Real-rooted atlas

A062145 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + n + n^{2}\right)\,P_{n}(x) = \left(-4\,x + 2\,n - 2\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-x - x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2516
32112134
48484211190
53304802163211059
61287247516504504515908
750051201210725440082560133028
819448560566306335035100101386771185076
975582254592346528244608955502038421849611039525
1029393011337301813968155937677968822932038220327611715851626
11114406649742009157050930240056977202159136499800672004725140133006438
12445740021573816449454505276205038372400179071205372136100980011220066001651186519138
13173838609271392021573816028765088024270543013507084850139936122791681918620179520897619211055789511
14678639153954828151016955810151915621014607271259465511774206894121280359082618916334641752771341193422115985405000
15265182525167663790047225300857816601520844472128562553491003247969725118782892830314384152720668599098541496015561252133979107336
Technical details
Node id
n-a50ebe9c8bf98776 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c43ba15ffe3d8a71

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      5,       1;
   3:     21,      12,       1;
   4:     84,      84,      21,       1;
   5:    330,     480,     216,      32,      1;
   6:   1287,    2475,    1650,     450,     45,      1;
   7:   5005,   12012,   10725,    4400,    825,     60,     1;
   8:  19448,   56056,   63063,   35035,  10010,   1386,    77,    1;
   9:  75582,  254592,  346528,  244608,  95550,  20384,  2184,   96,   1;
  10: 293930, 1133730, 1813968, 1559376, 779688, 229320, 38220, 3276, 117, 1;
%F Formula
The row polynomials satisfy (n^2+n-2)*P(n,x) = (4*n^2+2*n+(3*n^2-2*n-4)*x)*P(n-1,x) - ((2*n+1)*x+(2*n+1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 2*n + (3*n^2 - 2*n - 4)*x)*P[n - 1, x] - ((2*n + 1)*x + (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 + n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^2 + 2*n + (3*n^2 - 2*n - 4)*x)*v[i - 1] - ((2*n + 1)*x + (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 + n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.