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

[Vladimir Shevelev]

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}
P(n)={ n<3 & return(k->1);
with a program 
if( n%2, k-> (k+(n-1))*P(n-1)(k)+(1/2)*(2*k+(n-1)+1)*P(n-1)(k+1)+(1/4)*(4*k+(n-1))*((n-1)/2-1)!*C(k+(n-1)/2-1,(n-1)/2-1)
, k-> ((k+(n-1))/(4*k+2*(n-1)))*P(n-1)(k)+(1/4)*P(n-1)(k+1)+((4*k+(n-1))/(8*k+4*(n-1)))*
(((n-1)-1)/2)!*C(k+((n-1)-1)/2,((n-1)-1)/2))}
C(a,b)=binomial(a,b)
for(n=1,19,print(P(n)(k))) (Hasler)
and Mathematica
p[1,k_]:=1;
p[2,k_]:=1;
p[t_?OddQ,k_]:=p[t,k]=FullSimplify[(k+t-1) p[t-1,k]+1/2 (2 k+t) p[t-1,k+1]+1/4 (4 k+t-1) ((t-1)/2-1)! Binomial[k+(t-1)/2-1,(t-1)/2-1]]
p[t_?EvenQ,k_]:=p[t,k]=FullSimplify[((-1+k+t) p[t-1,k])/(4 k+2 (-1+t))+1/4 p[t-1,k+1]+((-1+4 k+t) (1/2 (-2+t))! Binomial[k+1/2 (-2+t),1/2 (-2+t)])/(8 k+4 (-1+t))] (Moses)

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

----- Original Message -----
From: "D. S. McNeil" <dsm054 at gmail.com>
Date: Tuesday, September 27, 2011 16:54
Subject: [seqfan] Re: Sequence with a strange for OEIS conjecture
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> > A year ago, I introduced into OEIS sequence A174531, defined 
> by a recursion, with a strange for OEIS conjecture:  all terms 
> of this sequence are integers.  I wonder
> > 1) whether exist other sequences in OEIS with such a strange 
> for OEIS conjecture?
> 
> One reason it's a little unusual is because in general sequences that
> aren't known to be integral are disfavoured.  (Yes, there 
> are a fair
> number of exceptions, and we can always rescue them by the usual "or
> some-number-if-not-integral" or "integral terms generated by.."
> tricks, but still.)
> 
> > 2) Can anyone  suggest a program for the calculation ? (the 
> existing 42 calculated terms I found by handy).
> 
> I'm not sure about the last few of your terms: I find
> 
> [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]
> 
> but you have
> 
> [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, 3342, 22540, 81332, 123380]
> 
> I'm not totally sure of mine, or I'd make the correction.  Anyone
> confirm?  BTW, adding a brief explanation of the motivation 
> for the
> polynomials might be a good idea..  Right now it seems like 
> they fell
> out of the sky.
> 
> 
> Doug
> 
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 Shevelev Vladimir‎