Real-rooted atlas

A038322 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 30\,x + 19\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1111930
2121418361900
31331689711913685927000
414641101156262086301796130321810000
51610511390895480491082993907167655247609924300000
61771561183598147928101518258658023653261516342253447045881729000000
7194871712356176131220927631351479166560710037856291767559362253283789387173921870000000
8214358881296204999217906938588618603333041335620832701845585150641593914448287866071303216983563041656100000000
923579476913665536865125325527431610206954995162644529248746456782324783452599176793243893705295084168137274105932268769777919683000000000
10259374246014480100612903482260021845160395007066804848303622701010049211145234814464773618141014276919415308092475500758245354956467556906131066257801590490000000000
112853116706115420921741609468170514048052425974481885358380639119240302026590914289018350047521559012243187681231306703729845197249215214748662871924574185901719392111649025889821917714700000000000
1231384283767216505106089930861798507854342635580959067651801382805454674649538215714383735768770104547429826841140154784506496881230848915092240959448941166364678048962695134798786153767141745649082213314919066161531441000000000000
13345227121439317751918090500878033806021064538508807714667420742197124222427498656831058946092768531573213575463789116271736890307381756435202278971638091173377996466975372385233388846809207546610994350634813909223165040334264610234205298346225705915943230000000000000
1437974983358324191830414302856461031005106036615717123308005343890363383571302538347921116887008633142928183028436132767794048359782115867623986408903525160272044340031040422911828414694588985470602154490544156437666592044733996243708105738374987716476159453187587086799006685782884121478296900000000000000
Technical details
Node id
n-45f0d9f6e2fe1e90 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b53510a6d5d438b6

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,          19;
  2:        121,         418,          361;
  3:       1331,        6897,        11913,          6859;
  4:      14641,      101156,       262086,        301796,        130321;
  5:     161051,     1390895,      4804910,       8299390,       7167655,       2476099;
  6:    1771561,    18359814,     79281015,     182586580,     236532615,     163422534,      47045881;
  7:   19487171,   235617613,   1220927631,    3514791665,    6071003785,    6291767559,    3622532837,     893871739;
  8:  214358881,  2962049992,  17906938588,   61860333304,  133562083270,  184558515064,  159391444828,   78660713032,   16983563041;
  9: 2357947691, 36655368651, 253255274316, 1020695499516, 2644529248746, 4567823247834, 5259917679324, 3893705295084, 1681372741059, 322687697779;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+30*x+19*x^2)*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[Together[((11 + 30*x + 19*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; for(n = 1, N, my(i = n + 1); v[i] = ((11 + 30*x + 19*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

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.