Real-rooted atlas

A038259 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(15 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1151126
2225330121676
3337574255445133117576
4506251485001633507986014641456976
5759375278437540837502994750109807516105111881376
61139062550118750918843758984250049413375144945901771561308915776
7170859375877078125192957187523583656251729468125760965975186013905194871718031810176
825628906251503562500038591437500566007750005188404375030438639000111608343002338460520214358881208827064576
9384433593752537261718757442634375001273517437500140086918125010273040662505022375435001578460851002893844893523579476915429503678976
10576650390625422876953125013954939453125272896593750003502172953125030819121987500188339078812507892304255000217038367012535369215365025937424601141167095653376
118649755859375697746972656252558405566406255628492246093758255121960937508475258546562506215189600812503255575505187501193711018568752917960267612542796750591652853116706113670344486987776
121297463378906251141767773437500460513001953125011256984492187500185740244121093752179352197687500018645568802437500117200718186750005371699583559375175077616056750038517075532485051356100709980313842837672195428956661682176
131946195068359375185537263183593758163639580078125021951119759765625040243719559570312553121709818632812551941227378218750038090233410693750020949628375881562585350337827665625250360990961152505007219819223050611993533460595345227121439312481152873203736576
142919292602539062529971404052734375014286369265136718754190668317773437500845118110750976562512395065624347656250136345721867824218751142707002320812500073323699315585468753584714188761956250131439520254605062535050538734561350064259321013362475724976955022551037974983358324164509974703297150976
Technical details
Node id
n-155058107af1ac83 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c62c9860d860e562

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:           1;
  1:          15,           11;
  2:         225,          330,          121;
  3:        3375,         7425,         5445,          1331;
  4:       50625,       148500,       163350,         79860,         14641;
  5:      759375,      2784375,      4083750,       2994750,       1098075,        161051;
  6:    11390625,     50118750,     91884375,      89842500,      49413375,      14494590,      1771561;
  7:   170859375,    877078125,   1929571875,    2358365625,    1729468125,     760965975,    186013905,     19487171;
  8:  2562890625,  15035625000,  38591437500,   56600775000,   51884043750,   30438639000,  11160834300,   2338460520,   214358881;
  9: 38443359375, 253726171875, 744263437500, 1273517437500, 1400869181250, 1027304066250, 502237543500, 157846085100, 28938448935, 2357947691;
%F Formula
The row polynomials satisfy P(n,x) = (15+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(15 + 11*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (15 + 11*x)*v[i - 1]); 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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.