Real-rooted atlas

A038335 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(39 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1391857
2152114043243249
35931982134379085832185193
423134414270968295682490979210497610556001
59022419920820969019219356088704720204703201889568601692057
63518743761974421349211243323260691896816023950274404421589123401222434296447249
71372310066794433617138866138854499964722195769202179474970406035469148892853371526122200321954897493193
85352009260481197612649617763192204339979229466501599808169999047691206276887914752144851259571219101264998411019960576111429157112001
92087283611587598670255001979221600662461903856172379034358876811933933147922245507969145194881694759736983043352272007219238680061621761983592903686351461955384057
1081404060851916013757110500857662078032295017812980960397477142313607757056546149456042962159332520064165239074355846404357953609384960754261201624320773601232435203570467226624362033331456891249
113174758373224724391611800404867936998371953939584908538051501314711756566804753967511854452320307179439227518457614177512579731621124673905246065369601078593518322777601659374643573504015317304402216966426841007923220635899893042801193
121238155765557642512168574780861654046824174074443725737195784267806836501134147360278107099443485460720205371396512112340224110584598121906644736437477531031718594561261954416437649792025886244439746662403584249230118768643007761591708057611568313814261761176246293903439668001
13482880748567480579719289728449140488347831480232493608135234784081357780661060750127115215666699935316347620560130154122539551195616968009484463972381268736369668513721802212403212796217782677768890883281081482737889459206057381198900719001676246756349799260165865135103830712322082296486567116867046038752496061076057
14188323491941317426090411216859486390051060891883650578459170153182675646739529463083359721862728554018164682725800825767896016767399439200762245466473146661150215912322883414407030057256744961164455818223676969070083582940979149775289446482683253364994814371841387690965566346534912160118188334578446336113693388166564577283748133675820810243821624208892275481335249
Technical details
Node id
n-ad5d029cdf51f0fe · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a285c8a8f5f97b30

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:              39,              18;
  2:            1521,            1404,              324;
  3:           59319,           82134,            37908,             5832;
  4:         2313441,         4270968,          2956824,           909792,           104976;
  5:        90224199,       208209690,        192193560,         88704720,         20470320,         1889568;
  6:      3518743761,      9744213492,      11243323260,       6918968160,       2395027440,       442158912,        34012224;
  7:    137231006679,    443361713886,     613885449996,     472219576920,     217947497040,     60354691488,      9285337152,      612220032;
  8:   5352009260481,  19761264961776,   31922043399792,   29466501599808,   16999904769120,   6276887914752,   1448512595712,   191012649984,   11019960576;
  9: 208728361158759, 867025500197922, 1600662461903856, 1723790343588768, 1193393314792224, 550796914519488, 169475973698304, 33522720072192, 3868006162176, 198359290368;
%F Formula
The row polynomials satisfy P(n,x) = (39+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[(39 + 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] = (39 + 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.