Real-rooted atlas

A081577 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6\,n x^{2} + 7\,n x^{3} + 2\,n x^{4} + 3\,n^{2} + 3\,n^{2} x + 10\,n^{2} x^{2} + 2\,n^{2} x^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(-15 - 51\,x - 17\,x^{2} + 65\,x^{3} + 32\,x^{4} + 4\,x^{5} + 15\,n + 75\,n x + 83\,n x^{2} - 11\,n x^{3} - 6\,n x^{4} + 21\,n^{2} + 21\,n^{2} x - 38\,n^{2} x^{2} - 10\,n^{2} x^{3} - 9\,n^{3} - 15\,n^{3} x\right)\,P_{n-1}(x) + \left(15\,x + 93\,x^{2} + 203\,x^{3} + 177\,x^{4} + 48\,x^{5} + 4\,x^{6} + 9\,n + 66\,n x + 165\,n x^{2} + 160\,n x^{3} + 46\,n x^{4} + 4\,n x^{5} + 9\,n^{2} + 36\,n^{2} x + 39\,n^{2} x^{2} + 6\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
21820442
364128648264
4220640568176161620
5720277636802016448329672
62296110561971216192630410886456712
77168414729344010412860672181762560128327744

All 15 rows

Technical details
Node id
n-2b3e71234dd2d442 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ff7594681990bf9f

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:     1;
  1:     4,      2;
  2:    18,     20,       4;
  3:    64,    128,      64,       8;
  4:   220,    640,     568,     176,      16;
  5:   720,   2776,    3680,    2016,     448,      32;
  6:  2296,  11056,   19712,   16192,    6304,    1088,     64;
  7:  7168,  41472,   93440,  104128,   60672,   18176,   2560,    128;
  8: 22032, 148992,  406848,  577152,  457536,  204288,  49536,   5888,   256;
  9: 66880, 517920, 1664256, 2879232, 2917632, 1772160, 637440, 129536, 13312, 512;
%F Formula
The row polynomials satisfy (3*n^2+(3*n^3+3*n^2)*x+(10*n^2-6*n)*x^2+(2*n^2+7*n)*x^3+2*n*x^4)*P(n,x) = -(9*n^3-21*n^2-15*n+15+(15*n^3-21*n^2-75*n+51)*x+(38*n^2-83*n+17)*x^2+(10*n^2+11*n-65)*x^3+(6*n-32)*x^4-4*x^5)*P(n-1,x) + (9*n^2+9*n+(36*n^2+66*n+15)*x+(39*n^2+165*n+93)*x^2+(6*n^2+160*n+203)*x^3+(46*n+177)*x^4+(4*n+48)*x^5+4*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^3 - 21*n^2 - 15*n + 15 + (15*n^3 - 21*n^2 - 75*n + 51)*x + (38*n^2 - 83*n + 17)*x^2 + (10*n^2 + 11*n - 65)*x^3 + (6*n - 32)*x^4 - 4*x^5)*P[n - 1, x] + (9*n^2 + 9*n + (36*n^2 + 66*n + 15)*x + (39*n^2 + 165*n + 93)*x^2 + (6*n^2 + 160*n + 203)*x^3 + (46*n + 177)*x^4 + (4*n + 48)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(3*n^2 + (3*n^3 + 3*n^2)*x + (10*n^2 - 6*n)*x^2 + (2*n^2 + 7*n)*x^3 + 2*n*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(9*n^3 - 21*n^2 - 15*n + 15 + (15*n^3 - 21*n^2 - 75*n + 51)*x + (38*n^2 - 83*n + 17)*x^2 + (10*n^2 + 11*n - 65)*x^3 + (6*n - 32)*x^4 - 4*x^5)*v[i - 1] + (9*n^2 + 9*n + (36*n^2 + 66*n + 15)*x + (39*n^2 + 165*n + 93)*x^2 + (6*n^2 + 160*n + 203)*x^3 + (46*n + 177)*x^4 + (4*n + 48)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(3*n^2 + (3*n^3 + 3*n^2)*x + (10*n^2 - 6*n)*x^2 + (2*n^2 + 7*n)*x^3 + 2*n*x^4)); 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.