Real-rooted atlas

A081579 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-10\,n x^{2} + 15\,n x^{3} + 2\,n x^{4} + 5\,n^{2} + 5\,n^{2} x + 18\,n^{2} x^{2} + 2\,n^{2} x^{3} + 5\,n^{3} x\right)\,P_{n}(x) = \left(-45 - 145\,x - 27\,x^{2} + 193\,x^{3} + 56\,x^{4} + 4\,x^{5} + 35\,n + 205\,n x + 221\,n x^{2} - 91\,n x^{3} - 14\,n x^{4} + 55\,n^{2} + 35\,n^{2} x - 142\,n^{2} x^{2} - 18\,n^{2} x^{3} - 25\,n^{3} - 45\,n^{3} x\right)\,P_{n-1}(x) + \left(45\,x + 275\,x^{2} + 579\,x^{3} + 457\,x^{4} + 80\,x^{5} + 4\,x^{6} + 25\,n + 190\,n x + 475\,n x^{2} + 428\,n x^{3} + 78\,n x^{4} + 4\,n x^{5} + 25\,n^{2} + 100\,n^{2} x + 105\,n^{2} x^{2} + 10\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
22424452
396184808368
44001120880224162640
515365760704032645763218208
6582427392457603302410496140864123968
721504121984260864262784129536308483328128830976

All 15 rows

Technical details
Node id
n-5e823d9026d2d84c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-800decc186267c05

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:     24,      24,       4;
  3:     96,     184,      80,        8;
  4:    400,    1120,     880,      224,      16;
  5:   1536,    5760,    7040,     3264,     576,      32;
  6:   5824,   27392,   45760,    33024,   10496,    1408,      64;
  7:  21504,  121984,  260864,   262784,  129536,   30848,    3328,    128;
  8:  78336,  520704, 1360640,  1777152, 1230336,  451072,   85248,   7680,   256;
  9: 281600, 2147840, 6645760, 10728448, 9720832, 5004800, 1443840, 225280, 17408, 512;
%F Formula
The row polynomials satisfy (5*n^2+(5*n^3+5*n^2)*x+(18*n^2-10*n)*x^2+(2*n^2+15*n)*x^3+2*n*x^4)*P(n,x) = -(25*n^3-55*n^2-35*n+45+(45*n^3-35*n^2-205*n+145)*x+(142*n^2-221*n+27)*x^2+(18*n^2+91*n-193)*x^3+(14*n-56)*x^4-4*x^5)*P(n-1,x) + (25*n^2+25*n+(100*n^2+190*n+45)*x+(105*n^2+475*n+275)*x^2+(10*n^2+428*n+579)*x^3+(78*n+457)*x^4+(4*n+80)*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[(-(25*n^3 - 55*n^2 - 35*n + 45 + (45*n^3 - 35*n^2 - 205*n + 145)*x + (142*n^2 - 221*n + 27)*x^2 + (18*n^2 + 91*n - 193)*x^3 + (14*n - 56)*x^4 - 4*x^5)*P[n - 1, x] + (25*n^2 + 25*n + (100*n^2 + 190*n + 45)*x + (105*n^2 + 475*n + 275)*x^2 + (10*n^2 + 428*n + 579)*x^3 + (78*n + 457)*x^4 + (4*n + 80)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(5*n^2 + (5*n^3 + 5*n^2)*x + (18*n^2 - 10*n)*x^2 + (2*n^2 + 15*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] = (-(25*n^3 - 55*n^2 - 35*n + 45 + (45*n^3 - 35*n^2 - 205*n + 145)*x + (142*n^2 - 221*n + 27)*x^2 + (18*n^2 + 91*n - 193)*x^3 + (14*n - 56)*x^4 - 4*x^5)*v[i - 1] + (25*n^2 + 25*n + (100*n^2 + 190*n + 45)*x + (105*n^2 + 475*n + 275)*x^2 + (10*n^2 + 428*n + 579)*x^3 + (78*n + 457)*x^4 + (4*n + 80)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(5*n^2 + (5*n^3 + 5*n^2)*x + (18*n^2 - 10*n)*x^2 + (2*n^2 + 15*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.