Real-rooted atlas

A038319 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(16 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(231 + 160\,x + 25\,x^{2} + 55\,n + 25\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 231 + 160\,x + 25\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
023116025416
13146346512001257936
24126162920346508000625147456
352707610315257865002887505000031252686976
466030911581228015472875786500021656253000001562548234496
5814918062311081852767149001805168756881875015159375175000078125855638016
699384572132596722404622163700368953200018051687505505500001010625001000000039062515032385536
71200409733644723057445733426254006933245550041507235000162465187504129125000649687500562500001953125261993005056

All 15 rows

Technical details
Node id
n-d7561282674dc1d2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-42dc7898d6edde45

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:           231,           160,             25;
  1:          3146,          3465,           1200,            125;
  2:         41261,         62920,          34650,           8000,            625;
  3:        527076,       1031525,         786500,         288750,          50000,          3125;
  4:       6603091,      15812280,       15472875,        7865000,        2165625,        300000,         15625;
  5:      81491806,     231108185,      276714900,      180516875,       68818750,      15159375,       1750000,         78125;
  6:     993845721,    3259672240,     4622163700,     3689532000,     1805168750,     550550000,     101062500,      10000000,       390625;
  7:   12004097336,   44723057445,    73342625400,    69332455500,    41507235000,   16246518750,    4129125000,     649687500,     56250000,     1953125;
  8:  143834809151,  600204866800,  1118076436125,  1222377090000,   866655693750,  415072350000,  135387656250,   29493750000,   4060546875,   312500000,    9765625;
  9: 1711870023666, 7910914503305, 16505633837000, 20498067995625, 16807684987500, 9533212631250, 3804829875000, 1063760156250, 202769531250, 24814453125, 1718750000, 48828125;
%F Formula
The row polynomials satisfy (5*n+16+5*x)*P(n,x) = (55*n+231+(25*n+160)*x+25*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 231 + 160*x + 25*x^2.
%t Mathematica
P[0, x_] := 231 + 160*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((55*n + 231 + (25*n + 160)*x + 25*x^2)*P[n - 1, x])/(5*n + 16 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 231 + 160*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((55*n + 231 + (25*n + 160)*x + 25*x^2)*v[i - 1])/(5*n + 16 + 5*x)); 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.