Real-rooted atlas

A008292 → γ (shift +1) → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3 + 8\,x + 2\,n\right)\,P_{n}(x) = \left(-45 - 147\,x - 80\,x^{2} + 144\,x^{3} + 128\,x^{4} + 52\,n + 202\,n x + 218\,n x^{2} + 8\,n x^{3} - 64\,n x^{4} + 8\,n^{2} - 24\,n^{2} x - 80\,n^{2} x^{2} - 48\,n^{2} x^{3} - 8\,n^{3} - 16\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(45\,x + 237\,x^{2} + 464\,x^{3} + 400\,x^{4} + 128\,x^{5} + 30\,n + 166\,n x + 336\,n x^{2} + 296\,n x^{3} + 96\,n x^{4} + 12\,n^{2} + 40\,n^{2} x + 44\,n^{2} x^{2} + 16\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-24 - 79\,x - 36\,x^{2} + 32\,x^{3} + 34\,n x + 40\,n x^{2} + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-9 - 45\,x - 68\,x^{2} - 32\,x^{3} - 6\,n - 14\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2527
317825
4938616195
55135961361245
634775442235227211543
72556948240268803968104657
82127334826703602889779279361161419
9192939350657404605768164531217689613423109
10191760215759289063034272300648485803264353792176025087
11206004465695177640884667744518101056133794688111841282448929721
122385999261895558858213062714480922233120031740902404672442882236825637290336307
13295925076811222526497322009024283601655540121607040949120014049627648951878656603712595437
14391601791701176694414136232354867953443061924258320157245353856041784819148849078255616190375731210497240729703
Technical details
Node id
n-582f3fb7e2f120e7 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-6e5ef435951ceba9

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:       1;
  2:       5,       2;
  3:      17,       8;
  4:      93,      86,      16;
  5:     513,     596,     136;
  6:    3477,    5442,    2352,     272;
  7:   25569,   48240,   26880,    3968;
  8:  212733,  482670,  360288,   97792,   7936;
  9: 1929393, 5065740, 4605768, 1645312, 176896;
%F Formula
The row polynomials satisfy (2*n+3+8*x)*P(n,x) = -(8*n^3-8*n^2-52*n+45+(16*n^3+24*n^2-202*n+147)*x+(8*n^3+80*n^2-218*n+80)*x^2+(48*n^2-8*n-144)*x^3+(64*n-128)*x^4)*P(n-2,x) + (12*n^2+30*n+(40*n^2+166*n+45)*x+(44*n^2+336*n+237)*x^2+(16*n^2+296*n+464)*x^3+(96*n+400)*x^4+128*x^5)*d/dx P(n-2,x) + (8*n^2-24+(8*n^2+34*n-79)*x+(40*n-36)*x^2+32*x^3)*P(n-1,x) - (6*n+9+(14*n+45)*x+(8*n+68)*x^2+32*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 8*n^2 - 52*n + 45 + (16*n^3 + 24*n^2 - 202*n + 147)*x + (8*n^3 + 80*n^2 - 218*n + 80)*x^2 + (48*n^2 - 8*n - 144)*x^3 + (64*n - 128)*x^4)*P[n - 2, x] + (12*n^2 + 30*n + (40*n^2 + 166*n + 45)*x + (44*n^2 + 336*n + 237)*x^2 + (16*n^2 + 296*n + 464)*x^3 + (96*n + 400)*x^4 + 128*x^5)*D[P[n - 2, x], x] + (8*n^2 - 24 + (8*n^2 + 34*n - 79)*x + (40*n - 36)*x^2 + 32*x^3)*P[n - 1, x] - (6*n + 9 + (14*n + 45)*x + (8*n + 68)*x^2 + 32*x^3)*D[P[n - 1, x], x])/(2*n + 3 + 8*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 8*n^2 - 52*n + 45 + (16*n^3 + 24*n^2 - 202*n + 147)*x + (8*n^3 + 80*n^2 - 218*n + 80)*x^2 + (48*n^2 - 8*n - 144)*x^3 + (64*n - 128)*x^4)*v[i - 2] + (12*n^2 + 30*n + (40*n^2 + 166*n + 45)*x + (44*n^2 + 336*n + 237)*x^2 + (16*n^2 + 296*n + 464)*x^3 + (96*n + 400)*x^4 + 128*x^5)*deriv(v[i - 2], x) + (8*n^2 - 24 + (8*n^2 + 34*n - 79)*x + (40*n - 36)*x^2 + 32*x^3)*v[i - 1] - (6*n + 9 + (14*n + 45)*x + (8*n + 68)*x^2 + 32*x^3)*deriv(v[i - 1], x))/(2*n + 3 + 8*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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.