Real-rooted atlas

A145596 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(16\,n + 12\,n^{2} + 8\,n^{2} x + 2\,n^{3} + 6\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(24 + 16\,x + 4\,x^{2} + 52\,n + 60\,n x + 15\,n x^{2} + 36\,n^{2} + 70\,n^{2} x + 23\,n^{2} x^{2} + 8\,n^{3} + 30\,n^{3} x + 15\,n^{3} x^{2} + 4\,n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x - 12\,x^{2} - 4\,x^{3} - 12\,n x - 23\,n x^{2} - 11\,n x^{3} - 4\,n^{2} x - 13\,n^{2} x^{2} - 9\,n^{2} x^{3} - 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
21428951
3481449616304
4165660675240251765
55722860396022005003610128
6200212012210211601657759244957799
77072495041048321019205096013104156864329024

All 15 rows

Technical details
Node id
n-26531901527cd69e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b5eb3a902a460da6

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,      4;
  2:    14,     28,       9;
  3:    48,    144,      96,      16;
  4:   165,    660,     675,     240,      25;
  5:   572,   2860,    3960,    2200,     500,      36;
  6:  2002,  12012,   21021,   16016,    5775,     924,     49;
  7:  7072,  49504,  104832,  101920,   50960,   13104,   1568,    64;
  8: 25194, 201552,  501228,  594048,  382200,  137592,  26754,  2496,   81;
  9: 90440, 813960, 2325600, 3255840, 2570400, 1199520, 329280, 50400, 3780, 100;
%F Formula
The row polynomials satisfy (2*n^3+12*n^2+16*n+(n^4+6*n^3+8*n^2)*x)*P(n,x) = (8*n^3+36*n^2+52*n+24+(4*n^4+30*n^3+70*n^2+60*n+16)*x+(3*n^4+15*n^3+23*n^2+15*n+4)*x^2)*P(n-1,x) - ((4*n^2+12*n+8)*x+(2*n^3+13*n^2+23*n+12)*x^2+(2*n^3+9*n^2+11*n+4)*x^3)*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[((8*n^3 + 36*n^2 + 52*n + 24 + (4*n^4 + 30*n^3 + 70*n^2 + 60*n + 16)*x + (3*n^4 + 15*n^3 + 23*n^2 + 15*n + 4)*x^2)*P[n - 1, x] - ((4*n^2 + 12*n + 8)*x + (2*n^3 + 13*n^2 + 23*n + 12)*x^2 + (2*n^3 + 9*n^2 + 11*n + 4)*x^3)*D[P[n - 1, x], x])/(2*n^3 + 12*n^2 + 16*n + (n^4 + 6*n^3 + 8*n^2)*x)]];
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] = ((8*n^3 + 36*n^2 + 52*n + 24 + (4*n^4 + 30*n^3 + 70*n^2 + 60*n + 16)*x + (3*n^4 + 15*n^3 + 23*n^2 + 15*n + 4)*x^2)*v[i - 1] - ((4*n^2 + 12*n + 8)*x + (2*n^3 + 13*n^2 + 23*n + 12)*x^2 + (2*n^3 + 9*n^2 + 11*n + 4)*x^3)*deriv(v[i - 1], x))/(2*n^3 + 12*n^2 + 16*n + (n^4 + 6*n^3 + 8*n^2)*x)); 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.