Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + 16\,x + 4\,x^{2}\right)\,P_{n-1}(x) + \left(16\,x + 8\,x^{2}\right)\,P_{n-1}'(x) + \left(4\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1916429
281544616192161449
372914896435643744012080153664110309
465613840642495056448627231335369891841487361024025611653905
55904997065761315015244447660805685889603355269121013940481643520014246406144010241609465453
6531441243609184667676869640272927552871159134568577538252843877523712125810995202103205632206807040116469763440644096278865763897
7478296960987326563330104122843476738166720121786497050401883869520332815024327910592678720080896018406064934403097827532803277309235221603778568535654418350081638458830135647413
84304672115254484390416463898917536292296519628032161401818262777637822216278712324484155310089728299841280088064012132347140193283110115169075205196027411660857355840389124179965870081974468608057737216094371846553614770101345135905
93874204893814309845136810392295399844242034730346073602070539164787523207178375139548835841227195428661488896116865612581173248067180007466474240024620850728128512059708937351624704981697240714444811090613484421128636323489382445962033233921631594741763673227264471859202621444334495344305735613
1034867844019536394485622439800767460557576198642912649537171226024904384494924496131349941741493866496317146087896223491584419153477849745510400331571849727062700544166996582187473059840559395767984215285761285249976934639206420683066027638210562361142411998003201921678497480704011114476323471364507420517990412461801472002226388992023068672010485761465386041035100771785
113138105960923841544099560161952526354191008204162174765260642376000322862979865354296280023455281281274201300480788175661317119218281601419581044909007314329601511450725082362056115201018478546342396174745604561402275777086868480014070998057239854022656306821379662053643878448163241543596559564855074265715820888064461457191320053350428323991924382105612642496696811520403579579662336893472735232012974974566411072962564194304563911490680574200057221
1228242953648159604362345854144957310111867225848161319864636606378454259239715900635292115757161641195654659317526514995201906036265058115185601408046164037555424971186329600650996249633449705762444805757434211232217068093440033697009536706820694466560135807113890578706277990403882567746720399356968960804976798832106563174400122995114391701890662400140013044868443170406401194847744132236902400765524140018473369603668463715557048320130161996212469760335525979357184060817139564544732006711296523449139216777216244531923283336236886097457
1325418658283291490113577518937296469225005397542530136410721539864265910540806404858456854739075833820848071532299063287066880502681645208658984983490007174275841457774832501873740224015360026901016766360456910864044800307829110188233748011369472002315265865065563904960001536011953154902134312611766927360437757390996801695929933824011661559210067178301213900802303257917771599019353702403421070242363972951474176038598463543396248856166403328635478709483864064002199509121617201332224011119661661296787456004274494558423744512012343284161367244802622839567941632039638186065920040185626624002442762649667108864118465822154538758006139472909
1422876792454961372528801078266856642299560153808237575752568699463086273539906592467259216377226915613027876571361232259333231756057924998615041057447252911758845647855542016450516597539684092796853986304010778948395651467433786080942336157856560939100197897794779136001506425666291873186738191565824098165086883721802303204788633604526990124791275431151378083840151883802709160484502068264960037879209205852887321165496320071401779970847004209766727680103022704807896833066267443201148350881754793242578124800994891784335402865393664006719883911335305648537600353826085159729118576640144704901300213723955204560347369619377356801092487848243757056019464223633440768024900056792432642155362228633611274289152026843545663644797087931874846025467681881
Technical details
Node id
n-1f22c0bcbe735965 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-14e66869ea533ba5

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:         9,            16,               4;
  2:        81,           544,             616,               192,                 16;
  3:       729,         14896,           43564,             37440,              12080,               1536,                  64;
  4:      6561,        384064,         2495056,           4486272,            3133536,             989184,              148736,               10240,                256;
  5:     59049,       9706576,       131501524,         444766080,          568588960,          335526912,           101394048,            16435200,            1424640,              61440,              1024;
  6:    531441,     243609184,      6676768696,       40272927552,        87115913456,        85775382528,         43877523712,         12581099520,         2103205632,          206807040,          11646976,           344064,             4096;
  7:   4782969,    6098732656,    333010412284,     3476738166720,     12178649705040,     18838695203328,      15024327910592,       6787200808960,      1840606493440,       309782753280,       32773092352,       2160377856,         85356544,        1835008,         16384;
  8:  43046721,  152544843904,  16463898917536,   292296519628032,   1614018182627776,   3782221627871232,    4484155310089728,    2998412800880640,   1213234714019328,    311011516907520,    51960274116608,    5735584038912,     417996587008,    19744686080,     577372160,      9437184,      65536;
  9: 387420489, 3814309845136, 810392295399844, 24203473034607360, 207053916478752320, 717837513954883584, 1227195428661488896, 1168656125811732480, 671800074664742400, 246208507281285120, 59708937351624704, 9816972407144448, 1109061348442112, 86363234893824, 4596203323392, 163159474176, 3673227264, 47185920, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (9+16*x+4*x^2)*P(n-1,x) + (16*x+8*x^2)*d/dx P(n-1,x) + 4*x^2*d^2/dx^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[(9 + 16*x + 4*x^2)*P[n - 1, x] + (16*x + 8*x^2)*D[P[n - 1, x], x] + 4*x^2*D[P[n - 1, x], {x, 2}]];
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] = (9 + 16*x + 4*x^2)*v[i - 1] + (16*x + 8*x^2)*deriv(v[i - 1], x) + 4*x^2*deriv(deriv(v[i - 1], x), x)); 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 Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.