Real-rooted atlas

A038277 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(18 + 58\,x + 40\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1184058
2324144016003364
35832388808640064000195112
410497693312031104004608000256000011316496
518895682099520093312000207360000230400000102400000656356768
634012224453496320251942400074649600001244160000011059200000409600000038068692544
76122200329523422720634894848002351462400005225472000006967296000005160960000001638400000002207984167552
81101996057619591041024015237476352006772211712000188116992000003344302080000037158912000000235929600000006553600000000128063081718016
919835929036839671858073603526387384320018284971622400060949905408000013544423424000002006581248000000191102976000000010616832000000002621440000000007427658739644928
103570467226624793437161472007934371614720004701849845760000182849716224000004875992432640000090296156160000000114661785600000000955514880000000004718592000000000010485760000000000430804206899405824
116426841007923215710055797145601745561755238400011637078368256000051720348303360000016090775027712000003575727783936000000567575838720000000063063982080000000004671406080000000000207618048000000000041943040000000000024986644000165537792
121156831381426176308488368380313603770413391314944002792898808381440000139644940419072000004965153437122560000012872620022169600000024519276232704000000034054550323200000000033634123776000000000022422749184000000000090596966400000000000167772160000000000001449225352009601191936
132082296486567116860155231834161152080206975778881536006535383211612569600036307684508958720000014523073803583488000004303132978839552000000956251773075456000000015937529551257600000000196759624089600000000001748974436352000000000010599845068800000000000392586854400000000000067108864000000000000084055070416556869132288
143748133675820810241166086032477585408016843464913565122560014971968812057886720009149536496257597440000406646066500337664000001355486888334458880000003442506383071641600000006693762411528192000000009916685054115840000000001101853894901760000000000890386985779200000000000494659436544000000000000169114337280000000000000268435456000000000000004875194084160298409672704
Technical details
Node id
n-54d12f8ab10eab7d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9eb007f6301418ee

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:           18,            40;
  2:          324,          1440,           1600;
  3:         5832,         38880,          86400,           64000;
  4:       104976,        933120,        3110400,         4608000,         2560000;
  5:      1889568,      20995200,       93312000,       207360000,       230400000,        102400000;
  6:     34012224,     453496320,     2519424000,      7464960000,     12441600000,      11059200000,       4096000000;
  7:    612220032,    9523422720,    63489484800,    235146240000,    522547200000,     696729600000,     516096000000,     163840000000;
  8:  11019960576,  195910410240,  1523747635200,   6772211712000,  18811699200000,   33443020800000,   37158912000000,   23592960000000,    6553600000000;
  9: 198359290368, 3967185807360, 35263873843200, 182849716224000, 609499054080000, 1354442342400000, 2006581248000000, 1911029760000000, 1061683200000000, 262144000000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (18+58*x+40*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[((18 + 58*x + 40*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] = ((18 + 58*x + 40*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.