Real-rooted atlas

A201637 → reverse (shift +1) → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(-12\,x - 6\,n + 8\,n x - n^{2} - 7\,n^{2} x^{2} + 4\,n^{3} + 8\,n^{3} x - 2\,n^{3} x^{2} - n^{4} - 4\,n^{4} x + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-3\,x - 15\,x^{2} + 8\,n x + 10\,n x^{2} + n^{2} x - 4\,n^{2} x^{2} - 13\,n^{2} x^{3} - 2\,n^{3} x - 4\,n^{3} x^{2} - 14\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(3\,x^{2} - 3\,x^{3} - 2\,n x^{2} + 2\,n x^{3} - n^{2} x^{2} + 4\,n^{2} x^{3} - 3\,n^{2} x^{4} + 4\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-3\,n - 3\,n^{2} + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-3\,x + 2\,n x + 2\,n^{2} x - 4\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1044
208917
30247216112
409652235225995
504803996524813003611060
602880333727040036300410449147105
702016030585693024080300020196011760642273040
80161280307022412564864161005007048800977550316168139954915
9014515203360528017642073631100160020755800052062500430464081324100786485700
1001451520039905136025949905925972622600559286582421934850003408742401772668820260012117126334225
1101596672005115208320401196994561160540988001440970231688077083089620119662080204475622469503800492712144408550942800
12019160064007044947136065261282918423049947556003642495588864274602854041699513349017616617010485611487637600262645746117590416910591552246275
1302490808320010380247718401116687098880047076991545600917951570432648909124347289644014082968576108801970381441266981777600613783723969647562242764164196296416803582900
140348713164800162989330112002008144642867209922394513952002330221505035392281722429976500818111942442311686184214041007521084676512416009076942319588315335492352346584997264166482258904626416310625
Technical details
Node id
n-60da4f963068547b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3f265e996c6d51b0

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: 0,       4;
  2: 0,       8,        9;
  3: 0,      24,       72,        16;
  4: 0,      96,      522,       352,        25;
  5: 0,     480,     3996,      5248,      1300,        36;
  6: 0,    2880,    33372,     70400,     36300,      4104,       49;
  7: 0,   20160,   305856,    930240,    803000,    201960,    11760,      64;
  8: 0,  161280,  3070224,  12564864,  16100500,   7048800,   977550,   31616,    81;
  9: 0, 1451520, 33605280, 176420736, 311001600, 207558000, 52062500, 4304640, 81324, 100;
%F Formula
The row polynomials satisfy n^2*P(n,x) = -(n^4-4*n^3+n^2+6*n+(4*n^4-8*n^3-8*n+12)*x-(9*n^4-2*n^3-7*n^2)*x^2)*P(n-2,x) - ((2*n^3-n^2-8*n+3)*x-(4*n^4-4*n^3-4*n^2+10*n-15)*x^2-(4*n^4-14*n^3-13*n^2)*x^3)*d/dx P(n-2,x) - ((n^2+2*n-3)*x^2-(4*n^3+4*n^2+2*n-3)*x^3+(4*n^3+3*n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3-3*n^2-3*n+4*n^3*x)*P(n-1,x) + ((2*n^2+2*n-3)*x-(4*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - 4*n^3 + n^2 + 6*n + (4*n^4 - 8*n^3 - 8*n + 12)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*P[n - 2, x] - ((2*n^3 - n^2 - 8*n + 3)*x - (4*n^4 - 4*n^3 - 4*n^2 + 10*n - 15)*x^2 - (4*n^4 - 14*n^3 - 13*n^2)*x^3)*D[P[n - 2, x], x] - ((n^2 + 2*n - 3)*x^2 - (4*n^3 + 4*n^2 + 2*n - 3)*x^3 + (4*n^3 + 3*n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 - 3*n^2 - 3*n + 4*n^3*x)*P[n - 1, x] + ((2*n^2 + 2*n - 3)*x - (4*n^3 + 4*n^2)*x^2)*D[P[n - 1, x], x])/(n^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] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 - 4*n^3 + n^2 + 6*n + (4*n^4 - 8*n^3 - 8*n + 12)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*v[i - 2] - ((2*n^3 - n^2 - 8*n + 3)*x - (4*n^4 - 4*n^3 - 4*n^2 + 10*n - 15)*x^2 - (4*n^4 - 14*n^3 - 13*n^2)*x^3)*deriv(v[i - 2], x) - ((n^2 + 2*n - 3)*x^2 - (4*n^3 + 4*n^2 + 2*n - 3)*x^3 + (4*n^3 + 3*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 - 3*n^2 - 3*n + 4*n^3*x)*v[i - 1] + ((2*n^2 + 2*n - 3)*x - (4*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^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. [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.