Real-rooted atlas

A154986 → θ+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(3 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(-12 - 24\,x - 12\,x^{2} + 40\,n + 80\,n x + 40\,n x^{2} - 63\,n^{2} - 126\,n^{2} x - 63\,n^{2} x^{2} + 59\,n^{3} + 118\,n^{3} x + 59\,n^{3} x^{2} - 32\,n^{4} - 64\,n^{4} x - 32\,n^{4} x^{2} + 9\,n^{5} + 18\,n^{5} x + 9\,n^{5} x^{2} - n^{6} - 2\,n^{6} x - n^{6} x^{2}\right)\,P_{n-4}(x) + \left(-12 - 20\,x - 8\,x^{2} + 30\,n + 50\,n x + 20\,n x^{2} - 32\,n^{2} - 52\,n^{2} x - 20\,n^{2} x^{2} + 18\,n^{3} + 28\,n^{3} x + 10\,n^{3} x^{2} - 4\,n^{4} - 6\,n^{4} x - 2\,n^{4} x^{2}\right)\,P_{n-3}(x) + \left(-6 - 8\,x - 3\,x^{2} + 2\,n x + 3\,n x^{2} + 8\,n^{2} + 8\,n^{2} x - n^{2} x^{2} - 8\,n^{3} - 8\,n^{3} x + 2\,n^{4} + 2\,n^{4} x\right)\,P_{n-2}(x) + \left(10 + 6\,x - 10\,n - 6\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
\(P_{2}(x) = 12 + 14\,x + 3\,x^{2}\)
\(P_{3}(x) = 60 + 100\,x + 45\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
21214329
360100454209
436077652811651785
52520667261202216255617789
62016063264730083719277804987201909
71814406587529182886075201891002343688982579433

All 15 rows

Technical details
Node id
n-6c995bf2922f8cbd · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f6fb188a0e181d5f

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:        3,        2;
  2:       12,       14,         3;
  3:       60,      100,        45,         4;
  4:      360,      776,       528,       116,        5;
  5:     2520,     6672,      6120,      2216,      255,        6;
  6:    20160,    63264,     73008,     37192,     7780,      498,       7;
  7:   181440,   658752,    918288,    607520,   189100,    23436,     889,      8;
  8:  1814400,  7481088,  12227328,   9973440,  4176720,   824496,   62048,   1480,    9;
  9: 19958400, 92148480, 172938240, 168147072, 89112240, 24686688, 3118640, 147600, 2331, 10;
%F Formula
The row polynomials satisfy (n^2-3*n+3)*P(n,x) = -(n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12+(2*n^6-18*n^5+64*n^4-118*n^3+126*n^2-80*n+24)*x+(n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12)*x^2)*P(n-4,x) - (4*n^4-18*n^3+32*n^2-30*n+12+(6*n^4-28*n^3+52*n^2-50*n+20)*x+(2*n^4-10*n^3+20*n^2-20*n+8)*x^2)*P(n-3,x) + (2*n^4-8*n^3+8*n^2-6+(2*n^4-8*n^3+8*n^2+2*n-8)*x-(n^2-3*n+3)*x^2)*P(n-2,x) + (4*n^2-10*n+10+(2*n^2-6*n+6)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 12 + 14*x + 3*x^2, P(3,x) = 60 + 100*x + 45*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 12 + 14*x + 3*x^2;
P[3, x_] := 60 + 100*x + 45*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12 + (2*n^6 - 18*n^5 + 64*n^4 - 118*n^3 + 126*n^2 - 80*n + 24)*x + (n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2)*P[n - 4, x] - (4*n^4 - 18*n^3 + 32*n^2 - 30*n + 12 + (6*n^4 - 28*n^3 + 52*n^2 - 50*n + 20)*x + (2*n^4 - 10*n^3 + 20*n^2 - 20*n + 8)*x^2)*P[n - 3, x] + (2*n^4 - 8*n^3 + 8*n^2 - 6 + (2*n^4 - 8*n^3 + 8*n^2 + 2*n - 8)*x - (n^2 - 3*n + 3)*x^2)*P[n - 2, x] + (4*n^2 - 10*n + 10 + (2*n^2 - 6*n + 6)*x)*P[n - 1, x])/(n^2 - 3*n + 3)]];
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] = 3 + 2*x; v[3] = 12 + 14*x + 3*x^2; v[4] = 60 + 100*x + 45*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12 + (2*n^6 - 18*n^5 + 64*n^4 - 118*n^3 + 126*n^2 - 80*n + 24)*x + (n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2)*v[i - 4] - (4*n^4 - 18*n^3 + 32*n^2 - 30*n + 12 + (6*n^4 - 28*n^3 + 52*n^2 - 50*n + 20)*x + (2*n^4 - 10*n^3 + 20*n^2 - 20*n + 8)*x^2)*v[i - 3] + (2*n^4 - 8*n^3 + 8*n^2 - 6 + (2*n^4 - 8*n^3 + 8*n^2 + 2*n - 8)*x - (n^2 - 3*n + 3)*x^2)*v[i - 2] + (4*n^2 - 10*n + 10 + (2*n^2 - 6*n + 6)*x)*v[i - 1])/(n^2 - 3*n + 3)); 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.