Real-rooted atlas

A030528 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(4 + 4\,n\right)\,P_{n}(x) = \left(16 - 5\,x^{2} + 16\,n + 16\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-16\,x - 24\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21616335
36496405205
42565123368881200
5102425602304944180137025
640961228814080793623523522141125
71638457344798725760023424542466834240750
86553626214443008037888019712062464118561240551409375
92621441179648222822423224321477632598272155008249122264898250625
10104857652428801120665613500416101785605038080166528036454450784408014448300000
11419430423068672550502407530086465732608385638401556992043417608223361010727276233282753125
121677721610066329626528972840632320040343961627420262413117440044636160107545601794304197280128643771655265625
136710886443620761612582912002133852160237633536018384814081018888192410562560120591360255628803810048378912225806109690093750
14268435456187904819258888028161095552204813529776128117503426567414284288346003865612026470403104972805873484879091207179683939298756726796875
Technical details
Node id
n-edf15db11e2841e8 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-fe14b25fdaf2823c

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:     16,      16,       3;
  3:     64,      96,      40,       5;
  4:    256,     512,     336,      88,       8;
  5:   1024,    2560,    2304,     944,     180,     13;
  6:   4096,   12288,   14080,    7936,    2352,    352,     21;
  7:  16384,   57344,   79872,   57600,   23424,   5424,    668,    34;
  8:  65536,  262144,  430080,  378880,  197120,  62464,  11856,  1240,   55;
  9: 262144, 1179648, 2228224, 2322432, 1477632, 598272, 155008, 24912, 2264, 89;
%F Formula
The row polynomials satisfy (4*n+4)*P(n,x) = (16*n+16+16*n*x+(5*n-5)*x^2)*P(n-1,x) - (16*x+24*x^2+5*x^3)*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 + 16 + 16*n*x + (5*n - 5)*x^2)*P[n - 1, x] - (16*x + 24*x^2 + 5*x^3)*D[P[n - 1, x], x])/(4*n + 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] = ((16*n + 16 + 16*n*x + (5*n - 5)*x^2)*v[i - 1] - (16*x + 24*x^2 + 5*x^3)*deriv(v[i - 1], x))/(4*n + 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

Same fingerprint ?

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