Real-rooted atlas

A038320 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(17 + 24\,x + 12\,n x\right)\,P_{n}(x) = \left(289 + 816\,x + 432\,x^{2} + 204\,n x + 144\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 816\,x + 432\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02898164321537
149132080822032691254665
2835214716487490884700161036801877953
314198571002252021224160199756808812800149299262948009
424137569204459408541216080679173120449452800152285184209018882071626049
541033867340551115921288094270420205400320178282944009060968448248732467228665446467215035273
66975757441787850252162919680346245495868887046061620096004107639029761691380776963898500710438698352642156234538625
711858787649715067636072566381587042496140144656619521854855749376015711719288832862604196249629823530434565920847953925159780352068533758575657
82015993900449284610903592801356087246530403403513089331205255424623232005341984558202883666067834060801690000057958405032720760832087716265984006810910064642161564750404481
9342718963076335322223897185362817647945568720795571184631168014039491493491200166491852063989761371109369938739279007502709555203137062607585280820147086950400127364018208768891610044825667733864561332361
10582622237229761987030613659830457480018089601888180329468516398080358007033084025600485204826015055872466177185779171328322350611054985216159990192986849280557700019126272001299112985729433618188844914442241159093058273282110688189467108417
1199045780329059371817781380156854321154825817982001568398528125421239756887910615901744064001340378331866591846414717879722456694784118732475071919554567071566530018738176308129260567265280095701323282068275220098673630458675225615956587839488149790487530700865455734893011772585
121683778265594009293327938219056394832229040453899763644328622699440932278374420922726584615087232035445560331583206604843785692174308666982440369041524452648550428050547235740994764814666952803001827328056942287352830623744159449477468305489923048298833952899072356501360323067904192587769682329602021343856452844631233
1328624230515098157936061601756138433444044927165957261330232018323236311981091545604850268435524406585600903861788455371768422412405946116054122311680128676319859192817254401021841363587707666432062334549412757766144002904056654994361810944101649041886044749824025910540088599642112045453923441191157760491098812689940480024651234519338188862189670629242087867529
14486611918756668684811099170451779769264512872870652883934415936038337848283529668464640109939417871886549273600223500369508964655464448337441734356672126877696388888433352227181035520347426063619820606586880242214249146715891302400131650568359744402096128552970787860083439042561761916726024775663616041211557253346649702406678943852583190528006705135789259987353631430324012156190721907050713219198761449729
Technical details
Node id
n-f325e89cf8a71448 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8f8e54bbd9560a6d

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,             816,              432;
  1:           4913,           20808,            22032,             6912;
  2:          83521,          471648,           749088,           470016,            103680;
  3:        1419857,        10022520,         21224160,         19975680,           8812800,           1492992;
  4:       24137569,       204459408,        541216080,        679173120,         449452800,         152285184,          20901888;
  5:      410338673,      4055111592,      12880942704,      20205400320,       17828294400,        9060968448,        2487324672,        286654464;
  6:     6975757441,     78785025216,     291968034624,     549586888704,      606162009600,      410763902976,      169138077696,      38985007104,       3869835264;
  7:   118587876497,   1506763607256,    6381587042496,   14014465661952,    18548557493760,    15711719288832,     8626041962496,    2982353043456,     592084795392,     51597803520;
  8:  2015993900449,  28461090359280,  135608724653040,  340351308933120,   525542462323200,   534198455820288,   366606783406080,  169000005795840,   50327207608320,   8771626598400,    681091006464;
  9: 34271896307633, 532222389718536, 2817647945568720, 7955711846311680, 14039491493491200, 16649185206398976, 13711093699387392, 7900750270955520, 3137062607585280, 820147086950400, 127364018208768, 8916100448256;
%F Formula
The row polynomials satisfy (17+(12*n+24)*x)*P(n,x) = (289+(204*n+816)*x+(144*n+432)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 816*x + 432*x^2.
%t Mathematica
P[0, x_] := 289 + 816*x + 432*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((289 + (204*n + 816)*x + (144*n + 432)*x^2)*P[n - 1, x])/(17 + (12*n + 24)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 289 + 816*x + 432*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((289 + (204*n + 816)*x + (144*n + 432)*x^2)*v[i - 1])/(17 + (12*n + 24)*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.