Real-rooted atlas

A376827 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-n x + n x^{2} - n^{2} + 2\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-3\,x - 9\,x^{2} - 9\,x^{3} - 3\,x^{4} - 3\,n - 9\,n x - 6\,n x^{2} + 3\,n x^{3} + 3\,n x^{4} + 6\,n^{2} x + 12\,n^{2} x^{2} + 6\,n^{2} x^{3} + 3\,n^{3} + 6\,n^{3} x + 3\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 + x - x^{2} - x^{3} - 2\,n - 4\,n x + 2\,n x^{3} - n^{2} + 3\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
2912324
32863427140
49530434215219912
5306127520401530510515712
6987507610575112806345169214136096
7314419257495186877555020247595502393226368
8996370848216972371952387450247968929881771211071416960
93139025425990403218457322373084197757010547043390125650231398839424
1098483895300362596585948801316091013536936940065042974401208655179060895355007232
11307836310401314109150380947056772392082961802711101164232745016930980423274556436625653341492736
12959257106256165357081416233580032872999546752710447726725235064532818262777564934320146102221770936737892115678208
132981174359870291993127766699123861522528150246649560329232540482557847292164433040276126407524360450449828194553646621294113083095040
1492434051207804927289965412691679536678527549712336864540166547671291691912851212953707767740211872430842161358972265121682299711725435661622780770105344
Technical details
Node id
n-0a3212c17664a261 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e99c3824c4864868

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     2,      1;
  2:     9,     12,      3;
  3:    28,     63,     42,       7;
  4:    95,    304,    342,     152,      19;
  5:   306,   1275,   2040,    1530,     510,      51;
  6:   987,   5076,  10575,   11280,    6345,    1692,     141;
  7:  3144,  19257,  49518,   68775,   55020,   24759,    5502,    393;
  8:  9963,  70848, 216972,  371952,  387450,  247968,   92988,  17712,  1107;
  9: 31390, 254259, 904032, 1845732, 2373084, 1977570, 1054704, 339012, 56502, 3139;
%F Formula
The row polynomials satisfy (n^3-n^2+(2*n^2-n)*x+n*x^2)*P(n,x) = (3*n^3-3*n+(6*n^3+6*n^2-9*n-3)*x+(3*n^3+12*n^2-6*n-9)*x^2+(6*n^2+3*n-9)*x^3+(3*n-3)*x^4)*P(n-2,x) + (2*n^3-n^2-2*n+1+(2*n^3+3*n^2-4*n+1)*x+(4*n^2-1)*x^2+(2*n-1)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^3 - 3*n + (6*n^3 + 6*n^2 - 9*n - 3)*x + (3*n^3 + 12*n^2 - 6*n - 9)*x^2 + (6*n^2 + 3*n - 9)*x^3 + (3*n - 3)*x^4)*P[n - 2, x] + (2*n^3 - n^2 - 2*n + 1 + (2*n^3 + 3*n^2 - 4*n + 1)*x + (4*n^2 - 1)*x^2 + (2*n - 1)*x^3)*P[n - 1, x])/(n^3 - n^2 + (2*n^2 - n)*x + n*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 + x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n^3 - 3*n + (6*n^3 + 6*n^2 - 9*n - 3)*x + (3*n^3 + 12*n^2 - 6*n - 9)*x^2 + (6*n^2 + 3*n - 9)*x^3 + (3*n - 3)*x^4)*v[i - 2] + (2*n^3 - n^2 - 2*n + 1 + (2*n^3 + 3*n^2 - 4*n + 1)*x + (4*n^2 - 1)*x^2 + (2*n - 1)*x^3)*v[i - 1])/(n^3 - n^2 + (2*n^2 - n)*x + n*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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.