Real-rooted atlas

A108838 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3 - 3\,x + 3\,x^{2} + 7\,n + 2\,n x + 4\,n x^{2} + 8\,n^{2} + 7\,n^{2} x + n^{2} x^{2} + 5\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(6 - 6\,x + 6\,x^{2} + 24\,n + 8\,n x + 10\,n x^{2} + 5\,n x^{3} + 32\,n^{2} + 29\,n^{2} x + 14\,n^{2} x^{2} + 3\,n^{2} x^{3} + 18\,n^{3} + 18\,n^{3} x + 6\,n^{3} x^{2} + 4\,n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(-6\,x - 9\,x^{2} - 6\,x^{3} - 3\,x^{4} - 12\,n x - 20\,n x^{2} - 10\,n x^{3} - 2\,n x^{4} - 8\,n^{2} x - 12\,n^{2} x^{2} - 4\,n^{2} x^{3} - 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
022
1729
22616244
39898272225
43725282404021182
51419264017254805526321
654341258410890440085072234232
72087858058630633403496251386912187137
8804442616643436162344168918018928212811221030490
931072611589241790100148512071853620638834398312013525707449
10120285250646409011700883728052264801919232435120588004410160231760676
114665412219006924416379050097300351747001586059246181528527209570060501872177435297
1218126760939025602118156482732683362227731001197218884297708810232640157528814960080962162994551222
1370538355399828780998123295144479192013438689558413788243612441691064194562120075127713402260831060824725590402785
1427487737016926058804634843850744438240077907529705581696120280047167498985744024477453041537496467867433196813650280231500824304
Technical details
Node id
n-f17451778b454895 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-51d076129291b50f

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:      2;
  1:      7,       2;
  2:     26,      16,       2;
  3:     98,      98,      27,       2;
  4:    372,     528,     240,      40,      2;
  5:   1419,    2640,    1725,     480,     55,      2;
  6:   5434,   12584,   10890,    4400,    850,     72,     2;
  7:  20878,   58058,   63063,   34034,   9625,   1386,    91,    2;
  8:  80444,  261664,  343616,  234416,  89180,  18928,  2128,  112,   2;
  9: 310726, 1158924, 1790100, 1485120, 718536, 206388, 34398, 3120, 135, 2;
%F Formula
The row polynomials satisfy (n^4+5*n^3+8*n^2+7*n+3+(2*n^3+7*n^2+2*n-3)*x+(n^2+4*n+3)*x^2)*P(n,x) = (4*n^4+18*n^3+32*n^2+24*n+6+(3*n^4+18*n^3+29*n^2+8*n-6)*x+(6*n^3+14*n^2+10*n+6)*x^2+(3*n^2+5*n)*x^3)*P(n-1,x) - ((2*n^3+8*n^2+12*n+6)*x+(2*n^3+12*n^2+20*n+9)*x^2+(4*n^2+10*n+6)*x^3+(2*n+3)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^4 + 18*n^3 + 32*n^2 + 24*n + 6 + (3*n^4 + 18*n^3 + 29*n^2 + 8*n - 6)*x + (6*n^3 + 14*n^2 + 10*n + 6)*x^2 + (3*n^2 + 5*n)*x^3)*P[n - 1, x] - ((2*n^3 + 8*n^2 + 12*n + 6)*x + (2*n^3 + 12*n^2 + 20*n + 9)*x^2 + (4*n^2 + 10*n + 6)*x^3 + (2*n + 3)*x^4)*D[P[n - 1, x], x])/(n^4 + 5*n^3 + 8*n^2 + 7*n + 3 + (2*n^3 + 7*n^2 + 2*n - 3)*x + (n^2 + 4*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^4 + 18*n^3 + 32*n^2 + 24*n + 6 + (3*n^4 + 18*n^3 + 29*n^2 + 8*n - 6)*x + (6*n^3 + 14*n^2 + 10*n + 6)*x^2 + (3*n^2 + 5*n)*x^3)*v[i - 1] - ((2*n^3 + 8*n^2 + 12*n + 6)*x + (2*n^3 + 12*n^2 + 20*n + 9)*x^2 + (4*n^2 + 10*n + 6)*x^3 + (2*n + 3)*x^4)*deriv(v[i - 1], x))/(n^4 + 5*n^3 + 8*n^2 + 7*n + 3 + (2*n^3 + 7*n^2 + 2*n - 3)*x + (n^2 + 4*n + 3)*x^2)); 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

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023