Real-rooted atlas

A256890 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 4\,n + n^{2}\right)\,P_{n}(x) = \left(-12 - 52\,x + 100\,x^{2} - 10\,n - 92\,n x + 236\,n x^{2} - 2\,n^{2} - 48\,n^{2} x + 197\,n^{2} x^{2} - 8\,n^{3} x + 70\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-27\,x + 127\,x^{2} - 100\,x^{3} - 20\,n x + 149\,n x^{2} - 92\,n x^{3} - 4\,n^{2} x + 60\,n^{2} x^{2} - n^{2} x^{3} + 8\,n^{3} x^{2} + 18\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-7\,x^{2} + 51\,x^{3} - 44\,x^{4} - 5\,n x^{2} + 65\,n x^{3} - 60\,n x^{4} - n^{2} x^{2} + 28\,n^{2} x^{3} - 27\,n^{2} x^{4} + 4\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(14 + 32\,x + 13\,n + 48\,n x + 3\,n^{2} + 24\,n^{2} x + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(11\,x - 48\,x^{2} + 9\,n x - 64\,n x^{2} + 2\,n^{2} x - 28\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 4 + 48\,x + 36\,x^{2}\)
\(P_{2}(x) = 8 + 208\,x + 468\,x^{2} + 128\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
14483688
28208468128812
316784374431364008080
4322736237964230417100115287120
56491041321204230403670008193631361016400
612829360671868352448055070002687472359660819212788160
725692688321595225983424660840005846270417509072148300820736172851840
8512288304147407401756551046806977009802046885379437561048230405838156512002500243200

All 15 rows

Technical details
Node id
n-83d40d2edb3e8402 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-5cf267279c5da1ec

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:    4,      48,        36;
   2:    8,     208,       468,        128;
   3:   16,     784,      3744,       3136,         400;
   4:   32,    2736,     23796,      42304,       17100,         1152;
   5:   64,    9104,    132120,     423040,      367000,        81936,         3136;
   6:  128,   29360,    671868,    3524480,     5507000,      2687472,       359660,         8192;
   7:  256,   92688,   3215952,   25983424,    66084000,     58462704,     17509072,      1483008,       20736;
   8:  512,  288304,  14740740,  175655104,   680697700,    980204688,    537943756,    104823040,     5838156,      51200;
   9: 1024,  887440,  65449800, 1114127360,  6279875000,  13722865632,  12308555000,   4456509440,   589048200,   22186000,   123904;
  10: 2048, 2711472, 283763340, 6734188800, 53346666000, 168404219424, 229216854216, 136567464960, 34091830800, 3152926000, 82022028, 294912;
%F Formula
The row polynomials satisfy (n^2+4*n+4)*P(n,x) = -(2*n^2+10*n+12+(8*n^3+48*n^2+92*n+52)*x-(9*n^4+70*n^3+197*n^2+236*n+100)*x^2)*P(n-2,x) - ((4*n^2+20*n+27)*x-(8*n^3+60*n^2+149*n+127)*x^2-(4*n^4+18*n^3-n^2-92*n-100)*x^3)*d/dx P(n-2,x) - ((n^2+5*n+7)*x^2-(4*n^3+28*n^2+65*n+51)*x^3+(4*n^3+27*n^2+60*n+44)*x^4)*d^2/dx^2 P(n-2,x) + (3*n^2+13*n+14+(4*n^3+24*n^2+48*n+32)*x)*P(n-1,x) + ((2*n^2+9*n+11)*x-(4*n^3+28*n^2+64*n+48)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 4 + 48*x + 36*x^2, P(2,x) = 8 + 208*x + 468*x^2 + 128*x^3.
%t Mathematica
P[1, x_] := 4 + 48*x + 36*x^2;
P[2, x_] := 8 + 208*x + 468*x^2 + 128*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 + 10*n + 12 + (8*n^3 + 48*n^2 + 92*n + 52)*x - (9*n^4 + 70*n^3 + 197*n^2 + 236*n + 100)*x^2)*P[n - 2, x] - ((4*n^2 + 20*n + 27)*x - (8*n^3 + 60*n^2 + 149*n + 127)*x^2 - (4*n^4 + 18*n^3 - n^2 - 92*n - 100)*x^3)*D[P[n - 2, x], x] - ((n^2 + 5*n + 7)*x^2 - (4*n^3 + 28*n^2 + 65*n + 51)*x^3 + (4*n^3 + 27*n^2 + 60*n + 44)*x^4)*D[P[n - 2, x], {x, 2}] + (3*n^2 + 13*n + 14 + (4*n^3 + 24*n^2 + 48*n + 32)*x)*P[n - 1, x] + ((2*n^2 + 9*n + 11)*x - (4*n^3 + 28*n^2 + 64*n + 48)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n + 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 4 + 48*x + 36*x^2; v[2] = 8 + 208*x + 468*x^2 + 128*x^3; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 + 10*n + 12 + (8*n^3 + 48*n^2 + 92*n + 52)*x - (9*n^4 + 70*n^3 + 197*n^2 + 236*n + 100)*x^2)*v[i - 2] - ((4*n^2 + 20*n + 27)*x - (8*n^3 + 60*n^2 + 149*n + 127)*x^2 - (4*n^4 + 18*n^3 - n^2 - 92*n - 100)*x^3)*deriv(v[i - 2], x) - ((n^2 + 5*n + 7)*x^2 - (4*n^3 + 28*n^2 + 65*n + 51)*x^3 + (4*n^3 + 27*n^2 + 60*n + 44)*x^4)*deriv(deriv(v[i - 2], x), x) + (3*n^2 + 13*n + 14 + (4*n^3 + 24*n^2 + 48*n + 32)*x)*v[i - 1] + ((2*n^2 + 9*n + 11)*x - (4*n^3 + 28*n^2 + 64*n + 48)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n + 4)); 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.