Real-rooted atlas

A038314 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 12\,x + 6\,n x\right)\,P_{n}(x) = \left(242 + 528\,x + 216\,x^{2} + 132\,n x + 72\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 1056\,x + 432\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
048410564321972
1106483484828512691280920
2234256102220812545286082561036803222928
35153632281107204599936033454080114048001492992125615584
411337990474212300815179788801471979520752716800197074944209018884816154944
5249435788819047823872467537495045667121152038639462400151747706883218890752286654464182286921088
6548758735364789167144961371443318784199482664550417001363456008902532136962832623861765045118566438698352646828035518720
71207269217792118531886837763879225387417665829279301632673253992857604406753407795218695317487616499466738073676622738227251597803520253582734495232
8265599227914242897446122700801066786981539840206892020662272024685979738112001938971499429888102824246181888036627560792064084285012049920113515167744006810910064649350416886035456
95843183014113287011819616935936286847166147379206258483625033728085342958523187200782051838103388164976693515203379222159674279198720679899097202688013735335297024001648240235642888916100448256342686707173922816
10128550026310492161682836708064624647572765186290810881835821863343226880281631763126517760029494526465613496322189745146689486848117003080194169241644873340415377408012087095061381120021756771110486016235385051833958411590930582732812493614167569764352
11282810057883082752401076082088735539219689189484356108288525045052916162887688949631583798231040010544293211456824934489466730279027605504557714682258873384962566755071759587737686422729688875008002074145512533000192336600624122560512331500614666158081497904875307008453419929901689200640
126221821273427820544950241794487158046725053558634318067793921470126148165256085504275648652780985516032036084914545874467553283444469115742562811904245394460193904289382413176009368365883719685323640148834700492801597092044650410147843455766407658287923251051094658588344324613547015945584641925877696823296016389937255604218052608
1313688006801541205196822398556584340153958401282826422557663363072040428469074544542351360826945958342956548096001190802180013857429258241262972009105606364364801012252148299855193702406211547273658202324992029280020818590852710400105408074946927069757442851007286318087536640561562041244471787520761235257631021465606355396399516876800246512345193381888590362294414734111637504
143011361496339065143296525619461179182279557123225392148144982170009601094678547249206069821440242570814447267254108160038105669760443437736263684445661472051734402564096395903062446165586870272027330808004096090229964801472366761163425736294400618394039688638809243648200710912956793362579456494174596295135173017608931827022870651863040111854976631497031680086772345508070424576314303240121561907221197868552723443522535424
Technical details
Node id
n-af505d67022dea0b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f7a814172b0ca510

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             484,             1056,               432;
  1:           10648,            34848,             28512,              6912;
  2:          234256,          1022208,           1254528,            608256,            103680;
  3:         5153632,         28110720,          45999360,          33454080,          11404800,           1492992;
  4:       113379904,        742123008,        1517978880,        1471979520,         752716800,         197074944,          20901888;
  5:      2494357888,      19047823872,       46753749504,       56671211520,       38639462400,       15174770688,        3218890752,         286654464;
  6:     54875873536,     478916714496,     1371443318784,     1994826645504,     1700136345600,      890253213696,      283262386176,       50451185664,       3869835264;
  7:   1207269217792,   11853188683776,    38792253874176,    65829279301632,    67325399285760,    44067534077952,    18695317487616,     4994667380736,     766227382272,      51597803520;
  8:  26559922791424,  289744612270080,  1066786981539840,  2068920206622720,  2468597973811200,  1938971499429888,  1028242461818880,   366275607920640,   84285012049920,   11351516774400,    681091006464;
  9: 584318301411328, 7011819616935936, 28684716614737920, 62584836250337280, 85342958523187200, 78205183810338816, 49766935152033792, 22159674279198720, 6798990972026880, 1373533529702400, 164824023564288, 8916100448256;
%F Formula
The row polynomials satisfy (11+(6*n+12)*x)*P(n,x) = (242+(132*n+528)*x+(72*n+216)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1056*x + 432*x^2.
%t Mathematica
P[0, x_] := 484 + 1056*x + 432*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((242 + (132*n + 528)*x + (72*n + 216)*x^2)*P[n - 1, x])/(11 + (6*n + 12)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 484 + 1056*x + 432*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((242 + (132*n + 528)*x + (72*n + 216)*x^2)*v[i - 1])/(11 + (6*n + 12)*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

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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.