Real-rooted atlas

A081579 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-10 + x^{2} + 10\,n - 5\,n x - n x^{2} + 5\,n^{2} x\right)\,P_{n}(x) = \left(-140 - 170\,x - 36\,x^{2} - 2\,x^{3} + 210\,n + 225\,n x + 24\,n x^{2} - 50\,n^{2} - 45\,n^{2} x\right)\,P_{n-1}(x) + \left(-100 - 220\,x - 145\,x^{2} - 26\,x^{3} - x^{4} + 100\,n + 200\,n x + 105\,n x^{2} + 5\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
28210
33628367
4160200604424
5640112060010452469
62496560043201344160613926
794082579226160120642540228776199
834816112768141888874242752042963088409028
912672047360071232055180823480054720672040092161097
1045568019302403379200315980817065605404809856099205041011280962
111622016768000015344640168314881106112044747521109920164800140046201158303371
12572620829970432673075208478412865786880325762561035686420883202600641908074812298876512
13200744961150894082870292484084756483659558402150891528404121621746176366710439184025256888131521586285
14699596804360192001195745280189776691219295744001315407360612469760195238400422363526090400568480326401040147701109918
Technical details
Node id
n-3264b366f920ab39 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b19535e309051ba4

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      0;
  1:      1;
  2:      8,      2;
  3:     36,     28,      3;
  4:    160,    200,     60,      4;
  5:    640,   1120,    600,    104,      5;
  6:   2496,   5600,   4320,   1344,    160,     6;
  7:   9408,  25792,  26160,  12064,   2540,   228,    7;
  8:  34816, 112768, 141888,  87424,  27520,  4296,  308,   8;
  9: 126720, 473600, 712320, 551808, 234800, 54720, 6720, 400, 9;
%F Formula
The row polynomials satisfy (10*n-10+(5*n^2-5*n)*x-(n-1)*x^2)*P(n,x) = -(50*n^2-210*n+140+(45*n^2-225*n+170)*x-(24*n-36)*x^2+2*x^3)*P(n-1,x) + (100*n-100+(200*n-220)*x+(105*n-145)*x^2+(5*n-26)*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(50*n^2 - 210*n + 140 + (45*n^2 - 225*n + 170)*x - (24*n - 36)*x^2 + 2*x^3)*P[n - 1, x] + (100*n - 100 + (200*n - 220)*x + (105*n - 145)*x^2 + (5*n - 26)*x^3 - x^4)*D[P[n - 1, x], x])/(10*n - 10 + (5*n^2 - 5*n)*x - (n - 1)*x^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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(50*n^2 - 210*n + 140 + (45*n^2 - 225*n + 170)*x - (24*n - 36)*x^2 + 2*x^3)*v[i - 1] + (100*n - 100 + (200*n - 220)*x + (105*n - 145)*x^2 + (5*n - 26)*x^3 - x^4)*deriv(v[i - 1], x))/(10*n - 10 + (5*n^2 - 5*n)*x - (n - 1)*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

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.