A208881 -id:A208881 - cp's OEIS frontend

A253283 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.

1, 0, 1, 0, 2, 3, 0, 3, 12, 10, 0, 4, 30, 60, 35, 0, 5, 60, 210, 280, 126, 0, 6, 105, 560, 1260, 1260, 462, 0, 7, 168, 1260, 4200, 6930, 5544, 1716, 0, 8, 252, 2520, 11550, 27720, 36036, 24024, 6435, 0, 9, 360, 4620, 27720, 90090, 168168, 180180, 102960, 24310

Offset: 0

Peter Luschny, Mar 20 2015

The rows give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1) / (n!*(n+1)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 31 2022

This is related to the cluster fans of type B (see Fomin and Zelevinsky reference) - F. Chapoton, Nov 17 2022.

			[1]
[0, 1]
[0, 2,   3]
[0, 3,  12,   10]
[0, 4,  30,   60,   35]
[0, 5,  60,  210,  280,  126]
[0, 6, 105,  560, 1260, 1260,  462]
[0, 7, 168, 1260, 4200, 6930, 5544, 1716]
.
R_0(x) = 1/(x-1)^0.
R_1(x) = 0/(x-1)^1 + 1/(x-1)^2.
R_2(x) = 0/(x-1)^2 + 2/(x-1)^3 + 3/(x-1)^4.
R_3(x) = 0/(x-1)^3 + 3/(x-1)^4 + 12/(x-1)^5 + 10/(x-1)^6.
Then k!*[x^k] R_n(x) is A001286(k+2) and A001754(k+3) for n = 2, 3 respectively.
.
Seen as an array A(n, k) = binomial(n + k, k)*binomial(n + 2*k - 1, n + k):
[0] 1, 1,   3,   10,    35,    126,     462, ...
[1] 0, 2,  12,   60,   280,   1260,    5544, ...
[2] 0, 3,  30,  210,  1260,   6930,   36036, ...
[3] 0, 4,  60,  560,  4200,  27720,  168168, ...
[4] 0, 5, 105, 1260, 11550,  90090,  630630, ...
[5] 0, 6, 168, 2520, 27720, 252252, 2018016, ...
[6] 0, 7, 252, 4620, 60060, 630630, 5717712, ...

Seiichi Manyama, Rows n = 0..139, flattened
F. Chapoton Enumerative Properties of Generalized Associahedra, Sém. Loth. Comb. B51b (2004).
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.
S. Fomin and A. Zelevinsky Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3.
Yunlong Shen and Lixin Shen, Orthogonal Fourier-Mellin moments for invariant pattern recognition, J. Opt. Soc. Am. A 11 (6) (1994) p 1748-1757, eq. (6).

T(n, n) = C(2*n-1, n) = A001700(n-1).
T(n, n-1) = A005430(n-1) for n >= 1.
T(n, n-2) = A051133(n-2) for n >= 2.
T(n, 2) = A027480(n-1) for n >= 2.
T(2*n, n) = A208881(n) for n >= 0.
A002002 (row sums).
Cf. A000142, A001286, A001754, A001755, A001777, A182626, A266732 (row k=3), A008316, A033282, A063007, A345013, A269944.

T_row := proc(n) local egf, k, F, t;
if n=0 then RETURN(1) fi;
egf := (x/(1-x))^n/n!; t := diff(egf,[x$n]);
F := convert(t,parfrac,x);
# print(seq(k!*coeff(series(F,x,20),x,k),k=0..7));
# gives A000142, A001286, A001754, A001755, A001777, ...
seq(coeff(F,(x-1)^(-k)),k=n..2*n) end:
seq(print(T_row(n)),n=0..7);
# 2nd version by R. J. Mathar, Dec 18 2016:
A253283 := proc(n,k)
    binomial(n,k)*binomial(n+k-1,k-1) ;
end proc:

Table[Binomial[n, k] Binomial[n + k - 1, k - 1], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 22 2017 *)

T(n,k) = binomial(n,k)*binomial(n+k-1,k-1);
tabl(nn) = for(n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Apr 29 2018

The exponential generating functions for the rows of the square array L(n,k) = ((n+k)!/n!)*C(n+k-1,n-1) (associated to the unsigned Lah numbers) are given by R_n(x) = Sum_{k=0..n} T(n,k)/(x-1)^(n+k).
T(n,k) = C(n,k)*C(n+k-1,k-1).
Sum_{k=0..n} T(n,k) = (-1)^n*hypergeom([-n,n],[1],2) = (-1)^n*A182626(n).
Row generating function: Sum_{k>=1} T(n,k)*z^k = z*n* 2F1(1-n,n+1 ; 2; -z). - R. J. Mathar, Dec 18 2016
From Peter Bala, Feb 22 2017: (Start)
G.f.: (1/2)*( 1 + (1 - t)/sqrt(1 - 2*(2*x + 1)*t + t^2) ) = 1 + x*t + (2*x + 3*x^2)*t^2 + (3*x + 12*x^2 + 10*x^3)*t^3 + ....
n-th row polynomial R(n,x) = (1/2)*(LegendreP(n, 2*x + 1) - LegendreP(n-1, 2*x + 1)) for n >= 1.
The row polynomials are the black diamond product of the polynomials x^n and x^(n+1) (see Dukes and White 2016 for the definition of this product).
exp(Sum_{n >= 1} R(n,x)*t^n/n) = 1 + x*t + x*(1 + 2*x)*t^2 + x*(1 + 5*x + 5*x^2)*t^3 + ... is a g.f. for A033282, but with a different offset.
The polynomials P(n,x) := (-1)^n/n!*x^(2*n)*(d/dx)^n(1 + 1/x)^n begin 1, 3 + 2*x , 10 + 12*x + 3*x^2, ... and are the row polynomials for the row reverse of this triangle. (End)
Let Q(n, x) = Sum_{j=0..n} (-1)^(n - j)*A269944(n, j)*x^(2*j - 1) and P(x, y) = (LegendreP(x, 2*y + 1) - LegendreP(x-1, 2*y + 1)) / 2 (see Peter Bala above). Then n!*(n - 1)!*[y^n] P(x, y) = Q(n, x) for n >= 1. - Peter Luschny, Oct 31 2022
From Peter Bala, Apr 18 2024: (Start)
G.f.: Sum_{n >= 0} binomial(2*n-1, n)*(x*t)^n/(1 - t)^(2*n) = 1 + x*t + (2*x + 3*x^2)*t^2 + (3*x + 12*x^2 + 10*x^3)*t^3 + ....
n-th row polynomial R(n, x) = [t^n] ( (1 - t)/(1 - (1 + x)*t) )^n.
It follows that for integer x, the sequence {R(n, x) : n >= 0} satisfies the Gauss congruences: R(n*p^r, x) == R(n*p^(r-1), x) (mod p^r) for all primes p and positive integers n and r.
R(n, -2) = (-1)^n * A002003(n) for n >= 1.
R(n, 3) = A299507(n). (End)

A208879 Number of words A(n,k), either empty or beginning with the first letter of the cyclic k-ary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 30, 10, 1, 1, 1, 2, 62, 560, 35, 1, 1, 1, 2, 114, 2830, 11550, 126, 1, 1, 1, 2, 202, 12622, 151686, 252252, 462, 1, 1, 1, 2, 346, 53768, 1754954, 8893482, 5717712, 1716, 1, 1

Offset: 0

Alois P. Heinz, Mar 02 2012

The first and the last letters are considered neighbors in a cyclic alphabet. The words are not considered cyclic here.

A(n,k) is also the number of (n*k-1)-step walks on k-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions such that the indices of dimensions used in consecutive steps differ by less than 2 or are in the set {1,k}.

			A(0,0) = A(n,0) = A(0,k) = 1: the empty word.
A(1,2) = 1 = |{ab}|.
A(1,3) = 2 = |{abc, acb}|.
A(1,4) = 2 = |{abcd, adcb}|.
A(2,2) = 3 = |{aabb, abab, abba}|.
A(2,4) = 62 = |{aabbccdd, aabbcdcd, aabbcddc, aabcbcdd, aabcddcb, aadcbbcd, aadcdcbb, aaddcbbc, aaddcbcb, aaddccbb, ababccdd, ababcdcd, ababcddc, abadcbcd, abadcdcb, abaddcbc, abaddccb, abbadccd, abbadcdc, abbaddcc, abbccdad, abbccdda, abbcdadc, abbcdcda, abcbadcd, abcbaddc, abcbcdad, abcbcdda, abccbadd, abccddab, abcdabcd, abcdadcb, abcdcbad, abcdcdab, abcddabc, abcddcba, adabbccd, adabbcdc, adabcbcd, adabcdcb, adadcbbc, adadcbcb, adadccbb, adcbabcd, adcbadcb, adcbbadc, adcbbcda, adcbcbad, adcbcdab, adccbbad, adccdabb, adcdabbc, adcdabcb, adcdcbab, adcdcbba, addabbcc, addabcbc, addabccb, addcbabc, addcbcba, addccbab, addccbba}|.
Square array A(n,k) begins:
  1,  1,   1,       1,         1,           1,             1, ...
  1,  1,   1,       2,         2,           2,             2, ...
  1,  1,   3,      30,        62,         114,           202, ...
  1,  1,  10,     560,      2830,       12622,         53768, ...
  1,  1,  35,   11550,    151686,     1754954,      19341130, ...
  1,  1, 126,  252252,   8893482,   276049002,    8151741752, ...
  1,  1, 462, 5717712, 552309938, 46957069166, 3795394507240, ...

Columns k=0+1, 2-6 give: A000012, A088218, A208881, A209183, A209184, A209185.
Rows n=0, 2 give: A000012, A208880.
Cf. A208673 (noncyclic alphabet).

b:= proc() option remember; local n; n:= nargs;
     `if`(n<2 or {0}={args}, 1,
     `if`(n=2, `if`(args[1]>0, b(args[1]-1, args[2]), 0)+
               `if`(args[2]>0, b(args[2]-1, args[1]), 0),
     `if`(args[1]>0, b(args[1]-1, seq(args[i], i=2..n)), 0)+
     `if`(args[2]>0, b(args[2]-1, seq(args[i], i=3..n), args[1]), 0)+
     `if`(args[n]>0, b(args[n]-1, seq(args[i], i=1..n-1)), 0)))
    end:
A:= (n, k)-> `if`(n=0 or k=0, 1, b(n-1, n$(k-1))):
seq(seq(A(n, d-n), n=0..d), d=0..9);

b[args__] := b[args] = With[{n = Length[{args}]}, If[n<2 || {0} == Union[ {args}], 1, If[n==2, If[{args}[[1]]>0, b[{args}[[1]]-1, {args}[[2]] ], 0] + If[{args}[[2]]>0, b[{args}[[2]]-1, {args}[[1]] ], 0], If[{args}[[1]]>0, b[{args}[[1]]-1, Sequence @@ {args}[[2;;n]] ], 0] + If[{args}[[2]]>0, b[{args}[[2]]-1, Sequence @@ {args}[[3;;n]], {args}[[1]] ], 0]+ If[{args}[[n]]>0, b[{args}[[n]]-1, Sequence @@ Most[{args}]] ],0]] /. Null -> 0];
a[n_,k_]:= If[n==0 || k==0, 1, b[n-1, Sequence @@ Array[n&, k-1]]];
Table[Table[a[n, d-n], {n,0,d}], {d,0,9}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

A361027 Table of generalized de Bruijn's numbers (A006480) read by ascending antidiagonals.

Original entry on oeis.org

2, 30, 3, 560, 20, 20, 11550, 210, 75, 210, 252252, 2772, 504, 504, 2772, 5717712, 42042, 4620, 2352, 4620, 42042, 133024320, 700128, 51480, 15840, 15840, 51480, 700128, 3155170590, 12471030, 656370, 135135, 81675, 135135, 656370, 12471030, 75957810500, 233716340, 9237800

Offset: 0

Peter Bala, Feb 28 2023

The Catalan numbers A000108 are given by the formula Catalan(n) = (2*n)!/(n!*(n + 1)!). Gessel (1992) considered generalized Catalan numbers defined by Catalan(r,n) = J(r) * (2*n)!/(n!*(n + r + 1)!), where J(r) = (2*r + 2)!/(2*(r + 1)!) = (2^r)*Product_{j = 0..r} (2*j + 1) is chosen so that these numbers are always integers. Gessel's generalized Catalan numbers are particular cases of super ballot numbers. See A135573 for a table of these generalized Catalan numbers.
For this table we carry out an analogous construction using the de Bruijn numbers B(n) = (3*n)!/n!^3 = A006480(n) in place of the central binomial numbers. We define the generalized de Bruijn number B(r,n), r = 0, 1, 2, ..., by B(r,n) = F(r) * (3*n)!/(n!*(n + r + 1)!^2), where choosing F(r) = (3*r + 3)!/(3*(r + 1)!) = (3^r)*Product_{j = 0..r} (3*j + 1)*(3*j + 2) appears to produce integer values for these quantities. We have verified this for rows 0, 1, 2 and 3 of the table.
An alternative expression for the generalized de Bruijn numbers is B(r,n) =  G(r,n) * B(n+r+1), where G(r) = (1/3)*Product_{j = 0..r} ( (3*j + 1)*(3*j + 2)/((3*n + 3*j + 1)*(3*n + 3*j + 2)) ).
The rows of the square array below are the sequences of generalized de Bruijn numbers {B(0,k)}, {B(1,k)}, {B(2,k)}, ....

			The square array with rows n >= 0 and columns k >= 0 begins:
  n\k|       0       1       2        3        4         5         6 ...
  ----------------------------------------------------------------------
   0 |       2       3      20      210     2772     42042    700128 ...
   1 |      30      20      75      504     4620     51480    656370 ...
   2 |     560     210     504     2352    15840    135135   1361360 ...
   3 |   11550    2772    4620    15840    81675    550550   4492488 ...
   4 |  252252   42042   51480   135135   550550   3006003  20271888 ...
   5 | 5717712  700128  656370  1361360  4492488  20271888  ...
  ...
As a triangle:
 Row
  0 |        2
  1 |       30       3
  2 |      560      20     20
  3 |    11550     210     75    210
  4 |   252252    2772    504    504   2772
  5 |  5717712   42042   4620   2352   4620   42042
  ...

N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapters 4 and 6.

Ira M. Gessel, Super ballot numbers, J. Symbolic Comp., 14 (1992), 179-194.

A208881 (column 1), A361028(row 0), A361029(row 1), A361030(row 2), A361031(row 3).

Cf. A006480, A024488, A135573.

# as a square array
T := proc (n,k) (1/3)*27^(n+k+1)*binomial(n+1/3, n+k+1)*binomial(n+2/3,
n+k+1); end proc:
for n from 0 to 10 do seq(T(n,k), k = 0..10); end do;
# as a triangle
T := proc (n,k) (1/3)*27^(n+k+1)*binomial(n+1/3, n+k+1)*binomial(n+2/3,
n+k+1); end proc:
for n from 0 to 10 do seq(T(n-k,k), k = 0..n); end do;

T(n,k) = (3*n + 3)!/(3*(n + 1)!) * (3*k)!/(k!*(k + n + 1)!^2), n, k >= 0.
T(n,k) = (1/3)*27^(n+1+k)*binomial(n+1/3, n+1+k)*binomial(n+2/3, n+1+k).
T(n,k) = (1/(2*Pi))^2 * 1/27^(n+k+1) * Integral_{x = 0..27} (27 - x)^(n+2/3)*x^(k-2/3) dx * Integral_{x = 0..27} (27 - x)^(n+1/3)*x^(k-1/3) dx.
P-recursive: (n + k + 1)^2*T(n,k) = 3*(3*k - 1)*(3*k - 2)*T(n,k-1) with T(n,0) = 1/(n+1)!^2 *  (3*n + 3)!/(3*(n + 1)!).
(n + k + 1)^2*T(n,k) = 3*(3*n + 1)*(3*n + 2)*T(n-1,k) with T(0,k) = 2*(k + 1)*(3*k)!/(k + 1)!^3.
T(n,0) = A208881(n+1).

A199127 Number of n X 2 0..2 arrays with values 0..2 introduced in row major order, the number of instances of each value within one of each other, and no element equal to any horizontal or vertical neighbor.

Original entry on oeis.org

1, 2, 2, 12, 30, 30, 210, 560, 560, 4200, 11550, 11550, 90090, 252252, 252252, 2018016, 5717712, 5717712, 46558512, 133024320, 133024320, 1097450640, 3155170590, 3155170590, 26293088250, 75957810500, 75957810500, 638045608200

Offset: 1

R. H. Hardin, Nov 03 2011

nonn

Column 2 of A199133.

a(n) is the last term in row n of triangle in A286030 (see also formulas below). Bob Selcoe, Sep 26 2021

			Some solutions for n=5:
  0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1
  1 0   1 2   1 2   1 2   1 0   1 2   1 0   1 2   1 2   1 0
  0 2   2 0   0 1   2 0   0 2   2 1   0 2   0 1   2 0   2 1
  2 1   0 2   2 0   0 1   2 1   1 0   2 0   1 2   0 1   1 2
  0 2   2 1   0 2   2 0   1 2   0 2   1 2   2 0   1 2   2 0

R. H. Hardin, Table of n, a(n) for n = 1..198

Cf. A286030.

Conjecture: a(3n+2) = a(3n+3) = A208881(n+1). - R. J. Mathar, Nov 01 2015
Conjecture: -(458*n-1205) *(n+2) *(n+1)*a(n) +(-208*n^3+2578*n^2-4613*n-2410) *a(n-1) +9*(-339*n-638) *a(n-2) +27*(n-2) *(458*n^2-289*n-1146) *a(n-3) +54*(n-2) *(n-3) *(104*n-1081) *a(n-4)=0. - R. J. Mathar, Nov 01 2015
Conjecture: (n+2)*(n+1)*a(n) +(5*n^2-2)*a(n-1) +3*(5*n^2-15*n+3) *a(n-2) +3*(n^2 -60*n +81)*a(n-3) +135*(-n^2+3*n-1)*a(n-4) -405*(n-2)*(n-4) *a(n-5) -810*(n-4) *(n-5) *a(n-6)=0. - R. J. Mathar, Nov 01 2015
From Bob Selcoe, Sep 26 2021: (Start)
When n == 0 (mod 3), a(n) = n!/(3*(n/3)!^3);
when n == 1 (mod 3), a(n) = n!/(((n+2)/3)!*((n-1)/3)!^2);
when n == 2 (mod 3), a(n) = n!/(((n-2)/3)!*((n+1)/3)!^2).
(End)

cp's OEIS Frontend

A253283 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A208879 Number of words A(n,k), either empty or beginning with the first letter of the cyclic k-ary alphabet, where each letter of the alphabet occurs n times and letters of neighboring word positions are equal or neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A361027 Table of generalized de Bruijn's numbers (A006480) read by ascending antidiagonals.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Formula

A199127 Number of n X 2 0..2 arrays with values 0..2 introduced in row major order, the number of instances of each value within one of each other, and no element equal to any horizontal or vertical neighbor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula