Real-rooted atlas

A038320 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(17 + 6\,x + 6\,n\right)\,P_{n}(x) = \left(253 + 204\,x + 36\,x^{2} + 66\,n + 36\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 253 + 204\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
025320436493
135094554183621610115
2465858421654648146881296201433
36002811397550126324054648011016077763925487
4756939721610116251559001515888049183207931524665675252421
5938927333179146744538124363521826001591682404131388855520642799361424116571
611497430894506851184762995217672609989764226191200152801510433051110438071296167961626672013745
71393332726562086126806121684981968137339139168980234861764564286496013752135936254965708825698124810077696495278778311
81674142860618359996359001862583804180243369963936020600870875201176281834112456428649600117875450880191224281601713208320604661769131266490269
91997181694277110493428800262758798798470040976843691960401560440494402719314955526412939100175232430347012480097247246976014023113984011307174912362797056167327493737267
102368086866071314379708198794439777634368093666211171163280073758318645528057824703431193632631779466316813308788751667238731231123200777977975808010096642068487401059942421767823363050198771379337
11279320125528169184710775553561456080861975298161034218493570433612911178376839600115062977087023687517211446055168363611256910387212976069032875523356706697344006068228211302471594371031044810688962561306069401655349112536827295
12327965765367344523462890544366196775785257324957881570264135308348482171858836497910562169077967309052801610881679218331529020653735266201638179181975590656121109977640171522819633625768960463391972499456501160597217283108445175808783641640961000362381323499637
133835473319190734129516918883061005010558300744964788202327355771974873640353309430444378408039093459056962390083253616950963579200207113358756642624010148235452174476803817918197559065601089989798761543682306972966538240034754397937459203469573365350401998286184448047018498457618016427441855899403
144469655541274746573682054386423104736141681210638692824003378656238388732224055856538527396967360678354106453206543366254953449113982412844621032470357657600248536030507971148801082478448231944192036652014696567029769512638243737108481845578373230592002566478616920064023791360219545601278903158046722821109907456323453804820609184609
Technical details
Node id
n-387f8733bacab0a3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-efae2a5576a476b8

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:           253,            204,             36;
  1:          3509,           4554,           1836,            216;
  2:         46585,          84216,          54648,          14688,           1296;
  3:        600281,        1397550,        1263240,         546480,         110160,           7776;
  4:       7569397,       21610116,       25155900,       15158880,        4918320,         793152,          46656;
  5:      93892733,      317914674,      453812436,      352182600,      159168240,       41313888,        5552064,        279936;
  6:    1149743089,     4506851184,     7629952176,     7260998976,     4226191200,     1528015104,      330511104,      38071296,      1679616;
  7:   13933327265,    62086126806,   121684981968,   137339139168,    98023486176,    45642864960,    13752135936,    2549657088,    256981248,     10077696;
  8:  167414286061,   835999635900,  1862583804180,  2433699639360,  2060087087520,  1176281834112,   456428649600,  117875450880,  19122428160,   1713208320,    60466176;
  9: 1997181694277, 11049342880026, 27587987984700, 40976843691960, 40156044049440, 27193149555264, 12939100175232, 4303470124800, 972472469760, 140231139840, 11307174912, 362797056;
%F Formula
The row polynomials satisfy (6*n+17+6*x)*P(n,x) = (66*n+253+(36*n+204)*x+36*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 253 + 204*x + 36*x^2.
%t Mathematica
P[0, x_] := 253 + 204*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((66*n + 253 + (36*n + 204)*x + 36*x^2)*P[n - 1, x])/(6*n + 17 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 253 + 204*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((66*n + 253 + (36*n + 204)*x + 36*x^2)*v[i - 1])/(6*n + 17 + 6*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.