A316649 → basis A013611

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(-8 - 32\,x - 32\,x^{2} + 12\,n + 48\,n x + 48\,n x^{2} - 4\,n^{2} - 16\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(12 + 56\,x + 64\,x^{2} - 30\,n - 140\,n x - 160\,n x^{2} + 12\,n^{2} + 56\,n^{2} x + 64\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 3 + 20\,x + 32\,x^{2}\)

\(P_{2}(x) = 26 + 300\,x + 1264\,x^{2} + 2304\,x^{3} + 1536\,x^{4}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	3	20	32																											55
2	26	300	1264	2304	1536																									5430
3	252	4128	27696	97472	189952	194560	81920																							595980
4	2568	54480	499488	2586048	8274176	16762880	21012480	14909440	4587520																					68689080
5	26928	701376	8143488	55522304	246258432	742712320	1543192576	2182053888	2010251264	1089994752	264241152																			8143098480
6	287648	8881600	124736512	1053918720	5968017920	23867730944	69144309760	146238898176	224164577280	242936709120	176733290496	77510737920	15502147584																	983254243680
7	3112896	111150080	1830950144	18446121984	127000011776	632239382528	2347403980800	6604617531392	14153422733312	22993104273408	27880379645952	24472631377920	14702679687168	5412464099328	921270484992															120267604543680
8	34013312	1378752768	26055394816	304753390592	2469639309312	14705116819456	66561157505024	233661014458368	643012724064256	1391968241319936	2362839624843264	3112476623241216	3119427398664192	2299836606971904	1176462409334784	373114546421760	55276229099520													14852143553604480
9	374416128	16985318400	362135973888	4822043394048	44942276960256	311378589548544	1662228653998080	6991486209490944	23491160237604864	63530400071548928	138637613209747456	243568896680919040	341992616897806336	378603380005666816	323231741831544832	205372687416033280	91457591946772480	25476199811645440	3341140958904320											1847718666337294080
10	4145895936	208098266112	4940590288896	73777471930368	777238185099264	6140935618068480	37760301403471872	185053911423123456	734171750133792768	2381423459489808384	6350738668292931584	13949580620154798080	25195673526604922880	37221055839527239680	44537420114212945920	42504860888123572224	31598420519496450048	17636463137148370944	6953248449575780352	1726701647561752576	203141370301382656									231222711307559892480
11	46127840256	2538077184000	66399927496704	1098848131399680	12909643792220160	114539542455779328	797042881440645120	4459192926595448832	20397092926284103680	77153648546938880000	243124048026997358592	640918947443695943680	1415442900712515698688	2615472115345029857280	4027360340793958072320	5130514226930226561024	5346833681096984494080	4483347294988466651136	2951058651461063802880	1468469722689432453120	519358614271434948608	116344602990791884800	12410090985684467712							29073590832036053191680
12	515268544512	30839581593600	881410070519808	16009853524819968	207500543957532672	2042419980247105536	15864819080024555520	99758029412249370624	516801980902635208704	2233194249110600810496	8118545952283394310144	24971095410448898457600	65195062997499999420416	144639119509022127947776	272413396115226030505984	434196566169557210759168	582445137416627819642880	652025258870811459059712	601735004892185757745152	449872536522651837923328	265650712351617167917056	119248047194317646725120	38237558675391459098624	7801810533000302034944	761152247121980686336					3670178890779411252295680
13	5775088103424	373533920772096	11579082203430912	229011995088912384	3245670506775773184	35094902472306327552	300976075482608959488	2101072974322121834496	12158039634798839857152	59082397347248332603392	243389211944144861134848	855611111556515251617792	2578168114184720801071104	6676071093583067960311808	14869091884012171918573568	28462189946064474520682496	46709240299205334321332224	65434291268415868484190208	77738845870309200233496576	77601361423936120827150336	64251762128540426443948032	43334883643212391564443648	23200552770503318884581376	9485406115918053891899392	2781777262539417260654592	521096538414279085260800	46840138284429580697600			464864259888199153177620480
14	64912164888576	4511973405081600	150774107477508096	3225300167629602816	49616614735359639552	584590818981696503808	5485798268136047247360	42094400527652521771008	269080975939184306946048	1452413057205515631722496	6686123383469708213747712	26443341191908446531747840	90322908948219510225108992	267399344050400899369009152	687544919718215831775084544	1536479179695768354898837504	2982424706870115991038197760	5018464726948221481788637184	7295052913267408202580885504	9113788584127202571728191488	9714944951972131079882539008	8749080703959844696348426240	6567938340508025666590998528	4034602175203294614942384128	1975519768940579001819201536	741503645542703270618726400	200345308614994837268070400	34688468123783278048051200	2890705676981939837337600	59047890528563294749746094080

Technical details

Node id: n-7f43c706f4df7398 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-dd4f339d9dffb950

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         1;
  1:         3,          20,           32;
  2:        26,         300,         1264,          2304,           1536;
  3:       252,        4128,        27696,         97472,         189952,          194560,            81920;
  4:      2568,       54480,       499488,       2586048,        8274176,        16762880,         21012480,         14909440,           4587520;
  5:     26928,      701376,      8143488,      55522304,      246258432,       742712320,       1543192576,       2182053888,        2010251264,        1089994752,          264241152;
  6:    287648,     8881600,    124736512,    1053918720,     5968017920,     23867730944,      69144309760,     146238898176,      224164577280,      242936709120,       176733290496,        77510737920,        15502147584;
  7:   3112896,   111150080,   1830950144,   18446121984,   127000011776,    632239382528,    2347403980800,    6604617531392,    14153422733312,    22993104273408,     27880379645952,     24472631377920,     14702679687168,      5412464099328,       921270484992;
  8:  34013312,  1378752768,  26055394816,  304753390592,  2469639309312,  14705116819456,   66561157505024,  233661014458368,   643012724064256,  1391968241319936,   2362839624843264,   3112476623241216,   3119427398664192,   2299836606971904,   1176462409334784,    373114546421760,    55276229099520;
  9: 374416128, 16985318400, 362135973888, 4822043394048, 44942276960256, 311378589548544, 1662228653998080, 6991486209490944, 23491160237604864, 63530400071548928, 138637613209747456, 243568896680919040, 341992616897806336, 378603380005666816, 323231741831544832, 205372687416033280, 91457591946772480, 25476199811645440, 3341140958904320;

%F Formula

The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(16*n^2-48*n+32)*x+(16*n^2-48*n+32)*x^2)*P(n-2,x) + (12*n^2-30*n+12+(56*n^2-140*n+56)*x+(64*n^2-160*n+64)*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 20*x + 32*x^2, P(2,x) = 26 + 300*x + 1264*x^2 + 2304*x^3 + 1536*x^4.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 20*x + 32*x^2;
P[2, x_] := 26 + 300*x + 1264*x^2 + 2304*x^3 + 1536*x^4;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (16*n^2 - 48*n + 32)*x + (16*n^2 - 48*n + 32)*x^2)*P[n - 2, x] + (12*n^2 - 30*n + 12 + (56*n^2 - 140*n + 56)*x + (64*n^2 - 160*n + 64)*x^2)*P[n - 1, x])/(n^2 - 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 20*x + 32*x^2; v[3] = 26 + 300*x + 1264*x^2 + 2304*x^3 + 1536*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (16*n^2 - 48*n + 32)*x + (16*n^2 - 48*n + 32)*x^2)*v[i - 2] + (12*n^2 - 30*n + 12 + (56*n^2 - 140*n + 56)*x + (64*n^2 - 160*n + 64)*x^2)*v[i - 1])/(n^2 - 2*n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] order…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003