,k) = k+1 and A(n,k) = (2*n-1+k)*A(n-1,k) - (n-1)*(n-2+k)*A(n-2,k) for n > 1.
From Seiichi Manyama, Jan 25 2025: (Start)
A(n,k) = n! * Sum_{j=0..n} binomial(n+k-1,j)/(n-j)!.
A(n,k) = n! * LaguerreL(n, k-1, -1). (End)

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A062140 Coefficient triangle of generalized Laguerre polynomials n!*L(n,4,x) (rising powers of x).

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A086885 Lower triangular matrix, read by rows: T(i,j) = number of ways i seats can be occupied by any number k (0<=k<=j<=i) of persons.

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A216294 Triangular array read by rows: T(n,k) is the number of partial permutations of {1,2,...,n} that have exactly k cycles, 0<=k<=n.

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A293985 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-x))/(1-x)^k.

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Mathematica

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