Real-rooted atlas

A038323 → x ↦ x+1 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(29 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1291847
284110443242209
32438945414281885832103823
4707281175600816349046765121049764879681
520511149636552907902036049047120152215201889568229345007
659482332122152040923437385660284473296013242722403287848323401222410779215329
7172498763097494773844613955785779614437019772089609088240333716604486904481472612220032506623120463
8500246412961248398218849653962371681126698777174208519732711792025807417413128009198507521420350474241101996057623811286661761
91450714597586981039918899682201202557268176291396807078048271300475555424168393398620608696800270154241853557368883228762097103361983592903681119130473102767
1042070723330020126112862756564207293592700971380120721534360905601311285631851216097668171199952645051801958618240179177212325376041705040799872057524194206720357046722662452599132235830049
1112200509765705829833000321934397982585173412899855804813771182641110805975715950864827205192691102130815363223049649598437121428938268294873604434636005053056091751089759718401138979045293056642684100792322472159215084012303
123538147832054690412635310109392459064899640347689149818418613248572878961760259943643862619983202581509290773605350418693687967670935296994541034733232025638581333243961587201064312641212733440198182353880991744223654067075727361156831381426176116191483108948578241
131026062871295860218982792659270079755594308331282798917710488701719471197536858352108887504151341926296012165362532770615213761006788623401706086656624903283490714122752290903252659470367488100311466434300126720249049158043779624964215879164377460736436125430797668352208229648656711685460999706120583177327
1429755823267579946348125856784356655677516281043187506803004920484425899827755109087680992442083266854448220761765487930209227633085318451094522637636583897792362443904424614191196161968445342995749486668881452910744651702896642527848954144363193344570548980245749686272885334624519266754568454123735462494208374813367582081024256666986187667409334369
Technical details
Node id
n-996ee77147adad39 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-dcbe8377329c0f7d

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:             29,             18;
  2:            841,           1044,             324;
  3:          24389,          45414,           28188,            5832;
  4:         707281,        1756008,         1634904,          676512,          104976;
  5:       20511149,       63655290,        79020360,        49047120,        15221520,         1889568;
  6:      594823321,     2215204092,      3437385660,      2844732960,      1324272240,       328784832,       34012224;
  7:    17249876309,    74947738446,    139557857796,    144370197720,     89609088240,     33371660448,     6904481472,      612220032;
  8:   500246412961,  2483982188496,   5396237168112,   6698777174208,   5197327117920,   2580741741312,   800919850752,   142035047424,   11019960576;
  9: 14507145975869, 81039918899682, 201202557268176, 291396807078048, 271300475555424, 168393398620608, 69680027015424, 18535573688832, 2876209710336, 198359290368;
%F Formula
The row polynomials satisfy P(n,x) = (29+18*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[(29 + 18*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] = (29 + 18*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.