Real-rooted atlas

A038301 → reverse → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(20 + 9\,x + 9\,n\right)\,P_{n}(x) = \left(319 + 360\,x + 81\,x^{2} + 99\,n + 81\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 319 + 360\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
031936081760
145988613486072918800
262557165528155034583206561448000
38198962815065372438023255106561005904910400000
410468315442743847600675567038840313943857085880531441236800000
51310955146595038451394643096159614185510558617303955692517440174047829695312000000
61617435193943887700823742138420334714343042873055339015204408912474683101276527504043046721117760000000
7197210170521310122506333822745188246410377373406777965446564654349649182052595203125492761599677484097803874204892585600000000
8238152716791177489153468058955512784851146823556472014423349090150122003378038086981524473770263905097544061793567995577484097800348678440156320000000000
92853116706110235771189623098785713096666019455319219000528383883022682028558231198497020130557376283298738703271890326582558210706797292479505767092568220313810596091218560000000000
1033952088802709308136604259880127316442396468631628567147997605252936189130135613091873289931251404816157294603105857423769408133297249417051539189906985284073410758778654753145430616028242953648126214400000000000
1140171883222028839723943899169531802599134920298049653412534622754925135589078929801229187068256451591195529152915365848591947843433526045423164822627592173286424242166954585219117278228780823525191138734316794850602541865828329561152000000000000
12472961156371854750616572859756288250260846564768039757091636666525160156408249484061675123313416844789030962581292843338548339215195247524765851213532317853407803456359243075167862882183408945451303575252160079791424881986470145069497117224319321202287679245496111960320000000000000
135544347570315318663849756110200384534166186680335494401126173809541456175525551842737495224150422302273606966522775245518790077531966049782076264386289395363141980198042373902029847678011170517585849781514766148882679638251235690745590868008976535290851397959198798556863037736488300205891132094649253952000000000000000
14647473466259425905798386050125405878445971824399344276840163997696065610373120405422571434924223180735893070839862455520101352545665671965464810790781511016637187208960773727589532091105810271683168677982402922980656336085545201123851685284211403523215565901482828894065449994840477755208757236151759070806588516227028768018530201888518415373952000000000000000
Technical details
Node id
n-cff98ce527c651f9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-83cfe7e8f5f2fb1e

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:           319,            360,             81;
  1:          4598,           8613,           4860,             729;
  2:         62557,         165528,         155034,           58320,            6561;
  3:        819896,        2815065,        3724380,         2325510,          656100,           59049;
  4:      10468315,       44274384,       76006755,        67038840,        31394385,         7085880,          531441;
  5:     131095514,      659503845,     1394643096,      1596141855,      1055861730,       395569251,        74401740,        4782969;
  6:    1617435193,     9438877008,    23742138420,     33471434304,     28730553390,     15204408912,      4746831012,      765275040,       43046721;
  7:   19721017052,   131012250633,   382274518824,    641037737340,    677796544656,    465434964918,    205259520312,    54927615996,     7748409780,     387420489;
  8:  238152716791,  1774891534680,  5895551278485,  11468235564720,  14423349090150,  12200337803808,   6981524473770,  2639050975440,   617935679955,   77484097800,   3486784401;
  9: 2853116706110, 23577118962309, 87857130966660, 194553192190005, 283838830226820, 285582311984970, 201305573762832, 98738703271890, 32658255821070, 6797292479505, 767092568220, 31381059609;
%F Formula
The row polynomials satisfy (9*n+20+9*x)*P(n,x) = (99*n+319+(81*n+360)*x+81*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 319 + 360*x + 81*x^2.
%t Mathematica
P[0, x_] := 319 + 360*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((99*n + 319 + (81*n + 360)*x + 81*x^2)*P[n - 1, x])/(9*n + 20 + 9*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 319 + 360*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((99*n + 319 + (81*n + 360)*x + 81*x^2)*v[i - 1])/(9*n + 20 + 9*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.