[seqfan] Re: polynomial-to-product transform

[Richard Mathar]

zs> From seqfan-bounces at list.seqfan.eu  Mon Nov 10 08:58:16 2008
zs> Date: Sun, 9 Nov 2008 23:31:02 -0800 (PST)
zs> From: zak seidov <zakseidov at yahoo.com>
zs> To: Maximilian Hasler <maximilian.hasler at gmail.com>, seqfan at list.seqfan.eu
zs> Subject: [seqfan] Re: polynomial-to-product transform
zs> 
zs> Thanks to Maximilian and Vladetta for Cf. A147557!!
zs> 
zs> My extended list differs with 
zs> 
zs> %V A147557 2,3,-1,9,-4,0,16,89,-52,60,-182,214,-620,966
zs> 
zs> only in sign of a(7)=+/-16 (A147557/me).
zs> 
zs> Anyone may wish to check this, plz 
zs> (and submit b147557.txt),
zs> thx, zak 
zs> 
zs> 2,3,-1,9,-4,0,-16,89,-52,60,-182,214,-620,966,-2142,10497,
zs> -7676,13684,-27530,48288,-98372,190928,-364464,619496,-1341508,
zs> 2649990,-4923220,9726940,-18510902,37055004,-69269976,213062855,
zs> -258284232,527143794,-981480012,1979517254,-3714536700,7551366306,
zs> -14016801198,28063025740,-53634593948,108757956004,-204560060586,
zs> 416896705290,-778522426720,1592078432226,-2994413652394,
zs> 5789062124454,-11488734188248,23383770149694,-44002174265850,
zs> 90058079530292,-169947147350468,344585755402348,-655029081099310,
zs> 1330399961685334,-2521381593853482,5138449375429832,-9770521153547216,
zs> 19862681215118990,-37800704921557716,76761272183181094,
zs> -146085259004502992,314629354289519543,-567573802060764702,
zs> 1148857463539727720,-2202596306028591264,4479728073233550184,
zs> -8538874003815347514,17345648087498134054,-33256101986609836428,
zs> 67450786459945302498,-129379903629067747422,262149774724535045168,
zs> -502905327648037910728,1022802038670879059774,-1962540968377372942408,
zs> 3966700017124863853724,-7651429237968671088252,15485239525185936636646,
zs> -29810243436684273184168,60409436449126286018102,-116522970564843598344936,
zs> 235899903648746026365746,-455119684716975212741976,
zs> 920580501534094742399156,-1776535206654668848302410,
zs> 3603923762971721944633672,-6954719321824229669804934,
zs> 14040832662617308346509334,-27207468359655264354393112,
zs> 55095055381903738766267924,-106378672353461769216151592,
zs> 215195889017579047026362158,-416987166300169532047992942,
zs> 839111284171414880906234688,-1633570361118782811048640282,
zs> 3299793062436832046486425714,-6396532289230722406545500102,
zs> 12950649168890351222748680200,-25101992083727014802336506688,
zs> 50631048103159668185384183136

I confirm the first 50 entries in this list, including the sign flip at a(7):

[2, 3, -1, 9, -4, 0, -16, 89, -52, 60, -182, 214, -620, 966, -2142, 10497,

    -7676, 13684, -27530, 48288, -98372, 190928, -364464, 619496, -1341508,

    2649990, -4923220, 9726940, -18510902, 37055004, -69269976, 213062855,

    -258284232, 527143794, -981480012, 1979517254, -3714536700, 7551366306,

    -14016801198, 28063025740, -53634593948, 108757956004, -204560060586,

    416896705290, -778522426720, 1592078432226, -2994413652394, 5789062124454,

    -11488734188248, 23383770149694]

The comparison became boring after the 50th entry, so I did not try to compare
the others.  The A147558 seem to be correct so far:

[1, 1, 1, 2, 2, 3, 4, 8, 8, 14, 18, 29, 40, 68, 88, 174, 210, 344, 492, 852,

    1144, 1962, 2786, 4601, 6704, 11240, 16096, 27738, 39650, 64936, 97108,

    168408, 236880, 397110, 589298, 979496, 1459960, 2421132, 3604880, 6086790

    ]

A147559 seems to be incorrect, I get:

[1, 4, 5, 11, -6, -22, -4, 155, 16, -182, -158, 376, 56, -1456, 680, 23155,

    -4966, -28674, 6132, 117946, 15792, -415426, -162814, 512550, 333904,

    -4231332, 235968, 15171332, -5259270, -68578566, 15199212, 736983115,

    -4403208, -1097465342, -161039518, 4048823252, 635914664, -13587460600,

    -773551072, 42897471418]

A147654 seems to be correct so far:
[1, 2, 1, 3, 0, -2, 0, 9, 0, -6, 0, 4, 0, -18, 0, 93, 0, -54, 0, 72, 0, -186,

    0, 232, 0, -630, 0, 1020, 0, -2106, 0, 10881, 0, -7710, 0, 13824, 0,

    -27594, 0, 49440]

Outputs of the following Maple prog:

# Partition n into a set of distinct positive integers, the maximum one
# being m.
# Example: partitionsQ(7,5) returns [[2,5],[3,4],[1,2,4]] ;
# Richard J. Mathar, 2008-11-10
partitionsQ := proc(n,m)
	local p,t,rec,q;
	p := [] ;
	# take 't' of the n and recursively determine the partitions of
	# what has been left over.
	for t from min(m,n) to 1 by -1 do
		# Since we are only considering partitions into distinct parts,
		# the triangular numbers set a lower bound on the t.
		if t*(t+1)/2 >= n then
			rec := partitionsQ(n-t,t-1) ;
			if nops(rec) = 0 then
				p := [op(p),[t]] ;
			else
				for q in rec do
					p := [op(p),[op(q),t]] ;
				end do:
			end if;
		end if;
	end do:
	RETURN(p) ;
end:

# Power product expansion of L.
# L is a list starting with 1, which is considered L[0].
# Returns the list [a(1),a(2),..] such that
#   product_(i=1,2,..) (1+a(i)x^i) = sum_(j=0,1,2,...) L[j]x^j. 
# Richard J. Mathar, 2008-11-10
ppe := proc(L)
	local pro,i,par,swithi,snoti,m,p,k ;
	pro := [] ;
	for i from 1 to nops(L)-1 do
		par := partitionsQ(i,i) ;
		swithi := 0 ;
		snoti := 0 ;
		for p in par do
			if i in p then
				m := 1 ;
				for k from 1 to nops(p)-1 do
					m := m*op(op(k,p),pro) ;
				end do;
				swithi := swithi+m ;
			else
				snoti := snoti+mul( op(k,pro),k=p) ;
			end if;
		end do:
		pro := [op(pro), (op(i+1,L)-snoti)/swithi] ;
	end do:
	RETURN(pro) ;
end:

A147557 := proc(nmax)
	local L ;
	L := [1,seq(ithprime(n),n=1..nmax)] ;
	ppe(L) ;
end:

A147558 := proc(nmax)
	local L ;
	L := [1,seq(combinat[fibonacci](n),n=1..nmax)] ;
	ppe(L) ;
end:

A147559 := proc(nmax)
	local L ;
	L := [1,seq(n^2,n=1..nmax)] ;
	ppe(L) ;
end:

A147654 := proc(nmax)
	local L ;
	L := [1,seq(n,n=1..nmax)] ;
	ppe(L) ;
end:

A147557(50) ;
A147558(40) ;
A147559(40) ;
A147654(40) ;