Real-rooted atlas

A272866 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 5\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
1527
25070304154
33759008253607562541
42500850012000910040001020140837268
51562568750131250142500968754275012250220022510512435
6937505062501218750172500015937501008750446250138000292504050330126764142
754687535000001017187517762500207812501719375010346875458500014962503552505967567204551486806489

All 15 rows

Technical details
Node id
n-2b8510f8f6f9afcd · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a2bca5b09b60fe17

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        5,         2;
  2:       50,        70,        30,          4;
  3:      375,       900,       825,        360,         75,          6;
  4:     2500,      8500,     12000,       9100,       4000,       1020,        140,          8;
  5:    15625,     68750,    131250,     142500,      96875,      42750,      12250,       2200,        225,        10;
  6:    93750,    506250,   1218750,    1725000,    1593750,    1008750,     446250,     138000,      29250,      4050,       330,       12;
  7:   546875,   3500000,  10171875,   17762500,   20781250,   17193750,   10346875,    4585000,    1496250,    355250,     59675,     6720,     455,      14;
  8:  3125000,  23125000,  78750000,  163625000,  231875000,  237300000,  181125000,  104975000,   46575000,  15820000,   4081000,   785400,  109200,   10360,    600,    16;
  9: 17578125, 147656250, 576562500, 1389375000, 2313281250, 2823187500, 2614500000, 1876275000, 1056234375, 469068750, 164340000, 45171000, 9623250, 1556100, 184500, 15120, 765, 18;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+5*n*x+n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 5*n*x + n*x^2)*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] = 0; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 5*n*x + n*x^2)*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

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.