Real-rooted atlas

A272866 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 50\,x - 120\,x^{2} - 40\,x^{3} + 16\,x^{4} + 150\,n x + 160\,n x^{2} - 40\,n x^{3} - 64\,n x^{4} + 100\,n^{2} x^{2} + 160\,n^{2} x^{3} + 64\,n^{2} x^{4}\right)\,P_{n}(x) = \left(125 + 750\,x + 1800\,x^{2} + 2200\,x^{3} + 1440\,x^{4} + 480\,x^{5} + 64\,x^{6} + 750\,n x + 3300\,n x^{2} + 5600\,n x^{3} + 4560\,n x^{4} + 1760\,n x^{5} + 256\,n x^{6} + 500\,n^{2} x^{2} + 1800\,n^{2} x^{3} + 2320\,n^{2} x^{4} + 1280\,n^{2} x^{5} + 256\,n^{2} x^{6}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15403681
225400126012804003365
31253000162003520036000172803136110941
462520000153000512000910000921600533120163840207363234921
5312512500012375005600000142500002232000022344000143360005702400128000012390487321929
6156257500009112500520000001725000003672000005272400005222400003576960001664000005018112088473606922242234874829
778125437500063000000434000000177625000047880000008986600000121088000001188432000085120000004401689600159989760038764544056197120368640055006539285

All 15 rows

Technical details
Node id
n-71f55a5a0c64a1c9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-91642c520b1b8422

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       5,        40,         36;
  2:      25,       400,       1260,        1280,          400;
  3:     125,      3000,      16200,       35200,        36000,        17280,          3136;
  4:     625,     20000,     153000,      512000,       910000,       921600,        533120,        163840,         20736;
  5:    3125,    125000,    1237500,     5600000,     14250000,     22320000,      22344000,      14336000,       5702400,       1280000,        123904;
  6:   15625,    750000,    9112500,    52000000,    172500000,    367200000,     527240000,     522240000,     357696000,     166400000,      50181120,       8847360,        692224;
  7:   78125,   4375000,   63000000,   434000000,   1776250000,   4788000000,    8986600000,   12108800000,   11884320000,    8512000000,    4401689600,    1599897600,     387645440,      56197120,      3686400;
  8:  390625,  25000000,  416250000,  3360000000,  16362500000,  53424000000,  124028800000,  211968000000,  272095200000,  264960000000,  196016128000,  109412352000,   45306060800,   13487308800,   2727936000,    335544320,    18939904;
  9: 1953125, 140625000, 2657812500, 24600000000, 138937500000, 532980000000, 1475586000000, 3059712000000, 4863304800000, 6008800000000, 5811949440000, 4405985280000, 2605704192000, 1188569088000, 409743360000, 103179878400, 17898209280, 1911029760, 94633984;
%F Formula
The row polynomials satisfy (25+(150*n-50)*x+(100*n^2+160*n-120)*x^2+(160*n^2-40*n-40)*x^3+(64*n^2-64*n+16)*x^4)*P(n,x) = (125+(750*n+750)*x+(500*n^2+3300*n+1800)*x^2+(1800*n^2+5600*n+2200)*x^3+(2320*n^2+4560*n+1440)*x^4+(1280*n^2+1760*n+480)*x^5+(256*n^2+256*n+64)*x^6)*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[Together[((125 + (750*n + 750)*x + (500*n^2 + 3300*n + 1800)*x^2 + (1800*n^2 + 5600*n + 2200)*x^3 + (2320*n^2 + 4560*n + 1440)*x^4 + (1280*n^2 + 1760*n + 480)*x^5 + (256*n^2 + 256*n + 64)*x^6)*P[n - 1, x])/(25 + (150*n - 50)*x + (100*n^2 + 160*n - 120)*x^2 + (160*n^2 - 40*n - 40)*x^3 + (64*n^2 - 64*n + 16)*x^4)]];
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] = ((125 + (750*n + 750)*x + (500*n^2 + 3300*n + 1800)*x^2 + (1800*n^2 + 5600*n + 2200)*x^3 + (2320*n^2 + 4560*n + 1440)*x^4 + (1280*n^2 + 1760*n + 480)*x^5 + (256*n^2 + 256*n + 64)*x^6)*v[i - 1])/(25 + (150*n - 50)*x + (100*n^2 + 160*n - 120)*x^2 + (160*n^2 - 40*n - 40)*x^3 + (64*n^2 - 64*n + 16)*x^4)); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.