Real-rooted atlas

A038303 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 24\,x + 13\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1111324
2121286169576
3133147195577219713824
414641692121226949666828561331776
51610519516652249390265837015708553712937962624
6177156112561978371149355848414051838215245053384826809191102976
71948717116121205157156999911258196951330514185943455513371664293627485174586471424
821435888120266657848383026652198144266322927131207027674695048163532288925521869496815730721110075314176
923579476912507998907711855994836432693803942857957197898668494870243853965655343627333254005280757341379106044993732641807540224
102593742460130653319983016301992900055137597762440106254862814101506887145363614840555219490100221931352404441653775845116649493103013785849184963403380965376
1128531167061137090517179432191712378784577706166156905183669120007230303888907648326359141436311658303171342341010179146702292415705729433273151668087751372917921603940371521681143169024
12313842837672144508620615316289306033999554113969043696794030305404801192955730476544225576790111159885647680037234378026645911841175649695310520950640186011009379159061142365651720128842329808512248136520347436056576
13345227121439315303943956658493760978441994202162975732486415424815192096189546510243226822803217116140842266235372419075540860096349216907865762358127711101123985386649552478040658191434169144097989212063331626172514783302875106592253876488338465357824
1437974983358324162831336101954424826589000559225922816602548098158874153958281319016117527299230129949263107112136250309187419661898922119682443396855456719194433419146187499088046202040456534037020986827303634462190825653521528363829146642766415206962393737638569928921035720123168587776
Technical details
Node id
n-211798b35e7125e7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-61ed8ff44e45ac0c

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,          13;
  2:        121,         286,          169;
  3:       1331,        4719,         5577,         2197;
  4:      14641,       69212,       122694,        96668,        28561;
  5:     161051,      951665,      2249390,      2658370,      1570855,       371293;
  6:    1771561,    12561978,     37114935,     58484140,     51838215,     24505338,      4826809;
  7:   19487171,   161212051,    571569999,   1125819695,   1330514185,    943455513,    371664293,     62748517;
  8:  214358881,  2026665784,   8383026652,  19814426632,  29271312070,  27674695048,  16353228892,   5521869496,   815730721;
  9: 2357947691, 25079989077, 118559948364, 326938039428, 579571978986, 684948702438, 539656553436, 273332540052, 80757341379, 10604499373;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+24*x+13*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 + 24*x + 13*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 + 24*x + 13*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.