Real-rooted atlas

A013622 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(8 + 21\,x + 13\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181321
264208169441
35122496405621979261
4409626624648967030428561194481
532768266240865280140608011424403712934084101
6262144255590410383360224972802741856017822064482680985766121
7209715223855104116293632314961920511813120499017792270301304627485171801088541
816777216218103808124046540840315125768189009920106457128968649641728401590508881573072137822859361
9134217728196293427212759072768483781509121179217428481916228321282075914014721445725831685873261191210604499373794280046581
101073741824174483046401275907276805528931532801572289904640306596531404841518280294403855268884480234930447648084835994984013785849184916679880978201
11858993459215354508083212475537817606081824686080197659302297604496749127270473072173318144848159154585606891293131008037327837792960121315472827121792160394037350277500542221
1268719476736134002979635211976516304896648727966515202371911627571206166970231685121169154773090304162846557680435216539103514419201194490809374720582314269570176172047397827552232980851224817355827511386641
135497558138881161359156838411323251779174467467708517580827408756585267208017061301190656173702994859130882822673666460876834401335309991936310567610437427202018689467843276889464646870327042423000852738024302875106592253154472377739119461
1443980465111041000555581276161056836832722944686943941269913630697807375499264997678739703726082431841928027832324516277866337402886421582591198494726956714473798369285652330509961175043340013483158876161356880477533293443392201193833233639373763856992893243919932521508681
Technical details
Node id
n-60e14c585778a1ea · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0e62401a26557dd3

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:         8,         13;
  2:        64,        208,         169;
  3:       512,       2496,        4056,        2197;
  4:      4096,      26624,       64896,       70304,        28561;
  5:     32768,     266240,      865280,     1406080,      1142440,       371293;
  6:    262144,    2555904,    10383360,    22497280,     27418560,     17822064,      4826809;
  7:   2097152,   23855104,   116293632,   314961920,    511813120,    499017792,    270301304,     62748517;
  8:  16777216,  218103808,  1240465408,  4031512576,   8189009920,  10645712896,   8649641728,   4015905088,   815730721;
  9: 134217728, 1962934272, 12759072768, 48378150912, 117921742848, 191622832128, 207591401472, 144572583168, 58732611912, 10604499373;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (8+21*x+13*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[((8 + 21*x + 13*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] = ((8 + 21*x + 13*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.