%I #39 Feb 11 2019 02:53:31

%S 1,-3,4,25,-56,32,-427,1228,-1184,384,12465,-41840,52416,-29184,6144,

%T -555731,2079892,-3076288,2258688,-829440,122880,35135945,-142843304,

%U 237829600,-208562688,102279168,-26787840,2949120,-2990414715,12987478876,-23672564832,23581133952,-13947525120,4929576960,-970260480

%N Triangle of coefficients of the Pbar polynomials, read by rows.

%C Pbar(r,n) is a polynomial of degree r defined by the recurrence

%C Pbar(r+1,n) = (2*n-1)^2 * Pbar(r,n) - 4*(n-1)^2 * Pbar(r,n-1)

%C with initial condition Pbar(0,n) = 1.

%D B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, Chapter 10, p. 20 and p. 23.

%H P. Bala, <a href="/A245244/a245244.pdf">A245244 and A160485 and some hypergeometric series evaluations of Ramanujan</a>

%H R. P. Brent, <a href="http://arxiv.org/abs/1407.3533">Generalising Tuenter's binomial sums</a>, arXiv:1407.3533 [math.CO], 2014.

%H R. P. Brent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Brent/brent5.html"> Generalizing Tuenter's binomial sums</a>, Journal of Integer Sequences, 18 (2015), Article 15.3.2.

%H L. Carlitz, <a href="http://dx.doi.org/10.1016/0012-365X(80)90228-9">Explicit formulas for the Dumont-Foata polynomials</a>, Discrete Mathematics, 30 (1980), 211-255.

%H H. J. H. Tuenter, <a href="http://www.fq.math.ca/Scanned/40-2/tuenter.pdf">Walking into an absolute sum</a>, The Fibonacci Quarterly, 40 (2002), 175-180.

%F In terms of Dumont-Foata polynomials F(r,x,y,z),

%F Pbar(r,n) = (-4)^r F(r+1,1/2-n,1/2,1/2).

%F In terms of odd absolute moments of a symmetric Bernoulli random walk with an odd number of steps,

%F n*C(2*n,n)*Pbar(r,n) = Sum_{k} C(2*n-1,k) * |2*n-1-2*k|^(2*r+1).

%F In terms of the Pochhammer symbol or ascending factorial (x)_k,

%F Pbar(r,n) = Sum_{1 <= j <= k <= r+1} (-1)^(j+1)*(1-n)_{k-1}*(2j-1)^(2r+1)/((k-j)!(k)_j).

%F n*Pbar(r,n) = 1 + 3^(2*r+1)*(n-1)/(n+1) + 5^(2*r+1)*(n-1)*(n-2)/((n+1)*(n+2)) + 7^(2*r+1)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ... = Sum_{k = 0..n-1} binomial(n-1,k)/binomial(n+k,k)*(2*k + 1)^(2*r+1); this follows easily from the above recurrence. Examples are given below. - _Peter Bala_, Jan 22 2018

%e Pbar(1,n) = 4*n-3, Pbar(2,n) = 32*n^2 - 56*n + 25.

%e Triangle begins:

%e 1,

%e -3, 4,

%e 25, -56, 32,

%e -427, 1228, -1184, 384,

%e 12465, -41840, 52416, -29184, 6144,

%e ...

%e From _Peter Bala_, Jan 22 2018: (Start)

%e The polynomials Pbar(r,n) as hypergeometric series:

%e r = 0: n*Pbar(0,n) = n = 1 + 3*(n-1)/(n+1) + 5*(n-1)*(n-2)/((n+1)*(n+2)) + 7*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ..., for n a positive integer (when the series terminates). The identity is also valid for complex n with real part greater than 1/2.

%e r = 1: n*Pbar(1,n) = n*(4*n - 3) = 1 + 3^3*(n-1)/(n+1) + 5^3*(n-1)*(n-2)/((n+1)*(n+2)) + 7^3*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ..., for n a positive integer (when the series terminates). The identity is also valid for complex n with real part greater than 3/2.

%e r = 2: n*Pbar(2,n) = n*(32*n^2 - 56*n + 25) = 1 + 3^5*(n-1)/(n+1) + 5^5*(n-1)*(n-2)/((n+1)*(n+2)) + 7^5*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ..., for n a positive integer (when the series terminates). The identity is also valid for complex n with real part greater than 5/2.

%e The above identities when r = 0 and r = 1 were found by Ramanujan. See Example 5 and Example 13 in Chapter 10 of Berndt. (End)

%o (PARI)

%o N=10; P=vector(N+2); P[1]=1;

%o Pbar(r)=P[r+1];

%o for (r=0, N, P[r+2] = (2*n-1)^2 * Pbar(r) - 4*(n-1)^2 * subst(Pbar(r),n,n-1) );

%o seq=[]; for(r=1,N, seq=concat(seq, Vecrev(P[r])); ); seq

%o \\ _Joerg Arndt_, Jan 27 2015

%Y (-1)^r Pbar(r,0) is sequence A009843. The leading coefficient of Pbar(r,n) is sequence A047053. Cf. also A036970, A083061, A160485 for analogous moments of Bernoulli random walks.

%K sign,easy,tabl

%O 0,2

%A _Richard P. Brent_, Jul 14 2014

%E More terms from _Joerg Arndt_, Jan 27 2015