Real-rooted atlas

A102365 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(3 + 6\,x + 6\,x^{2} - 6\,n - 14\,n x - 11\,n x^{2} + 4\,n^{2} + 10\,n^{2} x + 6\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 2\,x + x^{2} + 2\,x^{3} + 2\,n + 3\,n x - 2\,n x^{2} - 3\,n x^{3} - n^{2} - n^{2} x + n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 8\,x - 4\,n - 8\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2 + 2\,x^{2} + n - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
211
3729
45136390
543149212341050
6420866462964352514175
7468249453061680143409156218295
85861411441488124543544734061125224473783780
9816123723670236255243271257346027980452402885299872972900
10125183235418330806541501128341121396107766015158652908940541219291550674125
11209807686179379393521199984626892374099443823163085826478604839579583199512275311036010099125
1238154915956161216501980278872566300254139931860130837687050379482652085851017270422422200111288156130011907454497950
1374838622402834929678093726806010687180717364065798044320305762901627470122544344658036366390149035202046126195378930601350031224666263171550
14157501665757578047484008224017443904044909820890590067928015088923608305567400023466196183893228345402925947051064248445636570962574707012693727229472013719432675836875
Technical details
Node id
n-c545f8e99849a2e4 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ee6650f1ca825d7b

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
  • 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:       0;
  2:       1;
  3:       7,        2;
  4:      51,       36,        3;
  5:     431,      492,      123,        4;
  6:    4208,     6646,     2964,      352,       5;
  7:   46824,    94530,    61680,    14340,     915,      6;
  8:  586141,  1441488,  1245435,   447340,   61125,   2244,    7;
  9: 8161237, 23670236, 25524327, 12573460, 2798045, 240288, 5299, 8;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(n^3-4*n^2+6*n-3+(2*n^3-10*n^2+14*n-6)*x+(n^3-6*n^2+11*n-6)*x^2)*P(n-2,x) - (n^2-2*n+1+(n^2-3*n+2)*x-(n^2-2*n+1)*x^2-(n^2-3*n+2)*x^3)*d/dx P(n-2,x) + (2*n^2-4*n+1+(2*n^2-8*n+8)*x)*P(n-1,x) + (n-2-(n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + 6*n - 3 + (2*n^3 - 10*n^2 + 14*n - 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*P[n - 2, x] - (n^2 - 2*n + 1 + (n^2 - 3*n + 2)*x - (n^2 - 2*n + 1)*x^2 - (n^2 - 3*n + 2)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 4*n + 1 + (2*n^2 - 8*n + 8)*x)*P[n - 1, x] + (n - 2 - (n - 2)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
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] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (-(n^3 - 4*n^2 + 6*n - 3 + (2*n^3 - 10*n^2 + 14*n - 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*v[i - 2] - (n^2 - 2*n + 1 + (n^2 - 3*n + 2)*x - (n^2 - 2*n + 1)*x^2 - (n^2 - 3*n + 2)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 4*n + 1 + (2*n^2 - 8*n + 8)*x)*v[i - 1] + (n - 2 - (n - 2)*x^2)*deriv(v[i - 1], x))/(n - 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

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.