Real-rooted atlas

A038305 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(19 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 361 + 228\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
036122836625
168596498205221615625
213032116461677976164161296390625
324760993909630246924077976012312077769765625
447045881891395647037334029630880701784088646446656244140625
5893871739197592700218719308449852267603111242405894985662052482799366103515625
616983563041429058434724742224804829950893504118227211202986792704471598848425502721679616152587890625
73226876977799171124042141158457773744853600464864404337062304127685388096268811343363638048256287214336100776963814697265625

All 15 rows

Technical details
Node id
n-553d0e2315fdc85b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8952bc02823d42a3

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:             361,             228,              36;
  1:            6859,            6498,            2052,             216;
  2:          130321,          164616,           77976,           16416,            1296;
  3:         2476099,         3909630,         2469240,          779760,          123120,            7776;
  4:        47045881,        89139564,        70373340,        29630880,         7017840,          886464,          46656;
  5:       893871739,      1975927002,      1871930844,       985226760,       311124240,        58949856,        6205248,         279936;
  6:     16983563041,     42905843472,     47422248048,     29950893504,     11822721120,      2986792704,      471598848,       42550272,       1679616;
  7:    322687697779,    917112404214,   1158457773744,    853600464864,    404337062304,    127685388096,    26881134336,     3638048256,     287214336,     10077696;
  8:   6131066257801,  19361261866740,  27513372126420,  23169155474880,  12804006972960,   4852044747648,  1276853880960,   230409722880,   27285361920,   1914762240,    60466176;
  9: 116490258898219, 404650373014866, 638921641602420, 605294186781240, 382291065335520, 169012892043072, 53372492224128, 12038908020480, 1900880213760, 200092654080, 12637430784, 362797056;
%F Formula
The row polynomials satisfy P(n,x) = (19+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 228*x + 36*x^2.
%t Mathematica
P[0, x_] := 361 + 228*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(19 + 6*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] = 361 + 228*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 6*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.