Real-rooted atlas

A121574 → 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

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(60 + 396\,x + 1080\,x^{2} + 1560\,x^{3} + 1260\,x^{4} + 540\,x^{5} + 96\,x^{6} - 86\,n - 571\,n x - 1569\,n x^{2} - 2285\,n x^{3} - 1861\,n x^{4} - 804\,n x^{5} - 144\,n x^{6} + 29\,n^{2} + 188\,n^{2} x + 510\,n^{2} x^{2} + 740\,n^{2} x^{3} + 605\,n^{2} x^{4} + 264\,n^{2} x^{5} + 48\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(80 + 652\,x + 2252\,x^{2} + 4280\,x^{3} + 4840\,x^{4} + 3260\,x^{5} + 1212\,x^{6} + 192\,x^{7} - 39\,n - 316\,n x - 1090\,n x^{2} - 2076\,n x^{3} - 2359\,n x^{4} - 1600\,n x^{5} - 600\,n x^{6} - 96\,n x^{7}\right)\,P_{n-1}'(x) + \left(13 + 127\,x + 536\,x^{2} + 1278\,x^{3} + 1885\,x^{4} + 1763\,x^{5} + 1022\,x^{6} + 336\,x^{7} + 48\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
210271855
33313418080427
4109591119510683563319
53602442659788725940158425795
611899684327455883959258317167048200479
7392737330151614341044458880369328164640313601558123
812970140949668337180603030420813270672219194483721613953612109735
9428375238382840265896198418135603244083962184890412543360419083262086494116883
101414811922721117332714233995110005159016177657518126841813898008869781008207189602762528731476591
1146728069864364738815019248339252021665998227525813222568521268920800850782096379560992101407680122918405685037435
12154332125175592187894590848387982258112153055743036578762572911101021283008477287596504983352020269484324922317445469241644184121591
13509724390088336733668156364536302012329359653299752891105389444930572556918416731424341885452839982429221834224106601258242372695104243353344343399075747
141683505032045953528278658201533318462057071605695154257750153312232231297480787006815565963794546506840889132339910126152165540458448553438731201136938118410827982082668898259775
Technical details
Node id
n-4879f89488a3cb9e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-fa5a7708dc1bee84

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,      4;
  2:    10,     27,      18;
  3:    33,    134,     180,      80;
  4:   109,    591,    1195,    1068,      356;
  5:   360,   2442,    6597,    8872,     5940,     1584;
  6:  1189,   9684,   32745,   58839,    59258,    31716,     7048;
  7:  3927,  37330,  151614,  341044,   458880,   369328,   164640,    31360;
  8: 12970, 140949,  668337, 1806030,  3042081,  3270672,  2191944,   837216,  139536;
  9: 42837, 523838, 2840265, 8961984, 18135603, 24408396, 21848904, 12543360, 4190832, 620864;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (29*n^2-86*n+60+(188*n^2-571*n+396)*x+(510*n^2-1569*n+1080)*x^2+(740*n^2-2285*n+1560)*x^3+(605*n^2-1861*n+1260)*x^4+(264*n^2-804*n+540)*x^5+(48*n^2-144*n+96)*x^6)*P(n-1,x) - (39*n-80+(316*n-652)*x+(1090*n-2252)*x^2+(2076*n-4280)*x^3+(2359*n-4840)*x^4+(1600*n-3260)*x^5+(600*n-1212)*x^6+(96*n-192)*x^7)*d/dx P(n-1,x) + (13+127*x+536*x^2+1278*x^3+1885*x^4+1763*x^5+1022*x^6+336*x^7+48*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[((29*n^2 - 86*n + 60 + (188*n^2 - 571*n + 396)*x + (510*n^2 - 1569*n + 1080)*x^2 + (740*n^2 - 2285*n + 1560)*x^3 + (605*n^2 - 1861*n + 1260)*x^4 + (264*n^2 - 804*n + 540)*x^5 + (48*n^2 - 144*n + 96)*x^6)*P[n - 1, x] - (39*n - 80 + (316*n - 652)*x + (1090*n - 2252)*x^2 + (2076*n - 4280)*x^3 + (2359*n - 4840)*x^4 + (1600*n - 3260)*x^5 + (600*n - 1212)*x^6 + (96*n - 192)*x^7)*D[P[n - 1, x], x] + (13 + 127*x + 536*x^2 + 1278*x^3 + 1885*x^4 + 1763*x^5 + 1022*x^6 + 336*x^7 + 48*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] = ((29*n^2 - 86*n + 60 + (188*n^2 - 571*n + 396)*x + (510*n^2 - 1569*n + 1080)*x^2 + (740*n^2 - 2285*n + 1560)*x^3 + (605*n^2 - 1861*n + 1260)*x^4 + (264*n^2 - 804*n + 540)*x^5 + (48*n^2 - 144*n + 96)*x^6)*v[i - 1] - (39*n - 80 + (316*n - 652)*x + (1090*n - 2252)*x^2 + (2076*n - 4280)*x^3 + (2359*n - 4840)*x^4 + (1600*n - 3260)*x^5 + (600*n - 1212)*x^6 + (96*n - 192)*x^7)*deriv(v[i - 1], x) + (13 + 127*x + 536*x^2 + 1278*x^3 + 1885*x^4 + 1763*x^5 + 1022*x^6 + 336*x^7 + 48*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