Real-rooted atlas

A370258 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(12 + 14\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(57 + 45\,x + 71\,n + 57\,n x + 22\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(28 + 43\,x + 15\,x^{2} + 17\,n + 26\,n x + 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 26 + 40\,x + 15\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
026401581
1190441336841051
2146245365220264049514353
311584449906930052910200203003201807
493536436332842205860860491400148512185642891409
5765314416776596700801239028094676404313988108528011628041976627
663232703937336010671444016441992015750126096047280364111447845024735471615371169
75263876036887265011436890402059079400237216672018134119089198290642985147005624190046868259089130967

All 15 rows

Technical details
Node id
n-76a27096afd2aba0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2e6724d61e5bb4e3

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         26,          40,           15;
  1:        190,         441,          336,           84;
  2:       1462,        4536,         5220,         2640,          495;
  3:      11584,       44990,        69300,        52910,        20020,         3003;
  4:      93536,      436332,       842205,       860860,       491400,       148512,        18564;
  5:     765314,     4167765,      9670080,     12390280,      9467640,      4313988,      1085280,       116280;
  6:    6323270,    39373360,    106714440,    164419920,    157501260,     96047280,     36411144,      7845024,      735471;
  7:   52638760,   368872650,   1143689040,   2059079400,   2372166720,   1813411908,    919829064,    298514700,    56241900,     4686825;
  8:  440815036,  3433281600,  11984624190,  24690148560,  33243015960,  30563988840,  19432563150,   8436285000,  2393240850,   400600200,   30045015;
  9: 3709445084, 31787462938, 123366281280, 286211461800, 441030232500, 473930066610, 362396754720, 197181141300, 74812087350, 18849670125, 2838538560, 193536720;
%F Formula
The row polynomials satisfy (4*n^2+14*n+12)*P(n,x) = (22*n^2+71*n+57+(18*n^2+57*n+45)*x)*P(n-1,x) + (17*n+28+(26*n+43)*x+(9*n+15)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 26 + 40*x + 15*x^2.
%t Mathematica
P[0, x_] := 26 + 40*x + 15*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((22*n^2 + 71*n + 57 + (18*n^2 + 57*n + 45)*x)*P[n - 1, x] + (17*n + 28 + (26*n + 43)*x + (9*n + 15)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 14*n + 12)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 26 + 40*x + 15*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((22*n^2 + 71*n + 57 + (18*n^2 + 57*n + 45)*x)*v[i - 1] + (17*n + 28 + (26*n + 43)*x + (9*n + 15)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 14*n + 12)); 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.