Real-rooted atlas

A059473 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3\,n x + 6\,n x^{2} + 2\,n x^{3} + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n}(x) = \left(6 + 33\,x + 64\,x^{2} + 52\,x^{3} + 16\,x^{4} - 6\,n - 12\,n x + 17\,n x^{2} + 48\,n x^{3} + 24\,n x^{4} - 9\,n^{2} x - 18\,n^{2} x^{2} + 8\,n^{2} x^{4} - 3\,n^{3} x^{2} - 4\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(3 + 21\,x + 56\,x^{2} + 70\,x^{3} + 40\,x^{4} + 8\,x^{5} + 6\,n x + 33\,n x^{2} + 64\,n x^{3} + 50\,n x^{4} + 12\,n x^{5} + 3\,n^{2} x^{2} + 12\,n^{2} x^{3} + 14\,n^{2} x^{4} + 4\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2187248138
3804806722561488
4356284860964864128015444
51584158404569656064307206144156048
6704884576307104510208430080178176286721545864
731360439040192230440345604613120293683297484813107215083136

All 15 rows

Technical details
Node id
n-4ff65f330739bb6a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c65e5f7d71274589

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,        8;
  2:     18,       72,       48;
  3:     80,      480,      672,       256;
  4:    356,     2848,     6096,      4864,      1280;
  5:   1584,    15840,    45696,     56064,     30720,      6144;
  6:   7048,    84576,   307104,    510208,    430080,    178176,     28672;
  7:  31360,   439040,  1922304,   4034560,   4613120,   2936832,    974848,    131072;
  8: 139536,  2232576, 11445120,  29048832,  42036480,  36311040,  18493440,   5111808,   589824;
  9: 620864, 11175552, 65654784, 195557376, 342789120, 375035904, 259080192, 109707264, 25952256, 2621440;
%F Formula
The row polynomials satisfy (3*n*x+(3*n^2+6*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(6*n-6+(9*n^2+12*n-33)*x+(3*n^3+18*n^2-17*n-64)*x^2+(4*n^3-48*n-52)*x^3-(8*n^2+24*n+16)*x^4)*P(n-1,x) + (3+(6*n+21)*x+(3*n^2+33*n+56)*x^2+(12*n^2+64*n+70)*x^3+(14*n^2+50*n+40)*x^4+(4*n^2+12*n+8)*x^5)*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[(-(6*n - 6 + (9*n^2 + 12*n - 33)*x + (3*n^3 + 18*n^2 - 17*n - 64)*x^2 + (4*n^3 - 48*n - 52)*x^3 - (8*n^2 + 24*n + 16)*x^4)*P[n - 1, x] + (3 + (6*n + 21)*x + (3*n^2 + 33*n + 56)*x^2 + (12*n^2 + 64*n + 70)*x^3 + (14*n^2 + 50*n + 40)*x^4 + (4*n^2 + 12*n + 8)*x^5)*D[P[n - 1, x], x])/(3*n*x + (3*n^2 + 6*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)]];
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] = (-(6*n - 6 + (9*n^2 + 12*n - 33)*x + (3*n^3 + 18*n^2 - 17*n - 64)*x^2 + (4*n^3 - 48*n - 52)*x^3 - (8*n^2 + 24*n + 16)*x^4)*v[i - 1] + (3 + (6*n + 21)*x + (3*n^2 + 33*n + 56)*x^2 + (12*n^2 + 64*n + 70)*x^3 + (14*n^2 + 50*n + 40)*x^4 + (4*n^2 + 12*n + 8)*x^5)*deriv(v[i - 1], x))/(3*n*x + (3*n^2 + 6*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^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.