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A245244 Triangle of coefficients of the Pbar polynomials, read by rows.

%I A245244 #41 Jan 05 2025 19:51:40
%S A245244 1,-3,4,25,-56,32,-427,1228,-1184,384,12465,-41840,52416,-29184,6144,
%T A245244 -555731,2079892,-3076288,2258688,-829440,122880,35135945,-142843304,
%U A245244 237829600,-208562688,102279168,-26787840,2949120,-2990414715,12987478876,-23672564832,23581133952,-13947525120,4929576960,-970260480
%N A245244 Triangle of coefficients of the Pbar polynomials, read by rows.
%C A245244 Pbar(r,n) is a polynomial of degree r defined by the recurrence
%C A245244 Pbar(r+1,n) = (2*n-1)^2 * Pbar(r,n) - 4*(n-1)^2 * Pbar(r,n-1)
%C A245244 with initial condition Pbar(0,n) = 1.
%D A245244 B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, Chapter 10, p. 20 and p. 23.
%H A245244 P. Bala, <a href="/A245244/a245244.pdf">A245244 and A160485 and some hypergeometric series evaluations of Ramanujan</a>
%H A245244 R. P. Brent, <a href="http://arxiv.org/abs/1407.3533">Generalising Tuenter's binomial sums</a>, arXiv:1407.3533 [math.CO], 2014.
%H A245244 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 A245244 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 A245244 H. J. H. Tuenter, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/40-2/tuenter.pdf">Walking into an absolute sum</a>, The Fibonacci Quarterly, 40 (2002), 175-180.
%F A245244 In terms of Dumont-Foata polynomials F(r,x,y,z),
%F A245244 Pbar(r,n) = (-4)^r F(r+1,1/2-n,1/2,1/2).
%F A245244 In terms of odd absolute moments of a symmetric Bernoulli random walk with an odd number of steps,
%F A245244 n*C(2*n,n)*Pbar(r,n) = Sum_{k} C(2*n-1,k) * |2*n-1-2*k|^(2*r+1).
%F A245244 In terms of the Pochhammer symbol or ascending factorial (x)_k,
%F A245244 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 A245244 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 A245244 Pbar(1,n) = 4*n-3, Pbar(2,n) = 32*n^2 - 56*n + 25.
%e A245244 Triangle begins:
%e A245244 1,
%e A245244 -3, 4,
%e A245244 25, -56, 32,
%e A245244 -427, 1228, -1184, 384,
%e A245244 12465, -41840, 52416, -29184, 6144,
%e A245244 ...
%e A245244 From _Peter Bala_, Jan 22 2018: (Start)
%e A245244 The polynomials Pbar(r,n) as hypergeometric series:
%e A245244 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 A245244 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 A245244 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 A245244 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 A245244 (PARI)
%o A245244 N=10; P=vector(N+2); P[1]=1;
%o A245244 Pbar(r)=P[r+1];
%o A245244 for (r=0, N, P[r+2] = (2*n-1)^2 * Pbar(r) - 4*(n-1)^2 * subst(Pbar(r),n,n-1) );
%o A245244 seq=[];  for(r=1,N, seq=concat(seq, Vecrev(P[r])); );  seq
%o A245244 \\ _Joerg Arndt_, Jan 27 2015
%Y A245244 (-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 A245244 sign,easy,tabl
%O A245244 0,2
%A A245244 _Richard P. Brent_, Jul 14 2014
%E A245244 More terms from _Joerg Arndt_, Jan 27 2015