Real-rooted atlas

A299989 → D

nonnegative coefficients

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

Recurrence

\(\left(3 - x^{2} + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(9 + 24\,x + 22\,x^{2} + 8\,x^{3} + x^{4} + 12\,n x + 22\,n x^{2} + 12\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
138314
294866325160
3272165135442857271664
4818642916480044302400756128916384
5243324014175316804215035664196707040157520011155648
6729116646269417712030847535683228445215859261695164002838288131441792
7218740824260253889056191551528168562950437225190412644735216401560793292846413921513107200

All 15 rows

Technical details
Node id
n-a99de495fe940cd0 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b1018962b1f2016f

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:     3,      8,       3;
  2:     9,     48,      66,       32,        5;
  3:    27,    216,     513,      544,      285,        72,         7;
  4:    81,    864,    2916,     4800,     4430,      2400,       756,       128,         9;
  5:   243,   3240,   14175,    31680,    42150,     35664,     19670,      7040,      1575,       200,       11;
  6:   729,  11664,   62694,   177120,   308475,    356832,    284452,    158592,     61695,     16400,     2838,      288,       13;
  7:  2187,  40824,  260253,   889056,  1915515,   2816856,   2950437,   2251904,   1264473,    521640,   156079,    32928,     4641,     392,     15;
  8:  6561, 139968, 1032264,  4136832, 10625580,  18978624,  24678360,  23999232,  17720406,   9999680,  4308920,  1405824,   341068,   59584,   7080,   512,    17;
  9: 19683, 472392, 3956283, 18195840, 54281340, 114143904, 177477804, 210117888, 192887082, 138755440, 78583626, 35019648, 12207468, 3288096, 670140, 99840, 10251, 648, 19;
%F Formula
The row polynomials satisfy (3+4*n*x+(2*n-1)*x^2)*P(n,x) = (9+(12*n+24)*x+(22*n+22)*x^2+(12*n+8)*x^3+(2*n+1)*x^4)*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[((9 + (12*n + 24)*x + (22*n + 22)*x^2 + (12*n + 8)*x^3 + (2*n + 1)*x^4)*P[n - 1, x])/(3 + 4*n*x + (2*n - 1)*x^2)]];
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] = ((9 + (12*n + 24)*x + (22*n + 22)*x^2 + (12*n + 8)*x^3 + (2*n + 1)*x^4)*v[i - 1])/(3 + 4*n*x + (2*n - 1)*x^2)); 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.