Real-rooted atlas

A013622 → x ↦ x/(x+1) → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(11 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112435
21215285761225
313318712190081382442875
4146411277764181766082563317761500625
5161051175692076665601672704018247680796262452521875
61771561231913441264982403679948806021734405255331841911029761838265625
71948717129762224819480728967083901440154557849602023302758414714929152458647142464339296875
82143588813741536832285717358081246766653443400272691205935021424646474568826884036094853121100753141762251875390625
92357947691463015182964040859778562057164978176673253992857614689178025984213660771287041997866952294410897456103424264180754022478815638671875
1025937424601565907445840555618219552032326878228480123429898690560323161916571648587567121039360732551215841280599360085688320290598829424640634033809653762758547353515625
1128531167061168474800946647469978285088048894403320576021335739630796806517098650861568142191243291525122215967427919872024174190122762240175812291801907207671809096810496152168114316902496549157373046875
12313842837672182169761135968986037133631616717117915368448035203970390814720122893860273389568312820735241355264585015400970846208797748274051153920773574083928391680506339400389492736200861910898311168365203474360565763379220508056640625
1334522712143931979189653536952128184827372110081025478618976880645593519739873894402196727752386838528639048073421625753613942867056471834624228156006378630021122765527350044000256024135511418565820416143616266292292485125222409683356090368876488338465357824118272717781982421875
14379749833583241115996312803608161645038617942079361435670066567632896861402039940579737637588452651952570368123016754133662957568306743075242380361728585600416371817054208851782423813552078848929217189614784086016737230166967101423616402125545618418958336134979204123665104896210357201231685877764139545122369384765625
Technical details
Node id
n-8331247311aa3063 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-83fc6524b6311133

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:         11,          24;
  2:        121,         528,          576;
  3:       1331,        8712,        19008,         13824;
  4:      14641,      127776,       418176,        608256,        331776;
  5:     161051,     1756920,      7666560,      16727040,      18247680,        7962624;
  6:    1771561,    23191344,    126498240,     367994880,     602173440,      525533184,      191102976;
  7:   19487171,   297622248,   1948072896,    7083901440,   15455784960,    20233027584,    14714929152,     4586471424;
  8:  214358881,  3741536832,  28571735808,  124676665344,  340027269120,   593502142464,   647456882688,   403609485312,   110075314176;
  9: 2357947691, 46301518296, 404085977856, 2057164978176, 6732539928576, 14689178025984, 21366077128704, 19978669522944, 10897456103424, 2641807540224;
%F Formula
The row polynomials satisfy P(n,x) = (11+24*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[(11 + 24*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] = (11 + 24*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.