Real-rooted atlas

A038320 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 12\,x + 6\,n x\right)\,P_{n}(x) = \left(121 + 264\,x + 108\,x^{2} + 66\,n x + 36\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 264\,x + 108\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0121264108493
113314356356486410115
2146416388878408380166480201433
316105187846014374801045440356400466563925487
4177156111595672237184202299968011761200307929632659275252421
5194871711488111243652636684427438403018708001185528962514758422394881424116571
62143588811870768416535720046477922915846641157600347755161611064936961970749441511654426672013745
723579476912315075914875766120848128572811136131494920480860694024963651429196897552097281496537856100776960495278778311
8259374246012829537229201041784161660202042988928024107402088001893526854912100414302912035769102336082309582080110854656006651279369131266490269
928531167061134237400473321400620928454030559002075360416713664664003818612490739224300261304704108201534566403319819810560670670668800804804802564353564672167327493737267
10313842837672141084880567984184881962555928448198697105280687577546695600720081212539392534605748703488285652051255296109554053748480295095094272005311711696896574670536704282981703683050198771379337
11345227121439314895948267684762403465513227064640924136860550410924843241941200128714516741416321092123172351411268080405549178883133245937206528105496496202240025319159088537641088943374336404663836262418284971622455349112536827295
12379749833583241579981564018040830844474086413988897293791604770561682425859258944802202448397575345922102337106776466561497768922081935368041997905496755232492920830289920974787624908697621092324265492483115911539220482815885629849611754624614401000362381323499637
1341772481694156516835497004498338039148755571217754012337789634565595202523638788888417200363403985599932076838542846957568553603089148401793991680189561379200994944089355532283297280032167991621987020887005837595156480171375134657126402323105644625920193951306137600752295975321618016427441855899403
14459497298635721618020316485278049924921557843238803360167034690437195750403701336890369678560058144637695989132288678354106453206543366041001319063805952041703503424218887680224665338312290304009435944209116192768306260548334950809675405059249135616013628886448472064017067714940108800132404091656601647958868426752323453804820609184609
Technical details
Node id
n-d0d262c0de3cfac6 · 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:          121,           264,            108;
  1:         1331,          4356,           3564,            864;
  2:        14641,         63888,          78408,          38016,           6480;
  3:       161051,        878460,        1437480,        1045440,         356400,          46656;
  4:      1771561,      11595672,       23718420,       22999680,       11761200,        3079296,         326592;
  5:     19487171,     148811124,      365263668,      442743840,      301870800,      118552896,       25147584,        2239488;
  6:    214358881,    1870768416,     5357200464,     7792291584,     6641157600,     3477551616,     1106493696,      197074944,      15116544;
  7:   2357947691,   23150759148,    75766120848,   128572811136,   131494920480,    86069402496,    36514291968,     9755209728,    1496537856,    100776960;
  8:  25937424601,  282953722920,  1041784161660,  2020429889280,  2410740208800,  1893526854912,  1004143029120,   357691023360,   82309582080,  11085465600,   665127936;
  9: 285311670611, 3423740047332, 14006209284540, 30559002075360, 41671366466400, 38186124907392, 24300261304704, 10820153456640, 3319819810560, 670670668800, 80480480256, 4353564672;
%F Formula
The row polynomials satisfy (11+(6*n+12)*x)*P(n,x) = (121+(66*n+264)*x+(36*n+108)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 264*x + 108*x^2.
%t Mathematica
P[0, x_] := 121 + 264*x + 108*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((121 + (66*n + 264)*x + (36*n + 108)*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] = 121 + 264*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((121 + (66*n + 264)*x + (36*n + 108)*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.