Real-rooted atlas

A257180 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 + 6\,n + n^{2}\right)\,P_{n}(x) = \left(-72 - 304\,x + 612\,x^{2} - 42\,n - 320\,n x + 876\,n x^{2} - 6\,n^{2} - 108\,n^{2} x + 461\,n^{2} x^{2} - 12\,n^{3} x + 106\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-80\,x + 493\,x^{2} - 171\,x^{3} - 42\,n x + 425\,n x^{2} - 24\,n x^{3} - 6\,n^{2} x + 124\,n^{2} x^{2} + 77\,n^{2} x^{3} + 12\,n^{3} x^{2} + 34\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-14\,x^{2} + 149\,x^{3} - 135\,x^{4} - 7\,n x^{2} + 133\,n x^{3} - 126\,n x^{4} - n^{2} x^{2} + 40\,n^{2} x^{3} - 39\,n^{2} x^{4} + 4\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(51 + 108\,x + 32\,n + 108\,n x + 5\,n^{2} + 36\,n^{2} x + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(23\,x - 144\,x^{2} + 13\,n x - 132\,n x^{2} + 2\,n^{2} x - 40\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 9 + 96\,x + 81\,x^{2}\)
\(P_{2}(x) = 27 + 564\,x + 1269\,x^{2} + 432\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
199681186
22756412694322292
38129041269011616202529316
424313884102654182496867758748394800
57296331273194321899522033175569808357215624640
621872794924799331222483363476302519197324342377713996884853440
76561120544829656368201747168486682350453931128161462448192871685314411354510080
819683511047617540442016849848965928616050853720711251602662441247320320103487139196830022844384640

All 15 rows

Technical details
Node id
n-26d307a2c7ce8f8f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e33bea9253c6f0a1

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:      9,       96,         81;
   2:     27,      564,       1269,         432;
   3:     81,     2904,      12690,       11616,         2025;
   4:    243,    13884,     102654,      182496,        86775,          8748;
   5:    729,    63312,     731943,     2189952,      2033175,        569808,         35721;
   6:   2187,   279492,    4799331,    22248336,     34763025,      19197324,       3423777,        139968;
   7:   6561,  1205448,   29656368,   201747168,    486682350,     453931128,     161462448,      19287168,       531441;
   8:  19683,  5110476,  175404420,  1684984896,   5928616050,    8537207112,    5160266244,    1247320320,    103487139,     1968300;
   9:  59049, 21386688, 1003506381, 13228210176,  65195412450,  136595313792,  127783008402,   52912840704,   9031557429,   534667200,    7144929;
  10: 177147, 88617300, 5594972481, 98993385840, 663058671150, 1937695055688, 2637418270242, 1697430198144, 501154015815, 62166360900, 2680673325, 25509168;
%F Formula
The row polynomials satisfy (n^2+6*n+9)*P(n,x) = -(6*n^2+42*n+72+(12*n^3+108*n^2+320*n+304)*x-(9*n^4+106*n^3+461*n^2+876*n+612)*x^2)*P(n-2,x) - ((6*n^2+42*n+80)*x-(12*n^3+124*n^2+425*n+493)*x^2-(4*n^4+34*n^3+77*n^2-24*n-171)*x^3)*d/dx P(n-2,x) - ((n^2+7*n+14)*x^2-(4*n^3+40*n^2+133*n+149)*x^3+(4*n^3+39*n^2+126*n+135)*x^4)*d^2/dx^2 P(n-2,x) + (5*n^2+32*n+51+(4*n^3+36*n^2+108*n+108)*x)*P(n-1,x) + ((2*n^2+13*n+23)*x-(4*n^3+40*n^2+132*n+144)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 9 + 96*x + 81*x^2, P(2,x) = 27 + 564*x + 1269*x^2 + 432*x^3.
%t Mathematica
P[1, x_] := 9 + 96*x + 81*x^2;
P[2, x_] := 27 + 564*x + 1269*x^2 + 432*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n^2 + 42*n + 72 + (12*n^3 + 108*n^2 + 320*n + 304)*x - (9*n^4 + 106*n^3 + 461*n^2 + 876*n + 612)*x^2)*P[n - 2, x] - ((6*n^2 + 42*n + 80)*x - (12*n^3 + 124*n^2 + 425*n + 493)*x^2 - (4*n^4 + 34*n^3 + 77*n^2 - 24*n - 171)*x^3)*D[P[n - 2, x], x] - ((n^2 + 7*n + 14)*x^2 - (4*n^3 + 40*n^2 + 133*n + 149)*x^3 + (4*n^3 + 39*n^2 + 126*n + 135)*x^4)*D[P[n - 2, x], {x, 2}] + (5*n^2 + 32*n + 51 + (4*n^3 + 36*n^2 + 108*n + 108)*x)*P[n - 1, x] + ((2*n^2 + 13*n + 23)*x - (4*n^3 + 40*n^2 + 132*n + 144)*x^2)*D[P[n - 1, x], x])/(n^2 + 6*n + 9)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 9 + 96*x + 81*x^2; v[2] = 27 + 564*x + 1269*x^2 + 432*x^3; for(n = 3, N, my(i = n); v[i] = (-(6*n^2 + 42*n + 72 + (12*n^3 + 108*n^2 + 320*n + 304)*x - (9*n^4 + 106*n^3 + 461*n^2 + 876*n + 612)*x^2)*v[i - 2] - ((6*n^2 + 42*n + 80)*x - (12*n^3 + 124*n^2 + 425*n + 493)*x^2 - (4*n^4 + 34*n^3 + 77*n^2 - 24*n - 171)*x^3)*deriv(v[i - 2], x) - ((n^2 + 7*n + 14)*x^2 - (4*n^3 + 40*n^2 + 133*n + 149)*x^3 + (4*n^3 + 39*n^2 + 126*n + 135)*x^4)*deriv(deriv(v[i - 2], x), x) + (5*n^2 + 32*n + 51 + (4*n^3 + 36*n^2 + 108*n + 108)*x)*v[i - 1] + ((2*n^2 + 13*n + 23)*x - (4*n^3 + 40*n^2 + 132*n + 144)*x^2)*deriv(v[i - 1], x))/(n^2 + 6*n + 9)); 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

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.