Real-rooted atlas

A034870 → D → basis A013609

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(4\,n + 8\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
1448
232969632256
3192960192019209601926144
4102471682150435840358402150471681024131072
55120460801843204300806451206451204300801843204608051202621440
62457627033613516804055040811008011354112113541128110080405504013516802703362457650331648
7114688149094489456643280076882001920147603456196804608196804608147603456820019203280076889456641490944114688939524096

All 15 rows

Technical details
Node id
n-188134e047d233d5 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-f001db5d15224ada

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: yes
  • γ-positive (gamma_positive): yes
  • 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:       0;
  1:       4,        4;
  2:      32,       96,        96,         32;
  3:     192,      960,      1920,       1920,        960,         192;
  4:    1024,     7168,     21504,      35840,      35840,       21504,        7168,        1024;
  5:    5120,    46080,    184320,     430080,     645120,      645120,      430080,      184320,       46080,        5120;
  6:   24576,   270336,   1351680,    4055040,    8110080,    11354112,    11354112,     8110080,     4055040,     1351680,      270336,       24576;
  7:  114688,  1490944,   8945664,   32800768,   82001920,   147603456,   196804608,   196804608,   147603456,    82001920,    32800768,     8945664,     1490944,     114688;
  8:  524288,  7864320,  55050240,  238551040,  715653120,  1574436864,  2624061440,  3373793280,  3373793280,  2624061440,  1574436864,   715653120,   238551040,   55050240,    7864320,    524288;
  9: 2359296, 40108032, 320864256, 1604321280, 5615124480, 14599323648, 29198647296, 45883588608, 57354485760, 57354485760, 45883588608, 29198647296, 14599323648, 5615124480, 1604321280, 320864256, 40108032, 2359296;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n+8*n*x+4*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 8*n*x + 4*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] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 8*n*x + 4*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

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