A272866 → basis A013611 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(41 + 52\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)

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

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	41	52	16																											109
2	1681	4264	4016	1664	256																									11881
3	68921	262236	413280	345280	161280	39936	4096																							1295029
4	2825761	14335568	31683488	39842816	31179520	15548416	4825088	851968	65536																					141158161
5	115856201	734697860	2089684720	3510465920	3857122560	2896305152	1505218560	534609920	124190720	17039360	1048576																			15386239549
6	4750104241	36147134712	125735061456	264347873920	374121208320	375486339072	274035789824	146531254272	56975032320	15710289920	2916089856	327155712	16777216																	1677100110841
7	194754273881	1729037943724	7110790192576	17954840181824	31096819968256	39078808717312	36746696347648	26265423921152	14340174184448	5951323635712	1848095277056	416414695424	64357400576	6106906624	268435456															182803912081669
8	7984925229121	81017777934496	384568439351360	1133574144568320	2322393951234560	3506543238668288	4036261723045888	3612971530321920	2541696327024640	1409940109393920	614683522367488	208395135090688	53861742018560	10259603128320	1358283407360	111669149696	4294967296													19925626416901921
9	327381934393961	3736945007228628	20107989269665488	67770383220523776	160317102547791360	282669944562690048	384985282275385344	414122134160277504	356664256553484288	247783297975582720	139186051337945088	63066785452130304	22879814805356544	6555776474677248	1450976854671360	239362822373376	27715423961088	2010044694528	68719476736											2171893279442309389
10	13422659310152401	170238605884859720	1023986811382477040	3883992274179738240	10418788960241329920	21010283191083823104	33048307331315343360	41520961291894456320	42317350011433451520	35331610704344514560	24297987704441798656	13787945640719810560	6444522071937515520	2467605772574392320	766467464610447360	190157095755055104	36798828820561920	5353419036426240	550786606039040	35734127902720	1099511627776									236736367459211723401
11	550329031716248441	7677761125407173372	51050629321656702496	215214815127415829440	645521734609360547840	1465342513153861714944	2614215949884149157888	3757035925253411733504	4422874254948328734720	4313433620142975877120	3510538852690963726336	2394106903169997996032	1369966381537937063936	656894114667817861120	262853019373896990720	87134441446890799104	23660400394673061888	5175542811243577344	889741301870755840	115760707340861440	10715840324304896	628920651087872	17592186044416							25804264053054077850709
12	22563490300366186081	343405315790939027184	2501124645216557792736	11601284322956712310784	38474371574755912832256	97089610281033732071424	193709012383000693415936	313423182539827424722944	418531167764205274791936	466952814487229692903424	438996629286937518145536	349721341292187645640704	236830802250950735036416	136476620992073227567104	66854929861663298420736	27751174941450252058624	9706715681402268942336	2836689138934238674944	684174025876136001536	133821421678151860224	20694767064953389056	2435180761012240384	204878598673268736	10977524091715584	281474976710656					2812664781782894485727281
13	925103102315013629321	15252919443047541790756	120764202719813557893040	611205623845141234399360	2220734013682206390967040	6166961892576997879178240	13606319187512877103083520	24476652892545500184412160	36557127568531453446717440	45923457038352330493788160	48976906838327666671616000	44637644747696119501291520	34919518708073738000138240	23506284638399442882396160	13627129251931214829322240	6797880461279123493355520	2910715317679518908416000	1065072269192425803612160	330866421187938290237440	86450753857311117148160	18753983790945161707520	3317113045429507850240	466145695157494743040	50066657794966159360	3860429305586647040	190277084256403456	4503599627370496			306580461214335498944273629
14	37929227194915558802161	673475058485329922145688	5759285772187868264803088	31583215830169856289463936	124765014244434823144704000	378102896288653905126983680	914072849320947153199165440	1809742756626267082988093440	2989329287722361634554183680	4175458818416809132425216000	4980986987462258774360719360	5111717902862216854377594880	4536488303324464138818682880	3493774958957349446206095360	2339751399855130645889024000	1363424374227258320470671360	690863180042274134168698880	303791246936690417079418880	115520726349583914222223360	37790691117001359753216000	10558215275015942365511680	2494420853708736105021440	491665592518914279997440	79366117844655464775680	10220074929387208704000	1009610209064476737536	71845924855441522688	3278620528725721088	72057594037927936	33417270272362569384925825561

Technical details

Node id: n-b58f73cba181d1b6 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-10850f2bbd2ef26a

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:              41,               52,                16;
  2:            1681,             4264,              4016,              1664,                256;
  3:           68921,           262236,            413280,            345280,             161280,              39936,               4096;
  4:         2825761,         14335568,          31683488,          39842816,           31179520,           15548416,            4825088,             851968,              65536;
  5:       115856201,        734697860,        2089684720,        3510465920,         3857122560,         2896305152,         1505218560,          534609920,          124190720,           17039360,            1048576;
  6:      4750104241,      36147134712,      125735061456,      264347873920,       374121208320,       375486339072,       274035789824,       146531254272,        56975032320,        15710289920,         2916089856,         327155712,          16777216;
  7:    194754273881,    1729037943724,     7110790192576,    17954840181824,     31096819968256,     39078808717312,     36746696347648,     26265423921152,     14340174184448,      5951323635712,      1848095277056,      416414695424,       64357400576,       6106906624,        268435456;
  8:   7984925229121,   81017777934496,   384568439351360,  1133574144568320,   2322393951234560,   3506543238668288,   4036261723045888,   3612971530321920,   2541696327024640,   1409940109393920,    614683522367488,   208395135090688,    53861742018560,   10259603128320,    1358283407360,    111669149696,     4294967296;
  9: 327381934393961, 3736945007228628, 20107989269665488, 67770383220523776, 160317102547791360, 282669944562690048, 384985282275385344, 414122134160277504, 356664256553484288, 247783297975582720, 139186051337945088, 63066785452130304, 22879814805356544, 6555776474677248, 1450976854671360, 239362822373376, 27715423961088, 2010044694528, 68719476736;

%F Formula

The row polynomials satisfy P(n,x) = (41+52*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(41 + 52*x + 16*x^2)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (41 + 52*x + 16*x^2)*v[i - 1]); 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - ↑ Geometric basis map (elementary) (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.