Real-rooted atlas

basis-map coefficients of A026729 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
25813
3258064169
41256009605122197
56254000960010240409628561
63125250008000012800010240032768371293
715625150000600000128000015360009830402621444826809
87812587500042000001120000017920000172032009175040209715262748517

All 15 rows

Technical details
Node id
n-2b4f883be76475ef · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-4b28493176e02e0b

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 -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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       5,        8;
   3:      25,       80,        64;
   4:     125,      600,       960,       512;
   5:     625,     4000,      9600,     10240,       4096;
   6:    3125,    25000,     80000,    128000,     102400,      32768;
   7:   15625,   150000,    600000,   1280000,    1536000,     983040,     262144;
   8:   78125,   875000,   4200000,  11200000,   17920000,   17203200,    9175040,    2097152;
   9:  390625,  5000000,  28000000,  89600000,  179200000,  229376000,  183500800,   83886080,  16777216;
  10: 1953125, 28125000, 180000000, 672000000, 1612800000, 2580480000, 2752512000, 1887436800, 754974720, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (5+8*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (5 + 8*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, v[i])))};
T(10)

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.