Real-rooted atlas

A059364 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(x + n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
227615
36294624105
424146329326120945
512087425213604255672010395
67206084212443927140564222125040135135
7504048348197380444849598116479996212976403202027025
84032043214420141725335212878856992230126023772223934436288034459425
93628804292496224489886815862813244924117089279814644406880391816256593603628800654729075
10362880046916640271707336928525148207423731831658696313344261458241480290811409035763185409603991680013749310575
11399168005596286403551780376134742646603395891227459715413757747773116106669396948441527574232171947934724261576320479001600316234143225
1247900160072344620804989653683220786432080858267238786811580508246461673628118161177243268473813653524940767461959866802746712309766114851136062270208007905853580625
136227020800100754029440749937447936340078768615210484841533596232120741505503796987708113746472418555848425636400093202881548274390414017469816208464032788134493703042944087178291200213458046676875
148717829120015039617241601201043471270458860089325168197799596762624482241661111640879759391394168122016201599892712929772384682001041871358554456628476818585472275226637581936827233442323201527595251840013076743680006190283353629375
Technical details
Node id
n-11442bd8c172894a · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-774e45ab36bcae1f

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:      1;
  1:      1,       2;
  2:      2,       7,        6;
  3:      6,      29,       46,       24;
  4:     24,     146,      329,      326,       120;
  5:    120,     874,     2521,     3604,      2556,       720;
  6:    720,    6084,    21244,    39271,     40564,     22212,      5040;
  7:   5040,   48348,   197380,   444849,    598116,    479996,    212976,    40320;
  8:  40320,  432144,  2014172,  5335212,   8788569,   9223012,   6023772,  2239344,   362880;
  9: 362880, 4292496, 22448988, 68158628, 132449241, 170892798, 146444068, 80391816, 25659360, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (n+(n+1)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + (n + 1)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + (n + 1)*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

Candidate equalities ?

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.