Real-rooted atlas

A038320 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(34 + 12\,x + 17\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 204 + 72\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
020472276
1520236726489522
2117912124848440645184292008
3250563035373601872720440640388808395230
4511148529020268063672480224726403965760279936231708348
51013777898214682378418942562808914147202359627203331238419595526217507338
61969625630448661339104515237708163030810048010696976640226524211226649907213436928163431621456
737669090181410635978404161313856155808927427874688409159356480115527347712203871790082055849984906992644228793205174

All 15 rows

Technical details
Node id
n-ef5644e239cb13e5 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-dde9c6f2b382eac4

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             204,              72;
  1:            5202,            3672,             648;
  2:          117912,          124848,           44064,            5184;
  3:         2505630,         3537360,         1872720,          440640,           38880;
  4:        51114852,        90202680,        63672480,        22472640,         3965760,          279936;
  5:      1013777898,      2146823784,      1894256280,       891414720,       235962720,        33312384,        1959552;
  6:     19696256304,     48661339104,     51523770816,     30308100480,     10696976640,      2265242112,      266499072,       13436928;
  7:    376690901814,   1063597840416,   1313856155808,    927427874688,    409159356480,    115527347712,    20387179008,     2055849984,      90699264;
  8:   7115272589820,  22601454108840,  31907935212480,  26277123116160,  13911418120320,   4909912277760,  1155273477120,   174747248640,   15418874880,    604661760;
  9: 133055597429634, 469607990928120, 745847985591720, 701974574674560, 433572531416640, 183630719188224, 54009035055360, 10892578498560, 1441664801280, 113071749120, 3990767616;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (17*n+34+(6*n+12)*x)*P(n-1,x) for n >= 1, with P(0,x) = 204 + 72*x.
%t Mathematica
P[0, x_] := 204 + 72*x;
P[n_, x_] := P[n, x] = Expand[Together[((17*n + 34 + (6*n + 12)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 204 + 72*x; for(n = 1, N, my(i = n + 1); v[i] = ((17*n + 34 + (6*n + 12)*x)*v[i - 1])/(n + 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.