Real-rooted atlas

A272867 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(13 + 8\,x + x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1138122
216920890161484
3219740563003113623124110648
428561703047368442848150943296436321234256
53712931142440154888512168006126902064484713072007054015153632
648268091782206429646318293519201924825588021442876964677088113895133601038481113379904
76274851727030130453280545563656756851468899329776583212706593940620000977430317619282342692228814355612494357888

All 15 rows

Technical details
Node id
n-7bcdd15d15e2eef8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-77ff08c622920f3c

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:          13,           8,            1;
  2:         169,         208,           90,           16,            1;
  3:        2197,        4056,         3003,         1136,          231,           24,            1;
  4:       28561,       70304,        73684,        42848,        15094,         3296,          436,          32,           1;
  5:      371293,     1142440,      1548885,      1216800,       612690,       206448,        47130,        7200,         705,         40,          1;
  6:     4826809,    17822064,     29646318,     29351920,     19248255,      8802144,      2876964,      677088,      113895,      13360,       1038,        48,        1;
  7:    62748517,   270301304,    532805455,    636567568,    514688993,    297765832,    127065939,    40620000,     9774303,    1761928,     234269,     22288,     1435,      56,      1;
  8:   815730721,  4015905088,   9151629864,  12808123328,  12316302908,   8625035328,   4548672856,  1842353344,   579091878,  141719488,   26915224,   3925824,   431228,   34496,   1896,    64,    1;
  9: 10604499373, 58732611912, 151914159657, 243734547264, 271728554292, 223464005856, 140449332660, 68965011648, 26815694022, 8317441712, 2062745694, 408076992, 63927780, 7824096, 731844, 50496, 2421, 72, 1;
%F Formula
The row polynomials satisfy P(n,x) = (13+8*x+x^2)*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[(13 + 8*x + x^2)*P[n - 1, 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] = (13 + 8*x + x^2)*v[i - 1]); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.