Real-rooted atlas

A081579 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(101 + 669\,x + 1830\,x^{2} + 2650\,x^{3} + 2145\,x^{4} + 921\,x^{5} + 164\,x^{6} - 144\,n - 960\,n x - 2647\,n x^{2} - 3867\,n x^{3} - 3159\,n x^{4} - 1369\,n x^{5} - 246\,n x^{6} + 45\,n^{2} + 300\,n^{2} x + 832\,n^{2} x^{2} + 1228\,n^{2} x^{3} + 1017\,n^{2} x^{4} + 448\,n^{2} x^{5} + 82\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(130 + 1073\,x + 3743\,x^{2} + 7170\,x^{3} + 8160\,x^{4} + 5525\,x^{5} + 2063\,x^{6} + 328\,x^{7} - 60\,n - 500\,n x - 1763\,n x^{2} - 3416\,n x^{3} - 3934\,n x^{4} - 2696\,n x^{5} - 1019\,n x^{6} - 164\,n x^{7}\right)\,P_{n-1}'(x) + \left(20 + 200\,x + 861\,x^{2} + 2087\,x^{3} + 3120\,x^{4} + 2950\,x^{5} + 1725\,x^{6} + 571\,x^{7} + 82\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
28241749
32410815875365
480480106010203613001
52561920568082805946168323765
68327488277605424058896336967937190849
7268828224125792308400449096388404184726372751524605
88704104448543808160416029319363399552244193299354017532112203401
9281603801602264320780864017179872250048802407504014785920525637082416397607525
109113613670409167360361881609311379216316409619718256016227072087029720274696083875057780919249
11294912486604836280320161326080475329792974316672141765510414641723201051928440500688828142099854182184756247176845
12954368171786241409617926971904002314689792543414988892498699521150189632010369201840660956380828276394767289940608565588149977946201
13308838460223488539332608293686272010848425472287042019845596724838481416486400883631150727066600828840476063248157244893203713714474402715443399821975285
14999424020987904020369612801210945536049260177408145044080640318773899520531213941760674474212160649305209664466529089680242601328800863125649601880639736018933933773198580585249
Technical details
Node id
n-e333b1133cf42178 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-63e7b172ef605fa7

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:     2,      3;
  2:     8,     24,      17;
  3:    24,    108,     158,      75;
  4:    80,    480,    1060,    1020,      361;
  5:   256,   1920,    5680,    8280,     5946,     1683;
  6:   832,   7488,   27760,   54240,    58896,    33696,     7937;
  7:  2688,  28224,  125792,  308400,   449096,   388404,   184726,    37275;
  8:  8704, 104448,  543808, 1604160,  2931936,  3399552,  2441932,   993540,  175321;
  9: 28160, 380160, 2264320, 7808640, 17179872, 25004880, 24075040, 14785920, 5256370, 824163;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (45*n^2-144*n+101+(300*n^2-960*n+669)*x+(832*n^2-2647*n+1830)*x^2+(1228*n^2-3867*n+2650)*x^3+(1017*n^2-3159*n+2145)*x^4+(448*n^2-1369*n+921)*x^5+(82*n^2-246*n+164)*x^6)*P(n-1,x) - (60*n-130+(500*n-1073)*x+(1763*n-3743)*x^2+(3416*n-7170)*x^3+(3934*n-8160)*x^4+(2696*n-5525)*x^5+(1019*n-2063)*x^6+(164*n-328)*x^7)*d/dx P(n-1,x) + (20+200*x+861*x^2+2087*x^3+3120*x^4+2950*x^5+1725*x^6+571*x^7+82*x^8)*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[((45*n^2 - 144*n + 101 + (300*n^2 - 960*n + 669)*x + (832*n^2 - 2647*n + 1830)*x^2 + (1228*n^2 - 3867*n + 2650)*x^3 + (1017*n^2 - 3159*n + 2145)*x^4 + (448*n^2 - 1369*n + 921)*x^5 + (82*n^2 - 246*n + 164)*x^6)*P[n - 1, x] - (60*n - 130 + (500*n - 1073)*x + (1763*n - 3743)*x^2 + (3416*n - 7170)*x^3 + (3934*n - 8160)*x^4 + (2696*n - 5525)*x^5 + (1019*n - 2063)*x^6 + (164*n - 328)*x^7)*D[P[n - 1, x], x] + (20 + 200*x + 861*x^2 + 2087*x^3 + 3120*x^4 + 2950*x^5 + 1725*x^6 + 571*x^7 + 82*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];
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] = ((45*n^2 - 144*n + 101 + (300*n^2 - 960*n + 669)*x + (832*n^2 - 2647*n + 1830)*x^2 + (1228*n^2 - 3867*n + 2650)*x^3 + (1017*n^2 - 3159*n + 2145)*x^4 + (448*n^2 - 1369*n + 921)*x^5 + (82*n^2 - 246*n + 164)*x^6)*v[i - 1] - (60*n - 130 + (500*n - 1073)*x + (1763*n - 3743)*x^2 + (3416*n - 7170)*x^3 + (3934*n - 8160)*x^4 + (2696*n - 5525)*x^5 + (1019*n - 2063)*x^6 + (164*n - 328)*x^7)*deriv(v[i - 1], x) + (20 + 200*x + 861*x^2 + 2087*x^3 + 3120*x^4 + 2950*x^5 + 1725*x^6 + 571*x^7 + 82*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*x^3)); 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. [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