Real-rooted atlas

A034870 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(x + 2\,n\right)\,P_{n}(x) = \left(8 + 12\,x + 6\,x^{2} + x^{3} + 8\,n + 8\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
186115
2488048121189
32566727204001201812187
412804608716862723360112022424124057
561442816057600691205376028224100802400360301255879
628672159744405504619520633600456192236544887042376044005283612657205
713107286016026091524845568615014456376323843840197683276876822422448048728072842127103491

All 15 rows

Technical details
Node id
n-f4310eecc20ba49e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-312ef1e8ce9b56f4

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       8,        6,        1;
  2:      48,       80,       48,        12,         1;
  3:     256,      672,      720,       400,       120,        18,         1;
  4:    1280,     4608,     7168,      6272,      3360,      1120,       224,        24,         1;
  5:    6144,    28160,    57600,     69120,     53760,     28224,     10080,      2400,       360,        30,        1;
  6:   28672,   159744,   405504,    619520,    633600,    456192,    236544,     88704,     23760,      4400,      528,       36,       1;
  7:  131072,   860160,  2609152,   4845568,   6150144,   5637632,   3843840,   1976832,    768768,    224224,    48048,     7280,     728,     42,      1;
  8:  589824,  4456448, 15728640,  34406400,  52183040,  58146816,  49201152,  32215040,  16473600,   6589440,  2050048,   489216,   87360,  11200,    960,    48,    1;
  9: 2621440, 22413312, 90243072, 227278848, 401080320, 526417920, 532267008, 423641088, 268849152, 136913920, 56010240, 18330624, 4752384, 959616, 146880, 16320, 1224, 54, 1;
%F Formula
The row polynomials satisfy (2*n+x)*P(n,x) = (8*n+8+(8*n+12)*x+(2*n+6)*x^2+x^3)*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[((8*n + 8 + (8*n + 12)*x + (2*n + 6)*x^2 + x^3)*P[n - 1, x])/(2*n + 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] = ((8*n + 8 + (8*n + 12)*x + (2*n + 6)*x^2 + x^3)*v[i - 1])/(2*n + 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.