Real-rooted atlas

A081579 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(9\,n + 27\,n x + 27\,n x^{2} + 9\,n x^{3}\right)\,P_{n}(x) = \left(289 + 2157\,x + 6450\,x^{2} + 10010\,x^{3} + 8565\,x^{4} + 3849\,x^{5} + 712\,x^{6} - 396\,n - 2958\,n x - 8967\,n x^{2} - 14181\,n x^{3} - 12381\,n x^{4} - 5673\,n x^{5} - 1068\,n x^{6} + 125\,n^{2} + 900\,n^{2} x + 2706\,n^{2} x^{2} + 4324\,n^{2} x^{3} + 3861\,n^{2} x^{4} + 1824\,n^{2} x^{5} + 356\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(230 + 2417\,x + 10059\,x^{2} + 22090\,x^{3} + 28060\,x^{4} + 20805\,x^{5} + 8387\,x^{6} + 1424\,x^{7} - 100\,n - 1060\,n x - 4485\,n x^{2} - 10052\,n x^{3} - 13058\,n x^{4} - 9912\,n x^{5} - 4093\,n x^{6} - 712\,n x^{7}\right)\,P_{n-1}'(x) + \left(20 + 280\,x + 1589\,x^{2} + 4801\,x^{3} + 8570\,x^{4} + 9370\,x^{5} + 6185\,x^{6} + 2269\,x^{7} + 356\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1257
28404189
324180414285903
4808002820410020819881
5256320015280346003698614965105287
683212480751202312003824163220801081211132249
7268847040341856133320030052563909420271771878004512137223
8870417408014823047002560200320963548096037975276224643805630161130250521
928160633600618624034339200119231712268196400390498720354830400182666610406315251397242567
10911362278400250905601600512006540558721787158080330401328040792128003218897880146662940029324020114990718809
1129491281100809944064071677440033710584321085144256024377447232381989304004090911372028520892900116549236622116305405160824734343
12954368286310403868200963109222400165446853126134874624016243621593630927674304042013693944039709923440024792222670891838103540152733702411725401892761
1330883841003724801481478144131382169607804625459232771624832010000917381122244702312960370431181267244383318984003752923997584212118532188071854214487411022773768518510802623047
149994240349798400559998976054315212800356335904768167170001920057801994713601495727454720029097197263680423377864752004535337211304034686375330400179076220782405588043261200795512612281198591694071769
Technical details
Node id
n-ccaff4710d37530a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ac683ecb1e0666c7

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,      5;
  2:     8,     40,      41;
  3:    24,    180,     414,      285;
  4:    80,    800,    2820,     4100,      2081;
  5:   256,   3200,   15280,    34600,     36986,     14965;
  6:   832,  12480,   75120,   231200,    382416,    322080,    108121;
  7:  2688,  47040,  341856,  1333200,   3005256,   3909420,   2717718,    780045;
  8:  8704, 174080, 1482304,  7002560,  20032096,  35480960,  37975276,  22464380,   5630161;
  9: 28160, 633600, 6186240, 34339200, 119231712, 268196400, 390498720, 354830400, 182666610, 40631525;
%F Formula
The row polynomials satisfy (9*n+27*n*x+27*n*x^2+9*n*x^3)*P(n,x) = (125*n^2-396*n+289+(900*n^2-2958*n+2157)*x+(2706*n^2-8967*n+6450)*x^2+(4324*n^2-14181*n+10010)*x^3+(3861*n^2-12381*n+8565)*x^4+(1824*n^2-5673*n+3849)*x^5+(356*n^2-1068*n+712)*x^6)*P(n-1,x) - (100*n-230+(1060*n-2417)*x+(4485*n-10059)*x^2+(10052*n-22090)*x^3+(13058*n-28060)*x^4+(9912*n-20805)*x^5+(4093*n-8387)*x^6+(712*n-1424)*x^7)*d/dx P(n-1,x) + (20+280*x+1589*x^2+4801*x^3+8570*x^4+9370*x^5+6185*x^6+2269*x^7+356*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[((125*n^2 - 396*n + 289 + (900*n^2 - 2958*n + 2157)*x + (2706*n^2 - 8967*n + 6450)*x^2 + (4324*n^2 - 14181*n + 10010)*x^3 + (3861*n^2 - 12381*n + 8565)*x^4 + (1824*n^2 - 5673*n + 3849)*x^5 + (356*n^2 - 1068*n + 712)*x^6)*P[n - 1, x] - (100*n - 230 + (1060*n - 2417)*x + (4485*n - 10059)*x^2 + (10052*n - 22090)*x^3 + (13058*n - 28060)*x^4 + (9912*n - 20805)*x^5 + (4093*n - 8387)*x^6 + (712*n - 1424)*x^7)*D[P[n - 1, x], x] + (20 + 280*x + 1589*x^2 + 4801*x^3 + 8570*x^4 + 9370*x^5 + 6185*x^6 + 2269*x^7 + 356*x^8)*D[P[n - 1, x], {x, 2}])/(9*n + 27*n*x + 27*n*x^2 + 9*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] = ((125*n^2 - 396*n + 289 + (900*n^2 - 2958*n + 2157)*x + (2706*n^2 - 8967*n + 6450)*x^2 + (4324*n^2 - 14181*n + 10010)*x^3 + (3861*n^2 - 12381*n + 8565)*x^4 + (1824*n^2 - 5673*n + 3849)*x^5 + (356*n^2 - 1068*n + 712)*x^6)*v[i - 1] - (100*n - 230 + (1060*n - 2417)*x + (4485*n - 10059)*x^2 + (10052*n - 22090)*x^3 + (13058*n - 28060)*x^4 + (9912*n - 20805)*x^5 + (4093*n - 8387)*x^6 + (712*n - 1424)*x^7)*deriv(v[i - 1], x) + (20 + 280*x + 1589*x^2 + 4801*x^3 + 8570*x^4 + 9370*x^5 + 6185*x^6 + 2269*x^7 + 356*x^8)*deriv(deriv(v[i - 1], x), x))/(9*n + 27*n*x + 27*n*x^2 + 9*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