Real-rooted atlas

A038320 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(17 + 46\,x + 29\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 986\,x + 841\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02899868412116
1491325143428912438997336
283521569908145829416584527072814477456
314198571211054541318330704842106011888520511149205962976
42413756924705511810536174152396463140306606313520921371985948233219474296896
5410338673489992650725076094477712947784151216205043551244821632817078397519917249876309435817657216
66975757441951985721365683914748121939217972888413509714807056431914020724813310313532234598317802450024641296120047612231936
7118587876497182067269210112423413663748494500583086441265339727309422158520711292542454788259901321794677131188367653770118303314507145975869922190162669056
820159939004493439048418413026399754035464512009299874956403585129227376690733897041839463610432850104580610101698370767340406505704600557805246621481589773042070723330020142420747482776576
9342718963076336431020542432315485282227368735280717384577105859577416650277729022873124470663282239018859391131481447543988333731637040552225343476984523059108528643775578672252627137587122005097657058291951354384207722496
105826222372297611192661991505628411189975743832219463629273837477326024422412458208208956665881988593299384132664121929847036761939794724016250789620681634925173262095156801937994777673408024569767968033874248890399220398911635381478320546904189762301673555234816
1199045780329059372196485834356198972248167853988109534140620695180824890465997059059182237975518414498993488989548341883958494994564462871449105667931904083691412826369265818459986633070742114664553559114330624031182871827502389113854079731878193067088408658061102606287129586021894129065876983540801536
121683778265594009294021258681359810422445886624374308390913042520495730574902681427300056085372638169486961195605597723610612460477652260882927683242926959270968473884243626042112647544132242741237341673246580327466351730267212985538087212181856203032423658722893049749835206303092592442029633684147320982297558232675799463481189937030341242876870656
1328624230515098157937324435455333940411587462376319575876678564653560534274716681953308741039107000206664512417510487942741952070335304686681405834961768577432968267621201050601513209153410358910767455551752587021112979663917305179382436278622624424477113909333294331700824357857909228735992535762870153113587782947287834252057587734933232886318765586291887475981844409498737103395697172336050176
14486611918756668684811328164295900554527952169926902563747417546801352751420409440225960807499813021975867135106203070511684291296191784656960287477734238710960103223401963743103296317515120449111215952202966093488706810052424896149551221528081320037779642256405762952756667731211644524845067045378275540474924318751702028229862565388743030551920103193195091967253935210802347139339346706167938128250246473680347348787521401906756202069927458308096
Technical details
Node id
n-f9f2375bf3345aa6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b635250d2eb0b07f

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:            289,             986,              841;
  1:           4913,           25143,            42891,             24389;
  2:          83521,          569908,          1458294,           1658452,            707281;
  3:        1419857,        12110545,         41318330,          70484210,          60118885,           20511149;
  4:       24137569,       247055118,       1053617415,        2396463140,        3066063135,         2092137198,          594823321;
  5:      410338673,      4899926507,      25076094477,       71294778415,      121620504355,       124482163281,        70783975199,        17249876309;
  6:     6975757441,     95198572136,     568391474812,     1939217972888,     4135097148070,      5643191402072,      4813310313532,      2345983178024,       500246412961;
  7:   118587876497,   1820672692101,   12423413663748,    49450058308644,   126533972730942,    215852071129254,    245478825990132,    179467713118836,     76537701183033,     14507145975869;
  8:  2015993900449,  34390484184130,  263997540354645,  1200929987495640,  3585129227376690,   7338970418394636,  10432850104580610,  10169837076734040,   6505704600557805,   2466214815897730,   420707233300201;
  9: 34271896307633, 643102054243231, 5485282227368735, 28071738457710585, 95774166502777290, 228731244706632822, 390188593911314814, 475439883337316370, 405522253434769845, 230591085286437755, 78672252627137587, 12200509765705829;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (17+46*x+29*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 986*x + 841*x^2.
%t Mathematica
P[0, x_] := 289 + 986*x + 841*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((17 + 46*x + 29*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 289 + 986*x + 841*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((17 + 46*x + 29*x^2)*v[i - 1])/(1 + 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

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.