Real-rooted atlas

A114655 → θ+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
122
2448
3624838
48729616192
510160480320321002
61230016002400960645336
71450442001120010080268812828814
81678494083920062720376327168256157184
91811521881611289628224030105612902418432512864146
10201620345602822401016064169344012902404147204608010244780008
112222005940063360031046407451136887040050688001267200112640204826572086
1224290496800130680083635202732083246835712418176001858560037171202703364096148321344
132637441510082516800203860808698060820295475226094182418120960064430080105431046389768192830764794
1428473222713645805764580576024735110475383193613192058881319205888732892160213204992290734081490944163844666890936
15305880331240794976096192096641280640247351104056537395207695367680615629414427983155206783795207827456034406403276826283115038
Technical details
Node id
n-78c1f32e7d3584ec · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c5faa49c64af63fa

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  2;
   2:  4,    4;
   3:  6,   24,     8;
   4:  8,   72,    96,     16;
   5: 10,  160,   480,    320,      32;
   6: 12,  300,  1600,   2400,     960,      64;
   7: 14,  504,  4200,  11200,   10080,    2688,     128;
   8: 16,  784,  9408,  39200,   62720,   37632,    7168,    256;
   9: 18, 1152, 18816, 112896,  282240,  301056,  129024,  18432,   512;
  10: 20, 1620, 34560, 282240, 1016064, 1693440, 1290240, 414720, 46080, 1024;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = (n^2+(6*n^2-12*n+4)*x)*P(n-1,x) + ((2*n-1)*x-(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + (6*n^2 - 12*n + 4)*x)*P[n - 1, x] + ((2*n - 1)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((n^2 + (6*n^2 - 12*n + 4)*x)*v[i - 1] + ((2*n - 1)*x - (4*n - 2)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.