Real-rooted atlas

A081578 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(8\,n + 8\,n x + 7\,n x^{2} + 14\,n x^{3} + n x^{4} + 8\,n^{2} + 12\,n^{2} x + 15\,n^{2} x^{2} + n^{2} x^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(-48 + 16\,x + 198\,x^{2} + 157\,x^{3} + 24\,x^{4} + x^{5} + 120\,n + 192\,n x - 7\,n x^{2} - 83\,n x^{3} - 6\,n x^{4} + 8\,n^{2} - 92\,n^{2} x - 101\,n^{2} x^{2} - 7\,n^{2} x^{3} - 32\,n^{3} - 28\,n^{3} x\right)\,P_{n-1}(x) + \left(128 + 624\,x + 1128\,x^{2} + 927\,x^{3} + 327\,x^{4} + 33\,x^{5} + x^{6} + 192\,n + 624\,n x + 700\,n x^{2} + 299\,n x^{3} + 32\,n x^{4} + n x^{5} + 64\,n^{2} + 128\,n^{2} x + 68\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2149124
35054151120
4183278129211612
563712638572392713024
6218853674836198738433114796
77380216282447413542386856439171496

All 15 rows

Technical details
Node id
n-b7c35f1a4a8e7640 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f28bc1a41e15e92b

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:     3,      1;
  2:    14,      9,      1;
  3:    50,     54,     15,      1;
  4:   183,    278,    129,     21,      1;
  5:   637,   1263,    857,    239,     27,     1;
  6:  2188,   5367,   4836,   1987,    384,    33,     1;
  7:  7380,  21628,  24474,  13542,   3868,   564,    39,    1;
  8: 24605,  83964, 114718,  80510,  31146,  6700,   779,   45,  1;
  9: 81191, 316461, 507678, 433842, 214302, 62538, 10683, 1029, 51, 1;
%F Formula
The row polynomials satisfy (8*n^2+8*n+(4*n^3+12*n^2+8*n)*x+(15*n^2+7*n)*x^2+(n^2+14*n)*x^3+n*x^4)*P(n,x) = -(32*n^3-8*n^2-120*n+48+(28*n^3+92*n^2-192*n-16)*x+(101*n^2+7*n-198)*x^2+(7*n^2+83*n-157)*x^3+(6*n-24)*x^4-x^5)*P(n-1,x) + (64*n^2+192*n+128+(128*n^2+624*n+624)*x+(68*n^2+700*n+1128)*x^2+(4*n^2+299*n+927)*x^3+(32*n+327)*x^4+(n+33)*x^5+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[(-(32*n^3 - 8*n^2 - 120*n + 48 + (28*n^3 + 92*n^2 - 192*n - 16)*x + (101*n^2 + 7*n - 198)*x^2 + (7*n^2 + 83*n - 157)*x^3 + (6*n - 24)*x^4 - x^5)*P[n - 1, x] + (64*n^2 + 192*n + 128 + (128*n^2 + 624*n + 624)*x + (68*n^2 + 700*n + 1128)*x^2 + (4*n^2 + 299*n + 927)*x^3 + (32*n + 327)*x^4 + (n + 33)*x^5 + x^6)*D[P[n - 1, x], x])/(8*n^2 + 8*n + (4*n^3 + 12*n^2 + 8*n)*x + (15*n^2 + 7*n)*x^2 + (n^2 + 14*n)*x^3 + 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] = (-(32*n^3 - 8*n^2 - 120*n + 48 + (28*n^3 + 92*n^2 - 192*n - 16)*x + (101*n^2 + 7*n - 198)*x^2 + (7*n^2 + 83*n - 157)*x^3 + (6*n - 24)*x^4 - x^5)*v[i - 1] + (64*n^2 + 192*n + 128 + (128*n^2 + 624*n + 624)*x + (68*n^2 + 700*n + 1128)*x^2 + (4*n^2 + 299*n + 927)*x^3 + (32*n + 327)*x^4 + (n + 33)*x^5 + x^6)*deriv(v[i - 1], x))/(8*n^2 + 8*n + (4*n^3 + 12*n^2 + 8*n)*x + (15*n^2 + 7*n)*x^2 + (n^2 + 14*n)*x^3 + 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.