Real-rooted atlas

A345013 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(8\,n - 8\,n x + 20\,n^{2} + 8\,n^{2} x + 16\,n^{3} + 36\,n^{3} x + 8\,n^{3} x^{2} + 4\,n^{4} + 27\,n^{4} x + 12\,n^{4} x^{2} + 6\,n^{5} x + 6\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(8 + 24\,x + 24\,x^{2} + 8\,x^{3} + 40\,n + 116\,n x + 120\,n x^{2} + 40\,n x^{3} + 72\,n^{2} + 258\,n^{2} x + 244\,n^{2} x^{2} + 84\,n^{2} x^{3} + 56\,n^{3} + 286\,n^{3} x + 298\,n^{3} x^{2} + 96\,n^{3} x^{3} + 16\,n^{4} + 144\,n^{4} x + 194\,n^{4} x^{2} + 69\,n^{4} x^{3} + 24\,n^{5} x + 52\,n^{5} x^{2} + 24\,n^{5} x^{3} + 4\,n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-8\,x - 24\,x^{2} - 24\,x^{3} - 8\,x^{4} - 24\,n x - 86\,n x^{2} - 94\,n x^{3} - 32\,n x^{4} - 24\,n^{2} x - 119\,n^{2} x^{2} - 147\,n^{2} x^{3} - 52\,n^{2} x^{4} - 8\,n^{3} x - 68\,n^{3} x^{2} - 104\,n^{3} x^{3} - 44\,n^{3} x^{4} - 12\,n^{4} x^{2} - 28\,n^{4} x^{3} - 16\,n^{4} x^{4} - 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141216
2158054149
3564205401601176
42102016378022403758621
579292402268020160700075660628
630034118412474014784078750181441372415033
7114401801806486489609606930002494804116023042787172

All 15 rows

Technical details
Node id
n-a97bca496951abfa · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9803a3cae47e3dde

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:      4,      12;
  2:     15,      80,       54;
  3:     56,     420,      540,      160;
  4:    210,    2016,     3780,     2240,      375;
  5:    792,    9240,    22680,    20160,     7000,      756;
  6:   3003,   41184,   124740,   147840,    78750,    18144,    1372;
  7:  11440,  180180,   648648,   960960,   693000,   249480,   41160,    2304;
  8:  43758,  777920,  3243240,  5765760,  5255250,  2594592,  679140,   84480,   3645;
  9: 167960, 3325608, 15752880, 32672640, 36036000, 22702680, 8240232, 1647360, 160380, 5500;
%F Formula
The row polynomials satisfy (4*n^4+16*n^3+20*n^2+8*n+(6*n^5+27*n^4+36*n^3+8*n^2-8*n)*x+(n^6+6*n^5+12*n^4+8*n^3)*x^2)*P(n,x) = (16*n^4+56*n^3+72*n^2+40*n+8+(24*n^5+144*n^4+286*n^3+258*n^2+116*n+24)*x+(4*n^6+52*n^5+194*n^4+298*n^3+244*n^2+120*n+24)*x^2+(3*n^6+24*n^5+69*n^4+96*n^3+84*n^2+40*n+8)*x^3)*P(n-1,x) - ((8*n^3+24*n^2+24*n+8)*x+(12*n^4+68*n^3+119*n^2+86*n+24)*x^2+(2*n^5+28*n^4+104*n^3+147*n^2+94*n+24)*x^3+(2*n^5+16*n^4+44*n^3+52*n^2+32*n+8)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^4 + 56*n^3 + 72*n^2 + 40*n + 8 + (24*n^5 + 144*n^4 + 286*n^3 + 258*n^2 + 116*n + 24)*x + (4*n^6 + 52*n^5 + 194*n^4 + 298*n^3 + 244*n^2 + 120*n + 24)*x^2 + (3*n^6 + 24*n^5 + 69*n^4 + 96*n^3 + 84*n^2 + 40*n + 8)*x^3)*P[n - 1, x] - ((8*n^3 + 24*n^2 + 24*n + 8)*x + (12*n^4 + 68*n^3 + 119*n^2 + 86*n + 24)*x^2 + (2*n^5 + 28*n^4 + 104*n^3 + 147*n^2 + 94*n + 24)*x^3 + (2*n^5 + 16*n^4 + 44*n^3 + 52*n^2 + 32*n + 8)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 16*n^3 + 20*n^2 + 8*n + (6*n^5 + 27*n^4 + 36*n^3 + 8*n^2 - 8*n)*x + (n^6 + 6*n^5 + 12*n^4 + 8*n^3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^4 + 56*n^3 + 72*n^2 + 40*n + 8 + (24*n^5 + 144*n^4 + 286*n^3 + 258*n^2 + 116*n + 24)*x + (4*n^6 + 52*n^5 + 194*n^4 + 298*n^3 + 244*n^2 + 120*n + 24)*x^2 + (3*n^6 + 24*n^5 + 69*n^4 + 96*n^3 + 84*n^2 + 40*n + 8)*x^3)*v[i - 1] - ((8*n^3 + 24*n^2 + 24*n + 8)*x + (12*n^4 + 68*n^3 + 119*n^2 + 86*n + 24)*x^2 + (2*n^5 + 28*n^4 + 104*n^3 + 147*n^2 + 94*n + 24)*x^3 + (2*n^5 + 16*n^4 + 44*n^3 + 52*n^2 + 32*n + 8)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 16*n^3 + 20*n^2 + 8*n + (6*n^5 + 27*n^4 + 36*n^3 + 8*n^2 - 8*n)*x + (n^6 + 6*n^5 + 12*n^4 + 8*n^3)*x^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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.