Real-rooted atlas

A030528 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-14 - 68\,x - 8\,x^{2} + 16\,x^{3} - 34\,n - 52\,n x + 8\,n x^{2} + 16\,n x^{3} - 12\,n^{2} + 16\,n^{2} x + 16\,n^{2} x^{2} + 8\,n^{3}\right)\,P_{n}(x) = \left(-76 - 544\,x - 224\,x^{2} + 128\,x^{3} + 64\,x^{4} - 252\,n - 8\,n x + 112\,n x^{2} + 32\,n x^{3} + 136\,n^{2} + 32\,n^{2} x - 32\,n^{2} x^{2} - 16\,n^{3}\right)\,P_{n-1}(x) + \left(-15 - 126\,x - 168\,x^{2} + 112\,x^{3} + 144\,x^{4} + 32\,x^{5} - 50\,n - 80\,n x + 96\,n x^{2} + 128\,n x^{3} + 32\,n x^{4} + 20\,n^{2} + 48\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
21116431
32568488149
45221227612816684
51035621100928320323045
619713523564451228007686413257
736830461014017328160807872179212856754

All 15 rows

Technical details
Node id
n-f6efb36447711c64 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4f465080c2af11cf

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:   11,    16,     4;
  3:   25,    68,    48,      8;
  4:   52,   212,   276,    128,     16;
  5:  103,   562,  1100,    928,    320,     32;
  6:  197,  1352,  3564,   4512,   2800,    768,     64;
  7:  368,  3046, 10140,  17328,  16080,   7872,   1792,   128;
  8:  675,  6548, 26388,  57088,  71520,  52160,  21056,  4096,  256;
  9: 1221, 13588, 64356, 168944, 268272, 263904, 158144, 54272, 9216, 512;
%F Formula
The row polynomials satisfy (8*n^3-12*n^2-34*n-14+(16*n^2-52*n-68)*x+(16*n^2+8*n-8)*x^2+(16*n+16)*x^3)*P(n,x) = -(16*n^3-136*n^2+252*n+76-(32*n^2-8*n-544)*x+(32*n^2-112*n+224)*x^2-(32*n+128)*x^3-64*x^4)*P(n-1,x) + (20*n^2-50*n-15+(48*n^2-80*n-126)*x+(16*n^2+96*n-168)*x^2+(128*n+112)*x^3+(32*n+144)*x^4+32*x^5)*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[(-(16*n^3 - 136*n^2 + 252*n + 76 - (32*n^2 - 8*n - 544)*x + (32*n^2 - 112*n + 224)*x^2 - (32*n + 128)*x^3 - 64*x^4)*P[n - 1, x] + (20*n^2 - 50*n - 15 + (48*n^2 - 80*n - 126)*x + (16*n^2 + 96*n - 168)*x^2 + (128*n + 112)*x^3 + (32*n + 144)*x^4 + 32*x^5)*D[P[n - 1, x], x])/(8*n^3 - 12*n^2 - 34*n - 14 + (16*n^2 - 52*n - 68)*x + (16*n^2 + 8*n - 8)*x^2 + (16*n + 16)*x^3)]];
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] = (-(16*n^3 - 136*n^2 + 252*n + 76 - (32*n^2 - 8*n - 544)*x + (32*n^2 - 112*n + 224)*x^2 - (32*n + 128)*x^3 - 64*x^4)*v[i - 1] + (20*n^2 - 50*n - 15 + (48*n^2 - 80*n - 126)*x + (16*n^2 + 96*n - 168)*x^2 + (128*n + 112)*x^3 + (32*n + 144)*x^4 + 32*x^5)*deriv(v[i - 1], x))/(8*n^3 - 12*n^2 - 34*n - 14 + (16*n^2 - 52*n - 68)*x + (16*n^2 + 8*n - 8)*x^2 + (16*n + 16)*x^3)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.