Real-rooted atlas

A038226 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(24 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 528\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
05765281211225
113824190088712133142875
2331776608256418176127776146411500625
3796262418247680167270407666560175692016105152521875
41911029765255331846021734403679948801264982402319134417715611838265625
54586471424147149291522023302758415455784960708390144019480728962976222481948717164339296875
61100753141764036094853126474568826885935021424643400272691201246766653442857173580837415368322143588812251875390625
72641807540224108974561034241997866952294421366077128704146891780259846732539928576205716497817640408597785646301518296235794769178815638671875

All 15 rows

Technical details
Node id
n-d78c72d7d8963455 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-83fc6524b6311133

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:              576,              528,               121;
  1:            13824,            19008,              8712,              1331;
  2:           331776,           608256,            418176,            127776,             14641;
  3:          7962624,         18247680,          16727040,           7666560,           1756920,            161051;
  4:        191102976,        525533184,         602173440,         367994880,         126498240,          23191344,          1771561;
  5:       4586471424,      14714929152,       20233027584,       15455784960,        7083901440,        1948072896,        297622248,         19487171;
  6:     110075314176,     403609485312,      647456882688,      593502142464,      340027269120,      124676665344,      28571735808,       3741536832,       214358881;
  7:    2641807540224,   10897456103424,    19978669522944,    21366077128704,    14689178025984,     6732539928576,    2057164978176,     404085977856,     46301518296,     2357947691;
  8:   63403380965376,  290598829424640,   599360085688320,   732551215841280,   587567121039360,   323161916571648,  123429898690560,   32326878228480,   5556182195520,   565907445840,   25937424601;
  9: 1521681143169024, 7671809096810496, 17581229180190720, 24174190122762240, 22159674279198720, 14219124329152512, 6517098650861568, 2133573963079680, 488944033205760, 74699782850880, 6847480094664, 285311670611;
%F Formula
The row polynomials satisfy P(n,x) = (24+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 528*x + 121*x^2.
%t Mathematica
P[0, x_] := 576 + 528*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 11*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] = 576 + 528*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 11*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.