Real-rooted atlas

A038221 → x ↦ x+1 → basis A013610

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19918
28116281324
3729218721877295832
465612624439366262446561104976
559049295245590490590490295245590491889568
653144131886467971615106288207971615318864653144134012224
7478296933480783100442349167403915167403915100442349334807834782969612220032
843046721344373768120530818824106163763013270470241061637612053081883443737684304672111019960576
938742048934867844011394713760432543321076488149816144881498161432543321076139471376043486784401387420489198359290368
103486784401348678440101569052980454184141281207322247242108786696690527322247242104184141281201569052980453486784401034867844013570467226624
11313810596093451916556991725958278495517787483548510355749670970144980495393581449804953935810355749670970517787483548517259582784953451916556993138105960964268410079232
1228242953648133891544377721864034940774662134498025820139802620558095223684192892952260964891708444223684192892952139802620558095621344980258201864034940774633891544377722824295364811156831381426176
1325418658283293304425576827719826553460966272697362690209418174340672552353271381321059423436184176141256443618417614125643271381321059423181743406725523572697362690209419826553460966233044255768277254186582832920822964865671168
142287679245496132027509436945420817881134014518327152453605804228996692474159614579933849483192268699007742247883785131517054261526869900774224788345799338494831922228996692474159618327152453605804208178811340145132027509436945422876792454961374813367582081024
Technical details
Node id
n-b07056c1b369241f · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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:         9,          9;
  2:        81,        162,          81;
  3:       729,       2187,        2187,         729;
  4:      6561,      26244,       39366,       26244,        6561;
  5:     59049,     295245,      590490,      590490,      295245,       59049;
  6:    531441,    3188646,     7971615,    10628820,     7971615,     3188646,      531441;
  7:   4782969,   33480783,   100442349,   167403915,   167403915,   100442349,    33480783,     4782969;
  8:  43046721,  344373768,  1205308188,  2410616376,  3013270470,  2410616376,  1205308188,   344373768,   43046721;
  9: 387420489, 3486784401, 13947137604, 32543321076, 48814981614, 48814981614, 32543321076, 13947137604, 3486784401, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (9+9*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[(9 + 9*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] = (9 + 9*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.