Real-rooted atlas

A145596 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-8 + 4\,x + 6\,n - 3\,n x + 2\,n^{2} - 5\,n^{2} x + 3\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(16 - 16\,x + 4\,x^{2} + 12\,n + 10\,n x - 6\,n x^{2} + 2\,n^{2} + 19\,n^{2} x + n^{2} x^{2} + 6\,n^{3} x + 8\,n^{3} x^{2} + n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(6\,x - 8\,x^{2} + 2\,x^{3} + 4\,n x - 4\,n x^{3} + 6\,n^{2} x^{2} - 6\,n^{2} x^{3} + 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
281220
3208036136
44030036080780
570840189011201504070
6112196070567840280025220020
71684032211683763225200604839294640
82407560544321411201512006804011760576434928
933013200124740443520693000498960161700211208101957380
104402178026136012196802613600274428014229603484803564011008669320
115723432050965230201608494200122316489249240362419269498057200145237917616
127285205293693668708642453880046378332480960482748345684324241301300880881872164180900
139107644016396381457456064414350154594440210420210164900736737837101821820023123101310402366705068910
14112010920027518402914912015615600046378332080160080082450368050594544018218200036996960393120018928029403007302900
Technical details
Node id
n-502f8df33d851e0c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b32537e26b122a9d

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:   0;
  1:   2;
  2:   8,    12;
  3:  20,    80,     36;
  4:  40,   300,    360,     80;
  5:  70,   840,   1890,   1120,    150;
  6: 112,  1960,   7056,   7840,   2800,    252;
  7: 168,  4032,  21168,  37632,  25200,   6048,    392;
  8: 240,  7560,  54432, 141120, 151200,  68040,  11760,   576;
  9: 330, 13200, 124740, 443520, 693000, 498960, 161700, 21120, 810;
%F Formula
The row polynomials satisfy (2*n^2+6*n-8+(n^4+3*n^3-5*n^2-3*n+4)*x)*P(n,x) = (2*n^2+12*n+16+(n^4+6*n^3+19*n^2+10*n-16)*x+(3*n^4+8*n^3+n^2-6*n+4)*x^2)*P(n-1,x) + ((4*n+6)*x+(2*n^3+6*n^2-8)*x^2-(2*n^3+6*n^2+4*n-2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 12*n + 16 + (n^4 + 6*n^3 + 19*n^2 + 10*n - 16)*x + (3*n^4 + 8*n^3 + n^2 - 6*n + 4)*x^2)*P[n - 1, x] + ((4*n + 6)*x + (2*n^3 + 6*n^2 - 8)*x^2 - (2*n^3 + 6*n^2 + 4*n - 2)*x^3)*D[P[n - 1, x], x])/(2*n^2 + 6*n - 8 + (n^4 + 3*n^3 - 5*n^2 - 3*n + 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 12*n + 16 + (n^4 + 6*n^3 + 19*n^2 + 10*n - 16)*x + (3*n^4 + 8*n^3 + n^2 - 6*n + 4)*x^2)*v[i - 1] + ((4*n + 6)*x + (2*n^3 + 6*n^2 - 8)*x^2 - (2*n^3 + 6*n^2 + 4*n - 2)*x^3)*deriv(v[i - 1], x))/(2*n^2 + 6*n - 8 + (n^4 + 3*n^3 - 5*n^2 - 3*n + 4)*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.