Real-rooted atlas

A075497 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
21114429
34992528201
4257644556184161657
515394854571630246563215821
61029939290589764350415856240064170389
7759053404726232885934243081768236890241282032785

All 15 rows

Technical details
Node id
n-59f7f6e4da7ff8b1 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8e43e13b68ec28f7

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,        2;
  2:      11,       14,        4;
  3:      49,       92,       52,         8;
  4:     257,      644,      556,       184,       16;
  5:    1539,     4854,     5716,      3024,      656,       32;
  6:   10299,    39290,    58976,     43504,    15856,     2400,       64;
  7:   75905,   340472,   623288,    593424,   308176,    82368,     9024,     128;
  8:  609441,  3147464,  6815656,   7981168,  5418736,  2110016,   429248,   34688,    256;
  9: 5284451, 30919274, 77444632, 107937472, 90930064, 46926368, 14248256, 2259200, 135424, 512;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+1+2*n*x)*P(n-1,x) - (2+4*x+2*x^2)*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[(2*n + 1 + 2*n*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*D[P[n - 1, x], x]];
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] = (2*n + 1 + 2*n*x)*v[i - 1] - (2 + 4*x + 2*x^2)*deriv(v[i - 1], x)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.