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A253283 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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.

LINKS

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).

FORMULA

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

EXAMPLE

[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.

MAPLE

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:

MATHEMATICA

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

PROG

(PARI) 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

CROSSREFS

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.

Cf. A000142, A001286, A001754, A001755, A001777, A182626, A266732 (row k=3), A008316, A033282, A063007, A345013, A269944.

Sequence in context: A316607 A194365 A216217 * A261719 A137663 A257740

Adjacent sequences: A253280 A253281 A253282 * A253284 A253285 A253286

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Mar 20 2015

STATUS

approved

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Last modified December 8 06:47 EST 2022. Contains 358673 sequences. (Running on oeis4.)