Real-rooted atlas

A156289 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 12\,x + 37\,x^{2} + 48\,x^{3} + 28\,x^{4} + 6\,x^{5} - 3\,n x - 16\,n x^{2} - 28\,n x^{3} - 20\,n x^{4} - 5\,n x^{5} + n^{2} x^{2} + 3\,n^{2} x^{3} + 3\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}(x) + \left(3\,x + 20\,x^{2} + 48\,x^{3} + 54\,x^{4} + 29\,x^{5} + 6\,x^{6} - 2\,n x^{2} - 8\,n x^{3} - 12\,n x^{4} - 8\,n x^{5} - 2\,n x^{6}\right)\,P_{n-1}'(x) + \left(x^{2} + 5\,x^{3} + 10\,x^{4} + 10\,x^{5} + 5\,x^{6} + x^{7}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2145
31173149
4166339379785
5125929768329655618121
61102825222135343251102150349563045
714101215685202295067076259134586437346122458409
81163901875549297174951585503053707204163942993851562979641111477505
9165543165357584396478513590468155129081370312297168930819894869743669848637166519899761
1012621521471177206603977163805746414694210775858311119868377222159397737367511582038239333377894905994718242649765
11110485851315932795100808749920181977590739013396310668377501825781751701046918197919101233903003562807692948360950519738202807236390250597047169
1214194314118056461351560312124275416426857611904242192928291922161490083831218621880296984388510593070758174000106060551117544905773809805126316131970368193592937140366511220945
1311677722710607884839624410212506445967626600861870135154644441870429165754049234420435052283438008370281111289335591109511613339653835467201007329942671097566485560316250904452998967873425230814021921887229591801
141671088769539378831943848356959383592282369496668080043559060810459789738788814359281045121925307817718888869858266359615515536963112060619615160586750137854621421424590446105208334823005292336454194091451052389298484301665702298804490884405991802746565
15126843546985820183961616099184145365114545588556106113431142357773281130180521651152758538580530631046928371585943343872404097969062318526615310043048213018750098687164516038957501631728011768957444421707017909412028430546764045467211469509758973270828578980222167922097588663166977787180417351081529
Technical details
Node id
n-2f279ba4fe50fb9e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c0ca8bbbef2cb15d

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,      4;
   3: 1,     17,        31;
   4: 1,     66,       339,        379;
   5: 1,    259,      2976,       8329,        6556;
   6: 1,   1028,     25222,     135343,      251102,       150349;
   7: 1,   4101,    215685,    2022950,     6707625,      9134586,       4373461;
   8: 1,  16390,   1875549,   29717495,   158550305,    370720416,     394299385,     156297964;
   9: 1,  65543,  16535758,  439647851,  3590468155,  12908137031,   22971689308,   19894869743,    6698486371;
  10: 1, 262152, 147117720, 6603977163, 80574641469, 421077585831, 1119868377222, 1593977373675, 1158203823933, 337789490599;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(3*n-12)*x+(n^2-16*n+37)*x^2+(3*n^2-28*n+48)*x^3+(3*n^2-20*n+28)*x^4+(n^2-5*n+6)*x^5)*P(n-1,x) + (3*x-(2*n-20)*x^2-(8*n-48)*x^3-(12*n-54)*x^4-(8*n-29)*x^5-(2*n-6)*x^6)*d/dx P(n-1,x) + (x^2+5*x^3+10*x^4+10*x^5+5*x^6+x^7)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 - (3*n - 12)*x + (n^2 - 16*n + 37)*x^2 + (3*n^2 - 28*n + 48)*x^3 + (3*n^2 - 20*n + 28)*x^4 + (n^2 - 5*n + 6)*x^5)*P[n - 1, x] + (3*x - (2*n - 20)*x^2 - (8*n - 48)*x^3 - (12*n - 54)*x^4 - (8*n - 29)*x^5 - (2*n - 6)*x^6)*D[P[n - 1, x], x] + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*D[P[n - 1, x], {x, 2}])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 - (3*n - 12)*x + (n^2 - 16*n + 37)*x^2 + (3*n^2 - 28*n + 48)*x^3 + (3*n^2 - 20*n + 28)*x^4 + (n^2 - 5*n + 6)*x^5)*v[i - 1] + (3*x - (2*n - 20)*x^2 - (8*n - 48)*x^3 - (12*n - 54)*x^4 - (8*n - 29)*x^5 - (2*n - 6)*x^6)*deriv(v[i - 1], x) + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*deriv(deriv(v[i - 1], x), x))/(1 + 2*x + x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.