Real-rooted atlas

A345013 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + n^{2}\right)\,P_{n}(x) = \left(4 + 2\,x + 10\,n + 4\,n x + 7\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-4 - 6\,x - 2\,x^{2} - 4\,n - 6\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17310
24132679
32312559010586
412891824930200154258
5718312355819025903852130724
640081810246594027440609067228221275
722414352019150122825830076230127261092361593946
812564653290240366030022478408278201854722436016804511494222
97059735205825952595186018493860816354022979884084084356024755583004076
10397532731276718401798740901458679207505883025458048571956083160073755352066600312502
1122429823178665343912245161501113250710654280770259765506706545841305876015894451194054862784348191940
1212678548734820802336821532511882747064405468598135248821372879523043617871796827792765288288018618665529131540317508
1371784612152941127923554455105250601863362104417964050522667337693832821990022069733404190021555579073550029982811908645105229093439176
1440704778785178764096000357302804040429909611040347031544860198230864832823411308722515717776056501745309218809601065784728360352413140112001201666129426963
Technical details
Node id
n-d2990b6872a349e5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-3e366b2f98fb5223

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:       7,        3;
  2:      41,       32,        6;
  3:     231,      255,       90,       10;
  4:    1289,     1824,      930,      200,      15;
  5:    7183,    12355,     8190,     2590,     385,      21;
  6:   40081,    81024,    65940,    27440,    6090,     672,     28;
  7:  224143,   520191,   501228,   258300,   76230,   12726,   1092,    36;
  8: 1256465,  3290240,  3660300,  2247840,  827820,  185472,  24360,  1680,   45;
  9: 7059735, 20582595, 25951860, 18493860, 8163540, 2297988, 408408, 43560, 2475, 55;
%F Formula
The row polynomials satisfy (n^2+2*n)*P(n,x) = (7*n^2+10*n+4+(3*n^2+4*n+2)*x)*P(n-1,x) - (4*n+4+(6*n+6)*x+(2*n+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[Together[((7*n^2 + 10*n + 4 + (3*n^2 + 4*n + 2)*x)*P[n - 1, x] - (4*n + 4 + (6*n + 6)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n)]];
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] = ((7*n^2 + 10*n + 4 + (3*n^2 + 4*n + 2)*x)*v[i - 1] - (4*n + 4 + (6*n + 6)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n)); 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. [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.