[seqfan] Re: Sequence with a strange for OEIS conjecture

[Vladimir Shevelev]

Dear SeqFans,

Finally, I and Peter Moses found and proved the following simple formulas: for even n>=4,
P_n(x)=(n+x-1)P_(n-2)(x+1)+(x+n/2-1)(x+n/2-2)...(x+1),
for odd n>=3,
P_n(x)=2(n+x-1)P_(n-1)(x)+(x+(n-1)/2-1)(x+(n-1)/2-2)...x.
>From  the first formula, by a simple induction, we see that all coefficients of P_n(x) are integer
for n=2,4,6,... From the second formula we find that all coefficients of P_n(x) are integer for 1,3,5,...
This proves the " strange for OEIS conjecture" on  the integrality of all  coefficients of P_n(x) (see A174531).
Nevertheless, there are many other open problems.  I mention only one of them. To prove (or disprove) that the great common divisor of all coefficients is n/rad(n), where rad(n)=prod_{p|n}p (squarefree kernel: see A007947). 

Best regards,
Vladimir

----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Sunday, October 9, 2011 23:09
Subject: [seqfan] Re: Sequence with a strange for OEIS conjecture
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> 
> These days I was able to prove the following (a very strange for 
> polynomials) representation for polynomials of A174531:
> P_n(k)=c_n(k)*(2^(n+k-1)-R_k(n)/(2k-2)!!),
> where c_n(k)=((n-1)/2)!*prod{i=1,...,k-1}(n+i)/(n+2i), if n is odd,
> c_n(k)=(1/2)*((n-2)/2)!*prod{i=0,...,k-1}(n+i)/(n+2i+1), if n is even,
> and R_k(n), k=0,1,..., are polynomials in n of degree k-1(for 
> k>=1) with integer coefficients, defined by the recursion 
> R_0(n)=0, R_1(n)=1, and for k>=1,
> R_(k+1)(n)=4*k*(R_k(n+1)-R_k(n))+(n+4k)*prod{i=1,...,k-1}(n+k+i).
> 
> A few polynomials R's: 0, 1, n+4, n^2+11*n+32, 
> n^3+21*n^2+152*n+384, n^4+34*n^3+443*n^2+2642*n+6144.
> 
> Are the latter polynomials  in OEIS?
> 
> Best regards,
> Vladimir
> 
> ----- Original Message -----
> From: Vladimir Shevelev <shevelev at bgu.ac.il>
> Date: Wednesday, September 28, 2011 10:14
> Subject: [seqfan] Re: Sequence with a strange for OEIS conjecture
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> 
> > Maximilian Hasler and Peter J.C. Moses sent me the following 
> > extension with the correction:
> > 
> > {1, 1, 3, 4, 2, 4, 5, 25, 32, 3, 19, 32, 7, 77, 294, 384, 4, 52, 
> > 240, 384, 9, 174, 1323, 4614, 6144, 5, 110, 967, 3934, 6144, 11, 
> > 330, 4169, 27258, 90992, 122880, 6, 200, 2842, 21040, 79832, 
> > 122880, 13, 559, 10569, 110513, 664898, 2161848, 2949120, 7, 
> > 329, 6867, 79687, 533630, 1935048, 2949120, 15, 875, 23121, 
> > 352385, 3300120, 18813500, 60080304, 82575360, 8, 504, 14504, 
> > 241800, 2482832, 15542976, 54575616, 82575360, 17, 1292, 45458, 
> > 949688, 12710033, 110549708, 606539532, 1911575472, 2642411520, 
> > 9, 732, 27762, 627528, 9104601, 85954668, 512386668, 1756141872, 
> > 2642411520, 19, 1824, 82422, 2259024, 40822659, 499742256, 
> > 4119259508, 21958313376, 68505970752, 95126814720, 10, 1020, 
> > 49284, 1449048, 28137738, 370512540, 3287673896, 18879806352, 
> > 63499315392, 95126814720}
> 
> > I am very grateful to them! Now  I am sure that 
>  P_n(0)=floor((n-
> > 1)/2)!*4^floor((n-1)/2). Further, if n is even, then the 
> > coefficients of P_n do not exceed the corresponding coefficients 
> > of P_(n-1) and the equality holds only for the last ones. 
> > Although these statements have only status of conjectures...
> > 
> > Regards,
> > Vladimir
> 
> 
> 
>  Shevelev Vladimir‎
> 
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 Shevelev Vladimir‎