Real-rooted atlas

A038320 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(41 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1681 + 1968\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0168119685764225
1689211210327084813824274625
2282576166164165809536226713633177617850625
31158562013390913203969849602323814406801408079626241160290625
447501042411668329294424414575040190552780808365731840195880550419110297675418890625
5194754273881798017512488140139660729613672162022408003216793602810885898245484655411245864714244902227890625
67984925229121373928205851527660968119884889689382866944656263777075203073235248742489948348743681504362627072110075314176318644812890625
732738193439396117247438494901364038424623196416551589704631705648432266748149762835059516964864110636468954726427755490469478440617790930944264180754022420711912837890625
813422659310152401785716642545506402069692619388163203230739698557132803309538227790233602324748803911188481134023806785945603793250364162048083266471408435201083141091491840634033809653761346274334462890625
9550329031716248441354358205788023386410371459681600684480182132950506158361602132288201047707648017474361842732433408102288947372092293124276889785592709120125177262017347584024424831613140992028594924815384576152168114316902487507831740087890625
102256349030036618608115849476113427955100851027581633475367641699566012943366571008013113572436443402035201228198003803479605248838769368451156803584420845954902322577408153968032281337528320400567238455512268807034351504584605696748667122439159808365203474360565765688009063105712890625
119251031023150136293217039808973714250057272247251827369476099572485306868489881438234726477661490095825925386240818286920034068286996486386629619778093947289637385148993822988959744164129922411905805189125337558452419700981760124976978398119827865619951978813003608883219465345183418155008876488338465357824369720589101871337890625
123792922719491555880216131083464237784457945185611826879075839940096216962769220466538132315211776445776953150040811717017652188521344395021859553284582406752190782407180288306558221749348509469900815701762577405655363092486127517103377816727060481793419640013019529871363817478612888023832985655865540676410104872965031043062791153909762103572012316858777624031838291621636962890625
1315550983149915379108886011365452179016960116877796055950235628012024301334080141922548910079281154603520249229841988431908369059840320959406268029384436252672313131128066370131157319680235666632969811666654986240137951199787206829261455360628070503096226214523699202205906157216014021741772858693733673153366432153601145243583866407149895680154704574180827982725120129369678757486814822405048572829560461066241562069488955406402587890625
14637590309146530543464326415971577529567505577812227842621668183712563424405368320716163016038553911057076224013624564695367610990841937920191408518647115705627437957122054140200115388060392017100817177479025355161480630108160113119983825509599994393395205885917857587491381822095360241179073188950866377100492877006178579177216758985523218781994775409077258289152033828733554207718889226240424332546324556752617472033118637761916624594534412116574790945106558976101534516782101416168212890625
Technical details
Node id
n-34e4ae47eb2b0153 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-94dfba4114b12650

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:               1681,                1968,                  576;
  1:              68921,              121032,                70848,                13824;
  2:            2825761,             6616416,              5809536,              2267136,               331776;
  3:          115856201,           339091320,            396984960,            232381440,             68014080,              7962624;
  4:         4750104241,         16683292944,          24414575040,          19055278080,           8365731840,           1958805504,            191102976;
  5:       194754273881,        798017512488,        1401396607296,        1367216202240,         800321679360,         281088589824,          54846554112,          4586471424;
  6:      7984925229121,      37392820585152,       76609681198848,       89689382866944,       65626377707520,       30732352487424,        8994834874368,       1504362627072,        110075314176;
  7:    327381934393961,    1724743849490136,     4038424623196416,     5515897046317056,     4843226674814976,     2835059516964864,     1106364689547264,     277554904694784,      40617790930944,      2641807540224;
  8:  13422659310152401,   78571664254550640,   206969261938816320,   323073969855713280,   330953822779023360,   232474880391118848,   113402380678594560,   37932503641620480,    8326647140843520,   1083141091491840,    63403380965376;
  9: 550329031716248441, 3543582057880233864, 10371459681600684480, 18213295050615836160, 21322882010477076480, 17474361842732433408, 10228894737209229312, 4276889785592709120, 1251772620173475840, 244248316131409920, 28594924815384576, 1521681143169024;
%F Formula
The row polynomials satisfy P(n,x) = (41+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 1681 + 1968*x + 576*x^2.
%t Mathematica
P[0, x_] := 1681 + 1968*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(41 + 24*x)*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] = 1681 + 1968*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (41 + 24*x)*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

Proof chains ? JSON

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.