Real-rooted atlas

A038329 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(27 + 12\,x\right)\,P_{n-1}(x) + \left(12\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 729 + 792\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
07297921441665
1196833963616848172877895
253144117820001334880311040207363980097
3143489077587529289462880356140805287680248832220837671
438742048931313232725473110384331724160082394496085100544298598413221127233
51046035320312677065347631670450884827466354540810160277120017291086848131681894435831808848845569735
6282429536481506957972400017673177792960211042574668801091617038489627235578101763378581176321977915801642998169658127239972737
776255974849872011027636737729621670241289601541646353928960107196386846976036794357410406466130647367680624978395136029023764480051597803524225125010534695
82058911320946497934514953096952514837513946622241086697245870720009889704638237952044874657368690688109622569786490881487293787013120110696637726720417942208512061917364224324630074560834305
95559060566555523311916776755916916272088550056493684874639976657214826888721315173723328000509085974766464409616237593293105479682966366941022453763146321188454400018925816342118405925491756236874300837068826268345739854346311
10150094635296999121122314630202794260001425081618498827822405028824789472326515207436666100179073922565475605801315039723522218424905737199779845241178499707581235274296153910258565126330570632331264003142145249221017682919734168780889161004482562231348596193092342977
114052555153018976267478828193414461603452741469381058737500624033391694115874250432640618095854987098462374405656176979165573430169628533133528376325488640847981802156005490688015427841130610900992001746180882287823421441221563827515241267250889980723488358411475021276905472106993205379072198403713836717808975399
1210941898913151235920918722930205000230249832383895322660929664559664219265305502712627733120050364192417937294729996805662588997858567039096832350352145680912503499571212836574027336820005273602915202621652347410841604208685126761869737984038911158851927927685122275027286521754419208069026325169438720157280011907235840128391846454886418419301785714245180017345
1329543127065508336986437315073058645871280539561980333661816741565554084814274388633864011144658156840404327979485522580295040055308227371522008607742054441479969219840618293373747218452822918514820247848550451260881079128057159680000917996806711134103928832010770362503347649055293448286632447800940848742441098362454300793241601256105580810574233602137724243473858560154070215745863681781340148789995311054646471
14797664430768725098633612856423668119750799848800010187482551969132439692307209230504878752336009001128960320161726206778232725802588165296667717076241019837099212847702156812224853143369056256254602297459323246703167406088519427122954529460382466048185442626200956772622008320026848394560454653358427340826100180596020674815852544169717789170465646628044872304767073994460364800192418292445009149952028841944387625680896184884258895036416179093342433704131060657296129
Technical details
Node id
n-47b7c605b3a36a4e · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-65e248b1aaac025a

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:              729,                792,                 144;
  1:            19683,              39636,               16848,                1728;
  2:           531441,            1782000,             1334880,              311040,               20736;
  3:         14348907,           75875292,            89462880,            35614080,             5287680,              248832;
  4:        387420489,         3131323272,          5473110384,          3317241600,           823944960,            85100544,             2985984;
  5:      10460353203,       126770653476,        316704508848,        274663545408,        101602771200,         17291086848,          1316818944,           35831808;
  6:     282429536481,      5069579724000,      17673177792960,      21104257466880,      10916170384896,       2723557810176,        337858117632,        19779158016,         429981696;
  7:    7625597484987,    201102763673772,     962167024128960,    1541646353928960,    1071963868469760,     367943574104064,      66130647367680,      6249783951360,      290237644800,       5159780352;
  8:  205891132094649,   7934514953096952,   51483751394662224,  108669724587072000,   98897046382379520,   44874657368690688,   10962256978649088,   1487293787013120,   110696637726720,    4179422085120,    61917364224;
  9: 5559060566555523, 311916776755916916, 2720885500564936848, 7463997665721482688, 8721315173723328000, 5090859747664644096, 1623759329310547968, 296636694102245376, 31463211884544000, 1892581634211840, 59254917562368, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = (27+12*x)*P(n-1,x) + 12*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 729 + 792*x + 144*x^2.
%t Mathematica
P[0, x_] := 729 + 792*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 12*x)*P[n - 1, x] + 12*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 729 + 792*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (27 + 12*x)*v[i - 1] + 12*x*deriv(v[i - 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.