Real-rooted atlas

A027467 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(19 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
119625
236122836625
368596498205221615625
413032116461677976164161296390625
524760993909630246924077976012312077769765625
647045881891395647037334029630880701784088646446656244140625
7893871739197592700218719308449852267603111242405894985662052482799366103515625
816983563041429058434724742224804829950893504118227211202986792704471598848425502721679616152587890625
93226876977799171124042141158457773744853600464864404337062304127685388096268811343363638048256287214336100776963814697265625
10613106625780119361261866740275133721264202316915547488012804006972960485204474764812768538809602304097228802728536192019147622406046617695367431640625
1111649025889821940465037301486663892164160242060529418678124038229106533552016901289204307253372492224128120389080204801900880213760200092654080126374307843627970562384185791015625
12221331491906616183872986406717681456741342853517615334119398458080108952953620623205504991340831488202815470451686454897420573388810835017218432015207041710080144066710937682717728768217678233659604644775390625
13420529834622570591726385636871605583271046469861989523787527491419145762990153282699325601699666076481721927156488743080934422599438136045056535249850590540893903482559744011861492533862410215639502848537665236992130606940161490116119384765625
147990066857828841213532450610829592956725081967486074343691589301156135706567953807731980206096502345751493486700823795325070744106888587786491697121282372941004284730884995665272178380878878925350184969057867025858567150947651993634741446082567836416409637252902984619140625
Technical details
Node id
n-1915549f9358f0cb · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-8952bc02823d42a3

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:           19,            6;
  2:          361,          228,            36;
  3:         6859,         6498,          2052,          216;
  4:       130321,       164616,         77976,        16416,         1296;
  5:      2476099,      3909630,       2469240,       779760,       123120,         7776;
  6:     47045881,     89139564,      70373340,     29630880,      7017840,       886464,       46656;
  7:    893871739,   1975927002,    1871930844,    985226760,    311124240,     58949856,     6205248,     279936;
  8:  16983563041,  42905843472,   47422248048,  29950893504,  11822721120,   2986792704,   471598848,   42550272,   1679616;
  9: 322687697779, 917112404214, 1158457773744, 853600464864, 404337062304, 127685388096, 26881134336, 3638048256, 287214336, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (19+6*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[(19 + 6*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] = (19 + 6*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.