Real-rooted atlas

A154986 → D+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(3 + 6\,x + 3\,x^{2} - 3\,n - 6\,n x - 3\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(-12\,x^{2} - 48\,x^{3} - 72\,x^{4} - 48\,x^{5} - 12\,x^{6} + 40\,n x^{2} + 160\,n x^{3} + 240\,n x^{4} + 160\,n x^{5} + 40\,n x^{6} - 63\,n^{2} x^{2} - 252\,n^{2} x^{3} - 378\,n^{2} x^{4} - 252\,n^{2} x^{5} - 63\,n^{2} x^{6} + 59\,n^{3} x^{2} + 236\,n^{3} x^{3} + 354\,n^{3} x^{4} + 236\,n^{3} x^{5} + 59\,n^{3} x^{6} - 32\,n^{4} x^{2} - 128\,n^{4} x^{3} - 192\,n^{4} x^{4} - 128\,n^{4} x^{5} - 32\,n^{4} x^{6} + 9\,n^{5} x^{2} + 36\,n^{5} x^{3} + 54\,n^{5} x^{4} + 36\,n^{5} x^{5} + 9\,n^{5} x^{6} - n^{6} x^{2} - 4\,n^{6} x^{3} - 6\,n^{6} x^{4} - 4\,n^{6} x^{5} - n^{6} x^{6}\right)\,P_{n-4}(x) + \left(-4\,x - 24\,x^{2} - 48\,x^{3} - 40\,x^{4} - 12\,x^{5} + 10\,n x + 60\,n x^{2} + 120\,n x^{3} + 100\,n x^{4} + 30\,n x^{5} - 12\,n^{2} x - 68\,n^{2} x^{2} - 132\,n^{2} x^{3} - 108\,n^{2} x^{4} - 32\,n^{2} x^{5} + 8\,n^{3} x + 42\,n^{3} x^{2} + 78\,n^{3} x^{3} + 62\,n^{3} x^{4} + 18\,n^{3} x^{5} - 2\,n^{4} x - 10\,n^{4} x^{2} - 18\,n^{4} x^{3} - 14\,n^{4} x^{4} - 4\,n^{4} x^{5}\right)\,P_{n-3}(x) + \left(-1 - 6\,x - 15\,x^{2} - 16\,x^{3} - 6\,x^{4} + n - 3\,n x^{2} - 2\,n x^{3} - n^{2} + 6\,n^{2} x + 23\,n^{2} x^{2} + 24\,n^{2} x^{3} + 8\,n^{2} x^{4} - 8\,n^{3} x - 24\,n^{3} x^{2} - 24\,n^{3} x^{3} - 8\,n^{3} x^{4} + 2\,n^{4} x + 6\,n^{4} x^{2} + 6\,n^{4} x^{3} + 2\,n^{4} x^{4}\right)\,P_{n-2}(x) + \left(4 + 18\,x + 24\,x^{2} + 10\,x^{3} - 4\,n - 18\,n x - 24\,n x^{2} - 10\,n x^{3} + 2\,n^{2} + 8\,n^{2} x + 10\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 3\,x\)
\(P_{2}(x) = 5 + 16\,x + 12\,x^{2}\)
\(P_{3}(x) = 12 + 69\,x + 116\,x^{2} + 60\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
25161233
3126911660257
4252647849043602337
546903440887607728252024365
677269621588679761016647281620160286977
7120704993804452460104080012462247524481814403774345
8177164323588802645616899668816267968162455048485632181440054831297
9250348271207152137664806811267217724369626327692822540032010380672019958400872807445
10341682563604476635695844577131201678682208352484236844392815363326428800137048832023950080015104179809
11452125389969315625972975627517189761412747347240833186048716442900487823355955252111762560194287334403113510400282513783249
12585218232238235369432379441477966783210648546297641890398873699479520000014969888985601442352692736864777461760294486174720435891456005678125973217
1374236289554217384307552845670686122832722036481992385911669670412215330606592245424322667523228782340326427821123057664151644813465604753172551680653837184000122093169827517
14925580440115519300911229644030149953766443993178949923207913311724813456116416755235471976796800061657100436787272011170063872056108182726656028035908490240081410270054400104613949440002796065393256513
Technical details
Node id
n-b993c35e2c442094 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-19d36d71cbec1262

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:   5,    16,      12;
  3:  12,    69,     116,       60;
  4:  25,   264,     784,      904,      360;
  5:  46,   903,    4408,     8760,     7728,      2520;
  6:  77,  2696,   21588,    67976,   101664,     72816,     20160;
  7: 120,  7049,   93804,   452460,  1040800,   1246224,    752448,    181440;
  8: 177, 16432,  358880,  2645616,  8996688,  16267968,  16245504,   8485632,   1814400;
  9: 250, 34827, 1207152, 13766480, 68112672, 177243696, 263276928, 225400320, 103806720, 19958400;
%F Formula
The row polynomials satisfy (n^2-3*n+3+(2*n^2-6*n+6)*x+(n^2-3*n+3)*x^2)*P(n,x) = -((n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12)*x^2+(4*n^6-36*n^5+128*n^4-236*n^3+252*n^2-160*n+48)*x^3+(6*n^6-54*n^5+192*n^4-354*n^3+378*n^2-240*n+72)*x^4+(4*n^6-36*n^5+128*n^4-236*n^3+252*n^2-160*n+48)*x^5+(n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12)*x^6)*P(n-4,x) - ((2*n^4-8*n^3+12*n^2-10*n+4)*x+(10*n^4-42*n^3+68*n^2-60*n+24)*x^2+(18*n^4-78*n^3+132*n^2-120*n+48)*x^3+(14*n^4-62*n^3+108*n^2-100*n+40)*x^4+(4*n^4-18*n^3+32*n^2-30*n+12)*x^5)*P(n-3,x) - (n^2-n+1-(2*n^4-8*n^3+6*n^2-6)*x-(6*n^4-24*n^3+23*n^2-3*n-15)*x^2-(6*n^4-24*n^3+24*n^2-2*n-16)*x^3-(2*n^4-8*n^3+8*n^2-6)*x^4)*P(n-2,x) + (2*n^2-4*n+4+(8*n^2-18*n+18)*x+(10*n^2-24*n+24)*x^2+(4*n^2-10*n+10)*x^3)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 5 + 16*x + 12*x^2, P(3,x) = 12 + 69*x + 116*x^2 + 60*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[2, x_] := 5 + 16*x + 12*x^2;
P[3, x_] := 12 + 69*x + 116*x^2 + 60*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2 + (4*n^6 - 36*n^5 + 128*n^4 - 236*n^3 + 252*n^2 - 160*n + 48)*x^3 + (6*n^6 - 54*n^5 + 192*n^4 - 354*n^3 + 378*n^2 - 240*n + 72)*x^4 + (4*n^6 - 36*n^5 + 128*n^4 - 236*n^3 + 252*n^2 - 160*n + 48)*x^5 + (n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^6)*P[n - 4, x] - ((2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x + (10*n^4 - 42*n^3 + 68*n^2 - 60*n + 24)*x^2 + (18*n^4 - 78*n^3 + 132*n^2 - 120*n + 48)*x^3 + (14*n^4 - 62*n^3 + 108*n^2 - 100*n + 40)*x^4 + (4*n^4 - 18*n^3 + 32*n^2 - 30*n + 12)*x^5)*P[n - 3, x] - (n^2 - n + 1 - (2*n^4 - 8*n^3 + 6*n^2 - 6)*x - (6*n^4 - 24*n^3 + 23*n^2 - 3*n - 15)*x^2 - (6*n^4 - 24*n^3 + 24*n^2 - 2*n - 16)*x^3 - (2*n^4 - 8*n^3 + 8*n^2 - 6)*x^4)*P[n - 2, x] + (2*n^2 - 4*n + 4 + (8*n^2 - 18*n + 18)*x + (10*n^2 - 24*n + 24)*x^2 + (4*n^2 - 10*n + 10)*x^3)*P[n - 1, x])/(n^2 - 3*n + 3 + (2*n^2 - 6*n + 6)*x + (n^2 - 3*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; v[3] = 5 + 16*x + 12*x^2; v[4] = 12 + 69*x + 116*x^2 + 60*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-((n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2 + (4*n^6 - 36*n^5 + 128*n^4 - 236*n^3 + 252*n^2 - 160*n + 48)*x^3 + (6*n^6 - 54*n^5 + 192*n^4 - 354*n^3 + 378*n^2 - 240*n + 72)*x^4 + (4*n^6 - 36*n^5 + 128*n^4 - 236*n^3 + 252*n^2 - 160*n + 48)*x^5 + (n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^6)*v[i - 4] - ((2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x + (10*n^4 - 42*n^3 + 68*n^2 - 60*n + 24)*x^2 + (18*n^4 - 78*n^3 + 132*n^2 - 120*n + 48)*x^3 + (14*n^4 - 62*n^3 + 108*n^2 - 100*n + 40)*x^4 + (4*n^4 - 18*n^3 + 32*n^2 - 30*n + 12)*x^5)*v[i - 3] - (n^2 - n + 1 - (2*n^4 - 8*n^3 + 6*n^2 - 6)*x - (6*n^4 - 24*n^3 + 23*n^2 - 3*n - 15)*x^2 - (6*n^4 - 24*n^3 + 24*n^2 - 2*n - 16)*x^3 - (2*n^4 - 8*n^3 + 8*n^2 - 6)*x^4)*v[i - 2] + (2*n^2 - 4*n + 4 + (8*n^2 - 18*n + 18)*x + (10*n^2 - 24*n + 24)*x^2 + (4*n^2 - 10*n + 10)*x^3)*v[i - 1])/(n^2 - 3*n + 3 + (2*n^2 - 6*n + 6)*x + (n^2 - 3*n + 3)*x^2)); 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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.