Real-rooted atlas

A121574 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(9\,n + 27\,n x + 27\,n x^{2} + 9\,n x^{3}\right)\,P_{n}(x) = \left(164 + 1236\,x + 3720\,x^{2} + 5800\,x^{3} + 4980\,x^{4} + 2244\,x^{5} + 416\,x^{6} - 216\,n - 1641\,n x - 5055\,n x^{2} - 8103\,n x^{3} - 7149\,n x^{4} - 3300\,n x^{5} - 624\,n x^{6} + 79\,n^{2} + 540\,n^{2} x + 1578\,n^{2} x^{2} + 2492\,n^{2} x^{3} + 2223\,n^{2} x^{4} + 1056\,n^{2} x^{5} + 208\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(136 + 1396\,x + 5772\,x^{2} + 12680\,x^{3} + 16160\,x^{4} + 12036\,x^{5} + 4876\,x^{6} + 832\,x^{7} - 65\,n - 644\,n x - 2634\,n x^{2} - 5812\,n x^{3} - 7513\,n x^{4} - 5712\,n x^{5} - 2372\,n x^{6} - 416\,n x^{7}\right)\,P_{n-1}'(x) + \left(13 + 173\,x + 946\,x^{2} + 2798\,x^{3} + 4945\,x^{4} + 5393\,x^{5} + 3568\,x^{6} + 1316\,x^{7} + 208\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
210414091
333204408264909
41099012733360417449091
536037261516530330298081152090909
61189147847552520285330210023654476096909091
739275701235051411827682365536280411218243845026569090909
8129702153271547815629100215811681251640082476504013780928332032090909091
9428378004486586689313232829486248718972497425058625621077404810245734421932544909090909
1014148129385432723908514838136552598510412677362652103940416237410184017433014407522670081448765449090909091
11467280106791481101087906760371242746069155774844995615496899912219685532402163318796814093925120546772377695698944090909090909
1215433213848707243689898229850950741367127847844212952043103527961629176851808172218740455648191046084736111846139392394107617286321442816909090909091
135097243137736690170699302812845976780654910958132388494712246410850234571282170046914191079576979220957150660161644285027840873980430336282084741120417566146569090909090909
1416835050489997205658290685054103426780303893497895123400471327937356297334718564363705973149425189338841974407541246419449183865280138504772556806740896614400200702855168027582545920090909090909091
Technical details
Node id
n-39de342cb07ad664 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-db76e1900949a225

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:     3,      6;
  2:    10,     41,      40;
  3:    33,    204,     408,      264;
  4:   109,    901,    2733,     3604,     1744;
  5:   360,   3726,   15165,    30330,    29808,     11520;
  6:  1189,  14784,   75525,   202853,   302100,    236544,     76096;
  7:  3927,  57012,  350514,  1182768,  2365536,   2804112,   1824384,    502656;
  8: 12970, 215327, 1547815,  6291002, 15811681,  25164008,  24765040,  13780928,   3320320;
  9: 42837, 800448, 6586689, 31323282, 94862487, 189724974, 250586256, 210774048, 102457344, 21932544;
%F Formula
The row polynomials satisfy (9*n+27*n*x+27*n*x^2+9*n*x^3)*P(n,x) = (79*n^2-216*n+164+(540*n^2-1641*n+1236)*x+(1578*n^2-5055*n+3720)*x^2+(2492*n^2-8103*n+5800)*x^3+(2223*n^2-7149*n+4980)*x^4+(1056*n^2-3300*n+2244)*x^5+(208*n^2-624*n+416)*x^6)*P(n-1,x) - (65*n-136+(644*n-1396)*x+(2634*n-5772)*x^2+(5812*n-12680)*x^3+(7513*n-16160)*x^4+(5712*n-12036)*x^5+(2372*n-4876)*x^6+(416*n-832)*x^7)*d/dx P(n-1,x) + (13+173*x+946*x^2+2798*x^3+4945*x^4+5393*x^5+3568*x^6+1316*x^7+208*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[((79*n^2 - 216*n + 164 + (540*n^2 - 1641*n + 1236)*x + (1578*n^2 - 5055*n + 3720)*x^2 + (2492*n^2 - 8103*n + 5800)*x^3 + (2223*n^2 - 7149*n + 4980)*x^4 + (1056*n^2 - 3300*n + 2244)*x^5 + (208*n^2 - 624*n + 416)*x^6)*P[n - 1, x] - (65*n - 136 + (644*n - 1396)*x + (2634*n - 5772)*x^2 + (5812*n - 12680)*x^3 + (7513*n - 16160)*x^4 + (5712*n - 12036)*x^5 + (2372*n - 4876)*x^6 + (416*n - 832)*x^7)*D[P[n - 1, x], x] + (13 + 173*x + 946*x^2 + 2798*x^3 + 4945*x^4 + 5393*x^5 + 3568*x^6 + 1316*x^7 + 208*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] = ((79*n^2 - 216*n + 164 + (540*n^2 - 1641*n + 1236)*x + (1578*n^2 - 5055*n + 3720)*x^2 + (2492*n^2 - 8103*n + 5800)*x^3 + (2223*n^2 - 7149*n + 4980)*x^4 + (1056*n^2 - 3300*n + 2244)*x^5 + (208*n^2 - 624*n + 416)*x^6)*v[i - 1] - (65*n - 136 + (644*n - 1396)*x + (2634*n - 5772)*x^2 + (5812*n - 12680)*x^3 + (7513*n - 16160)*x^4 + (5712*n - 12036)*x^5 + (2372*n - 4876)*x^6 + (416*n - 832)*x^7)*deriv(v[i - 1], x) + (13 + 173*x + 946*x^2 + 2798*x^3 + 4945*x^4 + 5393*x^5 + 3568*x^6 + 1316*x^7 + 208*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