Real-rooted atlas

basis-map coefficients of A112227 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 16 + 72\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1167281169
2644329727292197
32562304777611664656128561
41024115205184011664013122059049371293
5409655296311040933120157464014171765314414826809
61638425804817418246531840146966401984046414880348478296962748517
765536117964892897284180377611757312021163161623808556815305500843046721815730721
8262144530841647775744250822656846526464190468454428570268162754990144154968195638742048910604499373

All 15 rows

Technical details
Node id
n-d09a0a5821de8b2b · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-12e1fe8c90bb4450

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:      16,        72,         81;
   2:      64,       432,        972,        729;
   3:     256,      2304,       7776,      11664,        6561;
   4:    1024,     11520,      51840,     116640,      131220,        59049;
   5:    4096,     55296,     311040,     933120,     1574640,      1417176,       531441;
   6:   16384,    258048,    1741824,    6531840,    14696640,     19840464,     14880348,      4782969;
   7:   65536,   1179648,    9289728,   41803776,   117573120,    211631616,    238085568,    153055008,     43046721;
   8:  262144,   5308416,   47775744,  250822656,   846526464,   1904684544,   2857026816,   2754990144,   1549681956,    387420489;
   9: 1048576,  23592960,  238878720, 1433272320,  5643509760,  15237476352,  28570268160,  36733201920,  30993639120,  15496819560,   3486784401;
  10: 4194304, 103809024, 1167851520, 7882997760, 35473489920, 111741493248, 251418359808, 404065221120, 454573373760, 340930030320, 153418513644, 31381059609;
%F Formula
The row polynomials satisfy P(n,x) = (4+9*x)*P(n-1,x) for n >= 2, with P(1,x) = 16 + 72*x + 81*x^2.
%t Mathematica
P[1, x_] := 16 + 72*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 9*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] = 16 + 72*x + 81*x^2; for(n = 2, N, my(i = n); v[i] = (4 + 9*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.