Real-rooted atlas

A330381 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(24\right)\,P_{n}(x) = \left(-152 - 876\,x + 540\,x^{2} + 368\,x^{3} - 24\,x^{4}\right)\,P_{n-4}(x) + \left(-27 - 119\,x + 45\,x^{2} + 432\,x^{3} + 128\,x^{4}\right)\,P_{n-4}'(x) + \left(-27\,x + 9\,x^{2} + 9\,x^{3} + 36\,x^{4}\right)\,P_{n-4}''(x) + \left(568 + 1400\,x - 1344\,x^{2} + 96\,x^{3}\right)\,P_{n-3}(x) + \left(92 + 396\,x - 650\,x^{2} - 1056\,x^{3}\right)\,P_{n-3}'(x) + \left(92\,x - 76\,x^{2} - 130\,x^{3} - 96\,x^{4}\right)\,P_{n-3}''(x) + \left(-596 - 336\,x - 144\,x^{2}\right)\,P_{n-2}(x) + \left(-103 - 300\,x + 960\,x^{2}\right)\,P_{n-2}'(x) + \left(-103\,x + 152\,x^{2} + 192\,x^{3}\right)\,P_{n-2}''(x) + \left(160 + 96\,x\right)\,P_{n-1}(x) + \left(38 - 32\,x\right)\,P_{n-1}'(x) + \left(38\,x - 96\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 8\,x\)
\(P_{2}(x) = 1 + 20\,x + 27\,x^{2}\)
\(P_{3}(x) = 1 + 36\,x + 117\,x^{2} + 64\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1189
21202748
313611764218
4156315416125913
5180675153611252163633
61108126042725400255634313940
71140214299681875015372514551252030
811763402206085307564800377309472729189993
9121651303897613027521765618904282944162811000681521
1012607425688322876256229087406354857601674272650013312408704
1113081039511510458492515812282436329218944011306793157004126117288407098
1213601415718409611137503655080702488581826565796846243100056301361920219729029961
131416188372837122009150783525618264701265333762437695014042600489674995846490077274499313033
141476245704236963464175157837684367590576922688881509236590220031612581933940815686581275963375337000020
Technical details
Node id
n-e7a0b023756a96cf · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-5078111138ba20c7

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: 1,   8;
  2: 1,  20,   27;
  3: 1,  36,  117,    64;
  4: 1,  56,  315,   416,    125;
  5: 1,  80,  675,  1536,   1125,    216;
  6: 1, 108, 1260,  4272,   5400,   2556,    343;
  7: 1, 140, 2142,  9968,  18750,  15372,   5145,   512;
  8: 1, 176, 3402, 20608,  53075,  64800,  37730,  9472,   729;
  9: 1, 216, 5130, 38976, 130275, 217656, 189042, 82944, 16281, 1000;
%F Formula
The row polynomials satisfy 24*P(n,x) = -(152+876*x-540*x^2-368*x^3+24*x^4)*P(n-4,x) - (27+119*x-45*x^2-432*x^3-128*x^4)*d/dx P(n-4,x) - (27*x-9*x^2-9*x^3-36*x^4)*d^2/dx^2 P(n-4,x) + (568+1400*x-1344*x^2+96*x^3)*P(n-3,x) + (92+396*x-650*x^2-1056*x^3)*d/dx P(n-3,x) + (92*x-76*x^2-130*x^3-96*x^4)*d^2/dx^2 P(n-3,x) - (596+336*x+144*x^2)*P(n-2,x) - (103+300*x-960*x^2)*d/dx P(n-2,x) - (103*x-152*x^2-192*x^3)*d^2/dx^2 P(n-2,x) + (160+96*x)*P(n-1,x) + (38-32*x)*d/dx P(n-1,x) + (38*x-96*x^2)*d^2/dx^2 P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 8*x, P(2,x) = 1 + 20*x + 27*x^2, P(3,x) = 1 + 36*x + 117*x^2 + 64*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[2, x_] := 1 + 20*x + 27*x^2;
P[3, x_] := 1 + 36*x + 117*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(152 + 876*x - 540*x^2 - 368*x^3 + 24*x^4)*P[n - 4, x] - (27 + 119*x - 45*x^2 - 432*x^3 - 128*x^4)*D[P[n - 4, x], x] - (27*x - 9*x^2 - 9*x^3 - 36*x^4)*D[P[n - 4, x], {x, 2}] + (568 + 1400*x - 1344*x^2 + 96*x^3)*P[n - 3, x] + (92 + 396*x - 650*x^2 - 1056*x^3)*D[P[n - 3, x], x] + (92*x - 76*x^2 - 130*x^3 - 96*x^4)*D[P[n - 3, x], {x, 2}] - (596 + 336*x + 144*x^2)*P[n - 2, x] - (103 + 300*x - 960*x^2)*D[P[n - 2, x], x] - (103*x - 152*x^2 - 192*x^3)*D[P[n - 2, x], {x, 2}] + (160 + 96*x)*P[n - 1, x] + (38 - 32*x)*D[P[n - 1, x], x] + (38*x - 96*x^2)*D[P[n - 1, x], {x, 2}])/(24)]];
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] = 1 + 8*x; v[3] = 1 + 20*x + 27*x^2; v[4] = 1 + 36*x + 117*x^2 + 64*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(152 + 876*x - 540*x^2 - 368*x^3 + 24*x^4)*v[i - 4] - (27 + 119*x - 45*x^2 - 432*x^3 - 128*x^4)*deriv(v[i - 4], x) - (27*x - 9*x^2 - 9*x^3 - 36*x^4)*deriv(deriv(v[i - 4], x), x) + (568 + 1400*x - 1344*x^2 + 96*x^3)*v[i - 3] + (92 + 396*x - 650*x^2 - 1056*x^3)*deriv(v[i - 3], x) + (92*x - 76*x^2 - 130*x^3 - 96*x^4)*deriv(deriv(v[i - 3], x), x) - (596 + 336*x + 144*x^2)*v[i - 2] - (103 + 300*x - 960*x^2)*deriv(v[i - 2], x) - (103*x - 152*x^2 - 192*x^3)*deriv(deriv(v[i - 2], x), x) + (160 + 96*x)*v[i - 1] + (38 - 32*x)*deriv(v[i - 1], x) + (38*x - 96*x^2)*deriv(deriv(v[i - 1], x), x))/(24)); 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.