Real-rooted atlas

A038323 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(38 + 54\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1385492
21444410429168464
354872233928332424157464778688
42085136118523522526422423934528850305671639296
5792351685629867201600067520227378016016155806404591650246590815232
63010936384256721944329120384864017280729216018417619296010468962547224794911296606355001344
7114415582592113813395315248520447476481149168492864016330289109120139237201877766595446404736133892520998455784660123648
8434779213849649427531679744245836933880832698694443661312124110197229312014109369790279681002507853519872407033263835136723019613391365132188731375616
9165216101262848211302697930905612010890198177792398255832886947848489137490484940812063511170689126411428589530126540869602688115808256247272707779845123904305912313344472161363286556672
106278211847988224892166946819379205705172844134451202161960235672002560537645374397379584091682684897237360641085716005362021376088163404946690457604698181447817057280148363624667907072021083251926492057643438845422363213824
1123857205022355251237292578377050050562649735832053556224011296242231386213376032105109499729238016063872270478408694579290765858048264987033692130758169291528192065461328172917664768031007997555592578048088127993052736800768113849560403057111043996373778857415671808
12906573790849499545615459468854486202777612082795394164216381445723429397235681443840182999124148456656691204160822191164909246873668982052116681390145536840232514503938737111047462591411712613783552047132156284500718632960200931824160239905751045191539954379404263424614787626176508399616367666387654882241806336
13344498040522809827328636414801176348681011254262735679246571749376282737412223442663325696100446185921486209339392025693077030443314599444484868161963662943818842112691791436941576226888089673730403147720624181493765820821301135838751170560330867737117195044803379212823103687317128530657283037050873311951494103043319853181353145357926433825307664249166246182912
141309092553986677343846426044051863524422945996824056479484465980142223361367420939117013608084275253437370910230663368556544151874633113287148521160704323732770583585763953000448525761492075597932434948096653742907909789534409244672619335386440853243124547584440054090365869409588494336227396372055090412610322432807855532300979097431408641766161892479873330416844817927207179306984932802563111928305110923294648827904
Technical details
Node id
n-f891f11e3691237f · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-5254523500bc8159

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:              38,               54;
  2:            1444,             4104,              2916;
  3:           54872,           233928,            332424,            157464;
  4:         2085136,         11852352,          25264224,          23934528,           8503056;
  5:        79235168,        562986720,        1600067520,        2273780160,        1615580640,          459165024;
  6:      3010936384,      25672194432,       91203848640,      172807292160,      184176192960,       104689625472,        24794911296;
  7:    114415582592,    1138133953152,     4852044747648,    11491684928640,    16330289109120,     13923720187776,      6595446404736,     1338925209984;
  8:   4347792138496,   49427531679744,   245836933880832,   698694443661312,  1241101972293120,   1410936979027968,   1002507853519872,   407033263835136,    72301961339136;
  9: 165216101262848, 2113026979309056, 12010890198177792, 39825583288694784, 84891374904849408, 120635111706891264, 114285895301265408, 69602688115808256, 24727270777984512, 3904305912313344;
%F Formula
The row polynomials satisfy P(n,x) = (38+54*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[(38 + 54*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] = (38 + 54*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.