Real-rooted atlas

A038291 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(9 + 19\,x + 10\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
191019
281180100361
37292430270010006859
4656129160486003600010000130321
5590493280507290008100004500001000002476099
65314413542940984150014580000121500005400000100000047045881
74782969372008701240029002296350002551500001701000006300000010000000893871739
8430467213826375201488034800330674400045927000004082400000226800000072000000010000000016983563041
9387420489387420489017218688400446410440007440174000082668600000612360000002916000000081000000001000000000322687697779
1034867844013874204890019371024450057395628000011160261000001488034800000137781000000087480000000036450000000090000000000100000000006131066257801
11313810596093835462841102130812689500710270896500015783797700000245525742000002728063800000021651300000000120285000000004455000000000990000000000100000000000116490258898219
1228242953648137657271530802301277704660085232507580000213081268950000378811144800000491051484000000467668080000000324769500000000160380000000000534600000000001080000000000010000000000002213314919066161
1325418658283293671583974253024477226495020099722033868600027700564963500005540112992700000820757480400000091195275600000007599606300000000469111500000000020849400000000006318000000000001170000000000001000000000000042052983462257059
14228767924549613558612159660602570108781977100114227056976760003490271185401000077561581897800000129269303163000000164151496080000000159591732300000000118216098000000000656756100000000002653560000000000073710000000000001260000000000000100000000000000799006685782884121
Technical details
Node id
n-45d245d96cab6cbb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-249d425e0def4c4f

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:         9,         10;
  2:        81,        180,         100;
  3:       729,       2430,        2700,        1000;
  4:      6561,      29160,       48600,       36000,       10000;
  5:     59049,     328050,      729000,      810000,      450000,      100000;
  6:    531441,    3542940,     9841500,    14580000,    12150000,     5400000,     1000000;
  7:   4782969,   37200870,   124002900,   229635000,   255150000,   170100000,    63000000,    10000000;
  8:  43046721,  382637520,  1488034800,  3306744000,  4592700000,  4082400000,  2268000000,   720000000,  100000000;
  9: 387420489, 3874204890, 17218688400, 44641044000, 74401740000, 82668600000, 61236000000, 29160000000, 8100000000, 1000000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (9+19*x+10*x^2)*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[Together[((9 + 19*x + 10*x^2)*P[n - 1, x])/(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 + 19*x + 10*x^2)*v[i - 1])/(1 + x)); 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.