Real-rooted atlas

A064861 → x ↦ x+1 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(27 + 21\,x + 4\,x^{2} + 22\,n + 19\,n x + 4\,n x^{2}\right)\,P_{n-1}(x) + \left(-60 - 74\,x - 30\,x^{2} - 4\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 15 + 11\,x + 2\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01511228
17585324196
2225330181444784
311252100156558210885488
43375742567953311906132821952
51687543875488253014511152247230416153664
6506251485001903501392606360118568338435216614656
725312584375012487501077000596525220042540568528784324302592

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(15 + 11\,x + 2\,x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 15 + 11\,x + 2\,x^{2}\)
\(P_{1}(x) = 75 + 85\,x + 32\,x^{2} + 4\,x^{3}\)
Technical details
Node id
n-5e277f929045261e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-ab734cd6d3e371af)
Fingerprint
f6-85be053e2490b2e9

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:      15,       11,        2;
  1:      75,       85,       32,        4;
  2:     225,      330,      181,       44,        4;
  3:    1125,     2100,     1565,      582,      108,        8;
  4:    3375,     7425,     6795,     3311,      906,      132,       8;
  5:   16875,    43875,    48825,    30145,    11152,     2472,     304,      16;
  6:   50625,   148500,   190350,   139260,    63601,    18568,    3384,     352,     16;
  7:  253125,   843750,  1248750,  1077000,   596525,   220042,   54056,    8528,    784,    32;
  8:  759375,  2784375,  4590000,  4479750,  2866575,  1256651,  382210,   79640,  10880,   880,   32;
  9: 3796875, 15440625, 28518750, 31578750, 23292375, 12016405, 4424352, 1162620, 213680, 26160, 1920, 64;
%F Formula
The row polynomials satisfy P(n,x) = (22*n+27+(19*n+21)*x+(4*n+4)*x^2)*P(n-1,x) - (60+74*x+30*x^2+4*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 15 + 11*x + 2*x^2.
%t Mathematica
P[0, x_] := 15 + 11*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[(22*n + 27 + (19*n + 21)*x + (4*n + 4)*x^2)*P[n - 1, x] - (60 + 74*x + 30*x^2 + 4*x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 15 + 11*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22*n + 27 + (19*n + 21)*x + (4*n + 4)*x^2)*v[i - 1] - (60 + 74*x + 30*x^2 + 4*x^3)*deriv(v[i - 1], 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

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.