Real-rooted atlas

A038277 → x ↦ x/(x+1) → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 7\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
118725
232425249625
358326804264634315625
410497616329695256246962401390625
51889568367416028576801111320216090168079765625
634012224793618567715736040007520116688601815156117649244140625
7612220032166659897619443654721260236880490092120114354828148237748235436103515625
81101996057634284321792466647713283629482214417643316320548903174410673117281185901925764801152587890625
91983592903686942575162881079956136448979960197888571643448768222305785632576348333129605805552933897762403536073814697265625
103570467226624138851503257602429901307008025198976517120171493034630408003008282752259356749904057634833312084050798580726364926028247524995367431640625
11642684100792322749259764500485345782875417606236746687987204850802979545602640992733308161027052729619842852924248944055473527062807191012767405593009930219773267432384185791015625
12115683138142617653985464466554881154689101090201614968192051169280130971680447731208149349005636608369738982663142412324632755438082995570461391205177529192528060404507246164271025764881384128720159604644775390625
132082296486567116810527165570978201624563386332282470435025569399736115234052636916410112023836845841487078412359845991882188848066067746208512140192697593108163028854577628880471155156520048499710014490963238861205034968890104071490116119384765625
1437481336758208102420406505568357744645158311129779318784802403953521227366485812645029353482246674316835616381952389335148744288947217303784388635064325888093298910542721526542707124955522968277486076302441975641217240644080965118342842441603062256467822307284937252902984619140625
Technical details
Node id
n-ef2b02719267ded3 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c53984b4362c13dd

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,            7;
  2:          324,          252,            49;
  3:         5832,         6804,          2646,          343;
  4:       104976,       163296,         95256,        24696,         2401;
  5:      1889568,      3674160,       2857680,      1111320,       216090,        16807;
  6:     34012224,     79361856,      77157360,     40007520,     11668860,      1815156,      117649;
  7:    612220032,   1666598976,    1944365472,   1260236880,    490092120,    114354828,    14823774,     823543;
  8:  11019960576,  34284321792,   46664771328,  36294822144,  17643316320,   5489031744,  1067311728,  118590192,   5764801;
  9: 198359290368, 694257516288, 1079956136448, 979960197888, 571643448768, 222305785632, 57634833312, 9605805552, 933897762, 40353607;
%F Formula
The row polynomials satisfy P(n,x) = (18+7*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[(18 + 7*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] = (18 + 7*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.