Real-rooted atlas

basis-map coefficients of A112227 → basis A013611 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 25 + 160\,x + 256\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
125160256441
21251200384040969261
36258000384008192065536194481
43125500003200001024000163840010485764084101
51562530000024000001024000024576000314572801677721685766121
678125175000016800000896000002867200005505024005872025602684354561801088541
739062510000000112000000716800000286720000073400320001174405120010737418240429496729637822859361
81953125562500007200000005376000000258048000008257536000017616076800024159191040019327352832068719476736794280046581

All 15 rows

Technical details
Node id
n-73557aaabcbf41d6 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-833b492313a93e6e

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       25,        160,         256;
   2:      125,       1200,        3840,         4096;
   3:      625,       8000,       38400,        81920,         65536;
   4:     3125,      50000,      320000,      1024000,       1638400,       1048576;
   5:    15625,     300000,     2400000,     10240000,      24576000,      31457280,       16777216;
   6:    78125,    1750000,    16800000,     89600000,     286720000,     550502400,      587202560,      268435456;
   7:   390625,   10000000,   112000000,    716800000,    2867200000,    7340032000,    11744051200,    10737418240,     4294967296;
   8:  1953125,   56250000,   720000000,   5376000000,   25804800000,   82575360000,   176160768000,   241591910400,   193273528320,    68719476736;
   9:  9765625,  312500000,  4500000000,  38400000000,  215040000000,  825753600000,  2202009600000,  4026531840000,  4831838208000,  3435973836800,  1099511627776;
  10: 48828125, 1718750000, 27500000000, 264000000000, 1689600000000, 7569408000000, 24222105600000, 55364812800000, 88583700480000, 94489280512000, 60473139527680, 17592186044416;
%F Formula
The row polynomials satisfy P(n,x) = (5+16*x)*P(n-1,x) for n >= 2, with P(1,x) = 25 + 160*x + 256*x^2.
%t Mathematica
P[1, x_] := 25 + 160*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 16*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] = 25 + 160*x + 256*x^2; for(n = 2, N, my(i = n); v[i] = (5 + 16*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.