Real-rooted atlas

A013622 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(8 + 26\,x + 18\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181826
264288324676
351234567776583217576
4409636864124416186624104976456976
532768368640165888037324804199040188956811881376
6262144353894419906560597196801007769609069926434012224308915776
72097152330301442229534728360755201881169920253957939219046845446122200328031810176
8167772163019898882378170368107017666563009871872054177693696609499054083918208204811019960576208827064576
9134217728271790899224461180928128421199872433421549568975198486528146279772979214105549537287934371614721983592903685429503678976
101073741824241591910402446118092801467670855680577895399424015603175784448292559545958403761479876608031737486458880158687432294403570467226624141167095653376
11858993459221260088115223917599129601614437941248072649707356160228846578171904514904800886784827525572853760930966269460480698224702095360314201115942912642684100792323670344486987776
126871947673618554258718722296089516441617220671373312087179648827392031384673577861128238476814188544158884909987921922234319046705152022343190467051520150816535652597766169767367606272115683138142617695428956661682176
135497558138881608035755622421708482700902417909498228244481007409275338752040800075651219456122400226953658368275400510645731328464738361714671616580922952143339520522830656929005568320827903115526144120310463668322304208229648656711682481152873203736576
1443980465111041385384650997762026125052084224182351254687580161128298388379402245077342747707310081713603177351217152440640817033170124886751160853405368321301267412801080524814639258394012155904119775750496463093766737385965426049024233217206495517081637481336758208102464509974703297150976
Technical details
Node id
n-1e0ba595f47d72f4 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-12e1fe8c90bb4450

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,         18;
  2:        64,        288,         324;
  3:       512,       3456,        7776,         5832;
  4:      4096,      36864,      124416,       186624,       104976;
  5:     32768,     368640,     1658880,      3732480,      4199040,      1889568;
  6:    262144,    3538944,    19906560,     59719680,    100776960,     90699264,      34012224;
  7:   2097152,   33030144,   222953472,    836075520,   1881169920,   2539579392,    1904684544,     612220032;
  8:  16777216,  301989888,  2378170368,  10701766656,  30098718720,  54177693696,   60949905408,   39182082048,  11019960576;
  9: 134217728, 2717908992, 24461180928, 128421199872, 433421549568, 975198486528, 1462797729792, 1410554953728, 793437161472, 198359290368;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (8+26*x+18*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 + 26*x + 18*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 + 26*x + 18*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

Candidate equalities ?

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.