Real-rooted atlas

A038335 → x ↦ x+1 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(30 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
130939
2900540811521
32700024300729072959319
48100009720004374008748065612313441
524300000364500002187000065610009841505904990224199
6729000000131220000098415000039366000088573500106288205314413518743761
721870000000459270000004133430000020667150000620014500011160261001116026104782969137231006679
86561000000001574640000000165337200000099202320000037200870000089282088000133923132001147912560430467215352009260481
919683000000000531441000000006377292000000044641044000000200884698000006026540940000120530818800015496819560011622614670387420489208728361158759
1059049000000000017714700000000002391484500000000191318760000000010044234900000003615924564000009039811410000015496819560000174339220050011622614670034867844018140406085191601
111771470000000000058458510000000000876877650000000007891898850000000047351393100000000198875851020000005966275530600000127848761370000019177314205500019177314205500115063885233031381059609317475837322472439
1253144100000000000019131876000000000003156759540000000000315675954000000000021308126895000000001022790090960000000357976531836000000920511081864000001725958278495000023012777046600002071149934194001129718145924028242953648112381557655576425121
131594323000000000000062178597000000000000111921474600000000000123113622060000000000923352165450000000004986101693430000000019944406773720000000598332203211600000013462474572261000002243745762043500002692494914452200022029503845518001101475192275902541865828329482880748567480579719
14478296900000000000000200884698000000000000039172516110000000000004700701933200000000000387807909489000000000023268474569340000000001047081355620300000000358999321926960000000942373220058270000001884746440116540000028271196601748100003084130538372520002313097903779390010675836478981802287679245496118832349194131742609041
Technical details
Node id
n-c3a62990f6820c27 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1024c0779b1dc97d

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:             30,              9;
  2:            900,            540,             81;
  3:          27000,          24300,           7290,            729;
  4:         810000,         972000,         437400,          87480,           6561;
  5:       24300000,       36450000,       21870000,        6561000,         984150,         59049;
  6:      729000000,     1312200000,      984150000,      393660000,       88573500,      10628820,        531441;
  7:    21870000000,    45927000000,    41334300000,    20667150000,     6200145000,    1116026100,     111602610,      4782969;
  8:   656100000000,  1574640000000,  1653372000000,   992023200000,   372008700000,   89282088000,   13392313200,   1147912560,    43046721;
  9: 19683000000000, 53144100000000, 63772920000000, 44641044000000, 20088469800000, 6026540940000, 1205308188000, 154968195600, 11622614670, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (30+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[(30 + 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] = (30 + 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.