Real-rooted atlas

A113413 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(32 + 32\,x + 10\,x^{2} + 40\,n x + 20\,n x^{2} - 2\,n x^{3} - n x^{4} + 16\,n^{2} x^{2} + 4\,n^{2} x^{3} - n^{2} x^{4} + 2\,n^{3} x^{3}\right)\,P_{n}(x) = \left(160 + 288\,x + 154\,x^{2} + 12\,x^{3} - 10\,x^{4} - 2\,x^{5} - 64\,n + 40\,n x + 160\,n x^{2} + 74\,n x^{3} + 3\,n x^{4} - 2\,n x^{5} - 48\,n^{2} x - 24\,n^{2} x^{2} + 24\,n^{2} x^{3} + 9\,n^{2} x^{4} - 8\,n^{3} x^{2} - 6\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(64 + 160\,x + 120\,x^{2} + 4\,x^{3} - 30\,x^{4} - 11\,x^{5} - x^{6} + 64\,n x + 144\,n x^{2} + 96\,n x^{3} + 10\,n x^{4} - 7\,n x^{5} - n x^{6} + 16\,n^{2} x^{2} + 32\,n^{2} x^{3} + 18\,n^{2} x^{4} + 2\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2712322
3174627494
441152144485390
5994666123287561586
623913642295171262510876350
757738707947762838751062147825114

All 15 rows

Technical details
Node id
n-577f3746211ab7ec · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9c2a3d20a0c057fc

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:    7,    12,     3;
  3:   17,    46,    27,      4;
  4:   41,   152,   144,     48,     5;
  5:   99,   466,   612,    328,    75,     6;
  6:  239,  1364,  2295,   1712,   625,   108,     7;
  7:  577,  3870,  7947,   7628,  3875,  1062,   147,    8;
  8: 1393, 10736, 26040,  30624, 20050,  7632,  1666,  192,   9;
  9: 3363, 29282, 81936, 114192, 91790, 45036, 13622, 2464, 243, 10;
%F Formula
The row polynomials satisfy (32+(40*n+32)*x+(16*n^2+20*n+10)*x^2+(2*n^3+4*n^2-2*n)*x^3-(n^2+n)*x^4)*P(n,x) = -(64*n-160+(48*n^2-40*n-288)*x+(8*n^3+24*n^2-160*n-154)*x^2+(6*n^3-24*n^2-74*n-12)*x^3-(9*n^2+3*n-10)*x^4+(2*n+2)*x^5)*P(n-1,x) + (64+(64*n+160)*x+(16*n^2+144*n+120)*x^2+(32*n^2+96*n+4)*x^3+(18*n^2+10*n-30)*x^4+(2*n^2-7*n-11)*x^5-(n+1)*x^6)*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[(-(64*n - 160 + (48*n^2 - 40*n - 288)*x + (8*n^3 + 24*n^2 - 160*n - 154)*x^2 + (6*n^3 - 24*n^2 - 74*n - 12)*x^3 - (9*n^2 + 3*n - 10)*x^4 + (2*n + 2)*x^5)*P[n - 1, x] + (64 + (64*n + 160)*x + (16*n^2 + 144*n + 120)*x^2 + (32*n^2 + 96*n + 4)*x^3 + (18*n^2 + 10*n - 30)*x^4 + (2*n^2 - 7*n - 11)*x^5 - (n + 1)*x^6)*D[P[n - 1, x], x])/(32 + (40*n + 32)*x + (16*n^2 + 20*n + 10)*x^2 + (2*n^3 + 4*n^2 - 2*n)*x^3 - (n^2 + n)*x^4)]];
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] = (-(64*n - 160 + (48*n^2 - 40*n - 288)*x + (8*n^3 + 24*n^2 - 160*n - 154)*x^2 + (6*n^3 - 24*n^2 - 74*n - 12)*x^3 - (9*n^2 + 3*n - 10)*x^4 + (2*n + 2)*x^5)*v[i - 1] + (64 + (64*n + 160)*x + (16*n^2 + 144*n + 120)*x^2 + (32*n^2 + 96*n + 4)*x^3 + (18*n^2 + 10*n - 30)*x^4 + (2*n^2 - 7*n - 11)*x^5 - (n + 1)*x^6)*deriv(v[i - 1], x))/(32 + (40*n + 32)*x + (16*n^2 + 20*n + 10)*x^2 + (2*n^3 + 4*n^2 - 2*n)*x^3 - (n^2 + n)*x^4)); 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.