Real-rooted atlas

A253284 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
224421278
31686246001201512
4132084001620010920168038520
511520111600352800463680241920302401211760
6110880151704070560001481760014817600632016066528045304560
7116928021450240137168640414892800651974400526901760190270080172972801961124480

All 15 rows

Technical details
Node id
n-854b531da9a10af4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-695598672c6e8bd2

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,         42,          12;
  3:       168,        624,         600,          120;
  4:      1320,       8400,       16200,        10920,         1680;
  5:     11520,     111600,      352800,       463680,       241920,         30240;
  6:    110880,    1517040,     7056000,     14817600,     14817600,       6320160,        665280;
  7:   1169280,   21450240,   137168640,    414892800,    651974400,     526901760,     190270080,     17297280;
  8:  13426560,  317882880,  2667168000,  10897286400,  24309331200,   30512401920,   20704844160,   6486480000,    518918400;
  9: 166924800, 4953312000, 52690176000, 278299929600, 828193766400, 1460236377600, 1525620096000, 892539648000, 247005158400, 17643225600;
%F Formula
The row polynomials satisfy (n^2-n+2+x-x^2)*P(n,x) = (n^3+2*n^2+3*n+2+(2*n^3+2*n^2+3*n+1)*x-(n+1)*x^2-2*n*x^3)*P(n-1,x) + ((2*n^2+2)*x+2*n^2*x^2-4*x^3-2*x^4)*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[((n^3 + 2*n^2 + 3*n + 2 + (2*n^3 + 2*n^2 + 3*n + 1)*x - (n + 1)*x^2 - 2*n*x^3)*P[n - 1, x] + ((2*n^2 + 2)*x + 2*n^2*x^2 - 4*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^2 - n + 2 + x - x^2)]];
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] = ((n^3 + 2*n^2 + 3*n + 2 + (2*n^3 + 2*n^2 + 3*n + 1)*x - (n + 1)*x^2 - 2*n*x^3)*v[i - 1] + ((2*n^2 + 2)*x + 2*n^2*x^2 - 4*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^2 - n + 2 + x - x^2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.