Real-rooted atlas

A013613 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(7 + 38\,x + 31\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
173138
2494349611444
33434557201812979154872
42401425322825348341489235212085136
516807372155329623014597590323232352862915179235168
611764931261023461041520436626067878793512024243428875036813010936384
782354325529833339182067250348668511086869605294593963794348768036927512614111114415582592
85764801204238664316569929228039050872155216174470549908732408121765505033215407063902168528910374414347792138496
9403536071608379479284912936282944100341561955723798322866106253542625570756056972485322512918045373213535878326439622160671165216101262848
10282475249125096181702492988192452944100341560228167776470901212548754959644474882309970101132419196808760188062473755740518507735512469708196282869808016278211847988224
11197732674396324059909213288989798528336965787515250984554117990155610423553153868913187573539542179906953856863048269368263973395712547817230083456311137809752167725408476896404831238572050223552512
1213841287201735565548396179162751430742644783473501402635337818238895186732508263784569647846260295535636622436824795298410136567335434412951995133888244233660265067788009591043421343120592980058047876627837885497619065737908494995456
1396889010407557803874200314821645800179424067529608862742664619349552660521240822815005493712542200138384196285554402918427286924184485525504906315694538929595755631576580403895696242616498971111986980592640827167731332475802825124417546297445042591344498040522809827328
146782230728494204983051663812104344070146512144198092425953226113269625616072923128895954117093141536419516952064044377761640857982016936301326358324680315627889630200768103788994196989544455794410420131722991574699359600123512188352913143384299239291953714961417391875694393522079632032113090925539866773438464
Technical details
Node id
n-7491f4800c3c086f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7a9fc1d0c935b2b6

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:        7,         31;
  2:       49,        434,         961;
  3:      343,       4557,       20181,        29791;
  4:     2401,      42532,      282534,       834148,        923521;
  5:    16807,     372155,     3296230,     14597590,      32323235,      28629151;
  6:   117649,    3126102,    34610415,    204366260,     678787935,    1202424342,      887503681;
  7:   823543,   25529833,   339182067,   2503486685,   11086869605,   29459396379,    43487680369,    27512614111;
  8:  5764801,  204238664,  3165699292,  28039050872,  155216174470,  549908732408,  1217655050332,  1540706390216,   852891037441;
  9: 40353607, 1608379479, 28491293628, 294410034156, 1955723798322, 8661062535426, 25570756056972, 48532251291804, 53732135358783, 26439622160671;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (7+38*x+31*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[((7 + 38*x + 31*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] = ((7 + 38*x + 31*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

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.