Real-rooted atlas

A056242 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-10 - 10\,x + 4\,x^{2} + 4\,x^{3} + 15\,n + 18\,n x - 4\,n x^{3} - 6\,n^{2} - 8\,n^{2} x - 4\,n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(-10 - 22\,x - 24\,x^{2} - 20\,x^{3} - 8\,x^{4} + 7\,n + 4\,n x - 16\,n x^{2} - 12\,n x^{3} + 2\,n^{2} - 2\,n^{2} x - 4\,n^{2} x^{2} + n^{3}\right)\,P_{n-1}(x) + \left(-24 - 92\,x - 144\,x^{2} - 116\,x^{3} - 48\,x^{4} - 8\,x^{5} - 20\,n - 62\,n x - 74\,n x^{2} - 40\,n x^{3} - 8\,n x^{4} + 4\,n^{2} + 6\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2527
32321448
499145648316
5408830593172162019
616324266425720024323212621
76388204402630517305605610406477598
8245969320814720912505761248170562432128470934
9934884096807677288012605070091976084568055682562828277
10351664175016037979844707476366129318439865965601178561254451216840035
1113115367309104180334482590292823936443148211016196172171294429542427904102499548028
124856432299652968286100013540360014506167910674232154867104196002004729600723712614402048584874432
13178734081209748803706440486791781288280839127065902184316708331895996365911510412654336174003213414440963418262775
1465436544482093920162121358432930164644503732096437449017430993756251618027874620255776171542208329935364119040290816819219886595849
152384809601899943680695812664015518971280235339594202564636778420686581445125252854895706894416194045116848214771284184320962355262668816384115231660938
Technical details
Node id
n-50e7e81aaf481dd2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4741a0ab4b883c71

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,       2;
   3:     23,      21,       4;
   4:     99,     145,      64,       8;
   5:    408,     830,     593,     172,      16;
   6:   1632,    4266,    4257,    2002,     432,      32;
   7:   6388,   20440,   26305,   17305,    6056,    1040,     64;
   8:  24596,   93208,  147209,  125057,   61248,   17056,   2432,    128;
   9:  93488,  409680,  767728,  801260,  507009,  197608,  45680,   5568,   256;
  10: 351664, 1750160, 3797984, 4707476, 3661293, 1843986, 596560, 117856, 12544, 512;
%F Formula
The row polynomials satisfy (n^3-6*n^2+15*n-10-(8*n^2-18*n+10)*x-(4*n^2-4)*x^2-(4*n-4)*x^3)*P(n,x) = (n^3+2*n^2+7*n-10-(2*n^2-4*n+22)*x-(4*n^2+16*n+24)*x^2-(12*n+20)*x^3-8*x^4)*P(n-1,x) + (4*n^2-20*n-24+(6*n^2-62*n-92)*x+(2*n^2-74*n-144)*x^2-(40*n+116)*x^3-(8*n+48)*x^4-8*x^5)*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[((n^3 + 2*n^2 + 7*n - 10 - (2*n^2 - 4*n + 22)*x - (4*n^2 + 16*n + 24)*x^2 - (12*n + 20)*x^3 - 8*x^4)*P[n - 1, x] + (4*n^2 - 20*n - 24 + (6*n^2 - 62*n - 92)*x + (2*n^2 - 74*n - 144)*x^2 - (40*n + 116)*x^3 - (8*n + 48)*x^4 - 8*x^5)*D[P[n - 1, x], x])/(n^3 - 6*n^2 + 15*n - 10 - (8*n^2 - 18*n + 10)*x - (4*n^2 - 4)*x^2 - (4*n - 4)*x^3)]];
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] = ((n^3 + 2*n^2 + 7*n - 10 - (2*n^2 - 4*n + 22)*x - (4*n^2 + 16*n + 24)*x^2 - (12*n + 20)*x^3 - 8*x^4)*v[i - 1] + (4*n^2 - 20*n - 24 + (6*n^2 - 62*n - 92)*x + (2*n^2 - 74*n - 144)*x^2 - (40*n + 116)*x^3 - (8*n + 48)*x^4 - 8*x^5)*deriv(v[i - 1], x))/(n^3 - 6*n^2 + 15*n - 10 - (8*n^2 - 18*n + 10)*x - (4*n^2 - 4)*x^2 - (4*n - 4)*x^3)); 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

References

  1. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.