Real-rooted atlas

A143491 → basis A007318 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 20 + 9\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0209130
112074151210
28406381792211680
367205944207035530115120
46048060216245745265625391151200
5604800662640305956772241151510154911663200
6665280078938404028156115542020388922680155460119958400
77983360010137888056231712178931963602088476049413282274721259459200

All 15 rows

Technical details
Node id
n-8fd867c061aa328b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6546c73f6027554e

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 2)
  • 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:          20,           9,           1;
  1:         120,          74,          15,          1;
  2:         840,         638,         179,         22,          1;
  3:        6720,        5944,        2070,        355,         30,         1;
  4:       60480,       60216,       24574,       5265,        625,        39,        1;
  5:      604800,      662640,      305956,      77224,      11515,      1015,       49,       1;
  6:     6652800,     7893840,     4028156,    1155420,     203889,     22680,     1554,      60,      1;
  7:    79833600,   101378880,    56231712,   17893196,    3602088,    476049,    41328,    2274,     72,    1;
  8:  1037836800,  1397759040,   832391136,  288843260,   64720340,   9790725,  1013313,   70890,   3210,   85,  1;
  9: 14529715200, 20606463360, 13051234944, 4876196776, 1194928020, 201790490, 23977107, 2005773, 115830, 4400, 99, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+5+x)*P(n-1,x) for n >= 1, with P(0,x) = 20 + 9*x + x^2.
%t Mathematica
P[0, x_] := 20 + 9*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n + 5 + x)*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] = 20 + 9*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 5 + x)*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

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.