Real-rooted atlas

A166979 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(9 + 9\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 13 + 13\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
11313127
2143174331351
3185226518616215427
42756346511215452695100198415
5473725926994550176103300654014712060883
692903962057382714719614378131635843513503203148737295
72037452355005555454170616391372814381800948110040382493426811277682579
849124903751321696573212548387151504656465786682342666545703241722642426342136660237039

All 15 rows

Technical details
Node id
n-a39aac1826daed13 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-205e3fa81fb7b7f3

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:            13,             13,              1;
   2:           143,            174,             33,             1;
   3:          1852,           2651,            861,            62,             1;
   4:         27563,          46511,          21545,          2695,           100,            1;
   5:        473725,         926994,         550176,        103300,          6540,          147,           1;
   6:       9290396,       20573827,       14719614,       3781316,        358435,        13503,         203,         1;
   7:     203745235,      500555545,      417061639,     137281438,      18009481,      1004038,       24934,       268,        1;
   8:    4912490375,    13216965732,    12548387151,    5046564657,     866823426,     66545703,     2417226,     42426,      342,      1;
   9:  128777672338,   376170082145,   400441954491,  190104405540,   40980411480,   4125001671,   204583197,   5199540,    67815,    425,   1;
  10: 3643086083981, 11474603756771, 13517828121545, 7388764528431, 1932985116975, 245902982655, 15909231723, 547435977, 10252275, 103180, 517, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (9+9*x)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 13 + 13*x + x^2.
%t Mathematica
P[1, x_] := 13 + 13*x + x^2;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (9 + 9*x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 13 + 13*x + x^2; for(n = 2, N, my(i = n); v[i] = (2 + x)*v[i - 1] + (9 + 9*x)*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, v[i])))};
T(10)

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.