Real-rooted atlas

A176230 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 21\,x + 40\,x^{2} + 30\,x^{3} + 8\,x^{4} - 2\,n - 22\,n x - 50\,n x^{2} - 42\,n x^{3} - 12\,n x^{4} + 4\,n^{2} x + 12\,n^{2} x^{2} + 12\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(2 + 28\,x + 88\,x^{2} + 116\,x^{3} + 70\,x^{4} + 16\,x^{5} - 8\,n x - 32\,n x^{2} - 48\,n x^{3} - 32\,n x^{4} - 8\,n x^{5}\right)\,P_{n-1}'(x) + \left(4\,x + 20\,x^{2} + 40\,x^{3} + 40\,x^{4} + 20\,x^{5} + 4\,x^{6}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
23121025
3159015076331
4105840210021287645937
594594503150047880343809496133651
610395124740519750105336011345406267361401523609673
713513518918909459450239639403441438028516488127538322390480113525595
820270253243240018918900057513456010324314001140659520765229920286857600462067364070168161
934459425620269650413513100014665931280315924008404363022664039026725920219446064007069630608997313824163716695587
1065472907513094581500982093612503980752776001000426026600165794861232018537694812001389825072000671614907760189489626560237586640967296866339961
11137493105753024848326502520706938750114944236407003301405887780063831021574320856441500314408026239790800051714347897520218860518676805488251406176618884638912356790528924523
123162341432257589619437400695715115095003524956583148001138985031284100251676599350176039396309014462404430484364521600356829000492888020135167718265607573786940522881708121603397121749219057760018983329135883665
13790585358062520555219309625020555219309625001145610889523100041130015018592500102243618486009000182911434710004000239984569744920000231938850320377200163598237710908000820493585223312002775697605520320056849619377200005329852082005761091556096587136051
142134580466768755976825306952500647489408253187503936735602179380001554714567702796500429423197641237800086425652900476890001295916676622568000014612147570183763600123680267709446448007753664380360298400349737898295560320010744578062290800002014684086998177281741127736739110467439846581526028457
Technical details
Node id
n-e2163acca2adac55 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9d419c98ea340a64

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        1,         2;
  2:        3,        12,         10;
  3:       15,        90,        150,          76;
  4:      105,       840,       2100,        2128,         764;
  5:      945,      9450,      31500,       47880,       34380,        9496;
  6:    10395,    124740,     519750,     1053360,     1134540,      626736,      140152;
  7:   135135,   1891890,    9459450,    23963940,    34414380,    28516488,    12753832,     2390480;
  8:  2027025,  32432400,  189189000,   575134560,  1032431400,  1140659520,   765229920,   286857600,   46206736;
  9: 34459425, 620269650, 4135131000, 14665931280, 31592400840, 43630226640, 39026725920, 21944606400, 7069630608, 997313824;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(2*n-3-(4*n^2-22*n+21)*x-(12*n^2-50*n+40)*x^2-(12*n^2-42*n+30)*x^3-(4*n^2-12*n+8)*x^4)*P(n-1,x) + (2-(8*n-28)*x-(32*n-88)*x^2-(48*n-116)*x^3-(32*n-70)*x^4-(8*n-16)*x^5)*d/dx P(n-1,x) + (4*x+20*x^2+40*x^3+40*x^4+20*x^5+4*x^6)*d^2/dx^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[(-(2*n - 3 - (4*n^2 - 22*n + 21)*x - (12*n^2 - 50*n + 40)*x^2 - (12*n^2 - 42*n + 30)*x^3 - (4*n^2 - 12*n + 8)*x^4)*P[n - 1, x] + (2 - (8*n - 28)*x - (32*n - 88)*x^2 - (48*n - 116)*x^3 - (32*n - 70)*x^4 - (8*n - 16)*x^5)*D[P[n - 1, x], x] + (4*x + 20*x^2 + 40*x^3 + 40*x^4 + 20*x^5 + 4*x^6)*D[P[n - 1, x], {x, 2}])/(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] = (-(2*n - 3 - (4*n^2 - 22*n + 21)*x - (12*n^2 - 50*n + 40)*x^2 - (12*n^2 - 42*n + 30)*x^3 - (4*n^2 - 12*n + 8)*x^4)*v[i - 1] + (2 - (8*n - 28)*x - (32*n - 88)*x^2 - (48*n - 116)*x^3 - (32*n - 70)*x^4 - (8*n - 16)*x^5)*deriv(v[i - 1], x) + (4*x + 20*x^2 + 40*x^3 + 40*x^4 + 20*x^5 + 4*x^6)*deriv(deriv(v[i - 1], x), x))/(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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.