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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!.

%I A253283 #86 Nov 12 2024 21:08:33
%S A253283 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,
%T A253283 1260,1260,462,0,7,168,1260,4200,6930,5544,1716,0,8,252,2520,11550,
%U A253283 27720,36036,24024,6435,0,9,360,4620,27720,90090,168168,180180,102960,24310
%N A253283 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.
%C A253283 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
%C A253283 This is related to the cluster fans of type B (see Fomin and Zelevinsky reference) - _F. Chapoton_, Nov 17 2022.
%H A253283 Seiichi Manyama, <a href="/A253283/b253283.txt">Rows n = 0..139, flattened</a>
%H A253283 F. Chapoton <a href="https://www.mat.univie.ac.at/~slc/wpapers/s51chapoton.html">Enumerative Properties of Generalized Associahedra</a>, Sém. Loth. Comb. B51b (2004).
%H A253283 Mark Dukes and Chris D. White, <a href="https://arxiv.org/abs/1603.01589">Web Matrices: Structural Properties and Generating Combinatorial Identities</a>, arXiv:1603.01589 [math.CO], 2016.
%H A253283 Mark Dukes and Chris D. White, <a href="https://doi.org/10.37236/5631">Web Matrices: Structural Properties and Generating Combinatorial Identities</a>, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.
%H A253283 S. Fomin and A. Zelevinsky <a href="https://dx.doi.org/10.4007/annals.2003.158.977">Y-systems and generalized associahedra</a>, Ann. of Math. (2) 158 (2003), no. 3.
%H A253283 Yunlong Shen and Lixin Shen, <a href="http://dx.doi.org/10.1364/JOSAA.11.001748">Orthogonal Fourier-Mellin moments for invariant pattern recognition</a>, J. Opt. Soc. Am. A 11 (6) (1994) p 1748-1757, eq. (6).
%F A253283 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).
%F A253283 T(n,k) = C(n,k)*C(n+k-1,k-1).
%F A253283 Sum_{k=0..n} T(n,k) = (-1)^n*hypergeom([-n,n],[1],2) = (-1)^n*A182626(n).
%F A253283 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
%F A253283 From _Peter Bala_, Feb 22 2017: (Start)
%F A253283 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 + ....
%F A253283 n-th row polynomial R(n,x) = (1/2)*(LegendreP(n, 2*x + 1) - LegendreP(n-1, 2*x + 1)) for n >= 1.
%F A253283 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).
%F A253283 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.
%F A253283 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)
%F A253283 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
%F A253283 From _Peter Bala_, Apr 18 2024: (Start)
%F A253283 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 + ....
%F A253283 n-th row polynomial R(n, x) = [t^n] ( (1 - t)/(1 - (1 + x)*t) )^n.
%F A253283 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.
%F A253283 R(n, -2) = (-1)^n * A002003(n) for n >= 1.
%F A253283 R(n, 3) = A299507(n). (End)
%e A253283 [1]
%e A253283 [0, 1]
%e A253283 [0, 2,   3]
%e A253283 [0, 3,  12,   10]
%e A253283 [0, 4,  30,   60,   35]
%e A253283 [0, 5,  60,  210,  280,  126]
%e A253283 [0, 6, 105,  560, 1260, 1260,  462]
%e A253283 [0, 7, 168, 1260, 4200, 6930, 5544, 1716]
%e A253283 .
%e A253283 R_0(x) = 1/(x-1)^0.
%e A253283 R_1(x) = 0/(x-1)^1 + 1/(x-1)^2.
%e A253283 R_2(x) = 0/(x-1)^2 + 2/(x-1)^3 + 3/(x-1)^4.
%e A253283 R_3(x) = 0/(x-1)^3 + 3/(x-1)^4 + 12/(x-1)^5 + 10/(x-1)^6.
%e A253283 Then k!*[x^k] R_n(x) is A001286(k+2) and A001754(k+3) for n = 2, 3 respectively.
%e A253283 .
%e A253283 Seen as an array A(n, k) = binomial(n + k, k)*binomial(n + 2*k - 1, n + k):
%e A253283 [0] 1, 1,   3,   10,    35,    126,     462, ...
%e A253283 [1] 0, 2,  12,   60,   280,   1260,    5544, ...
%e A253283 [2] 0, 3,  30,  210,  1260,   6930,   36036, ...
%e A253283 [3] 0, 4,  60,  560,  4200,  27720,  168168, ...
%e A253283 [4] 0, 5, 105, 1260, 11550,  90090,  630630, ...
%e A253283 [5] 0, 6, 168, 2520, 27720, 252252, 2018016, ...
%e A253283 [6] 0, 7, 252, 4620, 60060, 630630, 5717712, ...
%p A253283 T_row := proc(n) local egf, k, F, t;
%p A253283 if n=0 then RETURN(1) fi;
%p A253283 egf := (x/(1-x))^n/n!; t := diff(egf,[x$n]);
%p A253283 F := convert(t,parfrac,x);
%p A253283 # print(seq(k!*coeff(series(F,x,20),x,k),k=0..7));
%p A253283 # gives A000142, A001286, A001754, A001755, A001777, ...
%p A253283 seq(coeff(F,(x-1)^(-k)),k=n..2*n) end:
%p A253283 seq(print(T_row(n)),n=0..7);
%p A253283 # 2nd version by _R. J. Mathar_, Dec 18 2016:
%p A253283 A253283 := proc(n,k)
%p A253283     binomial(n,k)*binomial(n+k-1,k-1) ;
%p A253283 end proc:
%t A253283 Table[Binomial[n, k] Binomial[n + k - 1, k - 1], {n, 0, 9}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Feb 22 2017 *)
%o A253283 (PARI) T(n,k) = binomial(n,k)*binomial(n+k-1,k-1);
%o A253283 tabl(nn) = for(n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, Apr 29 2018
%Y A253283 T(n, n) = C(2*n-1, n) = A001700(n-1).
%Y A253283 T(n, n-1) = A005430(n-1) for n >= 1.
%Y A253283 T(n, n-2) = A051133(n-2) for n >= 2.
%Y A253283 T(n, 2) = A027480(n-1) for n >= 2.
%Y A253283 T(2*n, n) = A208881(n) for n >= 0.
%Y A253283 A002002 (row sums).
%Y A253283 Cf. A000142, A001286, A001754, A001755, A001777, A182626, A266732 (row k=3), A008316, A033282, A063007, A345013, A269944.
%K A253283 nonn,tabl
%O A253283 0,5
%A A253283 _Peter Luschny_, Mar 20 2015