Real-rooted atlas

A013609 → basis A008288 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-3 - 7\,x - 5\,x^{2} - x^{3}\right)\,P_{n-2}(x) + \left(4 + 10\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1358
29342972
327163297169656
481676193423089855984
52432593102221897816815574154592
672994784785912190816700711760633461498048
7218733535207023674371126289713690057997011950254543744
865611159128470843372984806130211910648106750205326856113668941453056
919683393661332641215689700457899148617743010502594880204660349280916625109378180608
105904913187621265615369094072238774930548099068849439418880383032585547813226195370386139653450181632
11177147436962746963621291544797116620919831676642065980774602787127343470938605834178276007145019955322505868131476371456
12531441143489081707696901188735324540874834316999421016379495941886078273551269475467951553751071322926428306692207808121311738121287162261504
13159432346766809610625690471270093424055634913859907999152220157130204201239242605838508347895895716924474211502117862018145713825822120307976453700452619811643392
14478296915146068621528674711825092423610334678924141447501739412167524365992661443970856436663994462753565818498104843262256149313406497441213736896199313654409653584456048214923900818341888
Technical details
Node id
n-99c2c2e86ca25e12 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-82e9b32c2a573e98

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     3,      5;
  2:     9,     34,      29;
  3:    27,    163,     297,      169;
  4:    81,    676,    1934,     2308,      985;
  5:   243,   2593,   10222,    18978,    16815,     5741;
  6:   729,   9478,   47859,   121908,   167007,   117606,     33461;
  7:  2187,  33535,  207023,   674371,  1262897,  1369005,    799701,   195025;
  8:  6561, 115912,  847084,  3372984,  8061302, 11910648,  10675020,  5326856,  1136689;
  9: 19683, 393661, 3326412, 15689700, 45789914, 86177430, 105025948, 80204660, 34928091, 6625109;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(3+7*x+5*x^2+x^3)*P(n-2,x) + (4+10*x+6*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 5*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(3 + 7*x + 5*x^2 + x^3)*P[n - 2, x] + (4 + 10*x + 6*x^2)*P[n - 1, x])/(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; v[2] = 3 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(3 + 7*x + 5*x^2 + x^3)*v[i - 2] + (4 + 10*x + 6*x^2)*v[i - 1])/(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

Proof chains ? JSON

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.