Real-rooted atlas

A094262 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 2\,x - x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n-1}(x) + \left(-x - 2\,x^{2} - x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11214
2310126132
3157013112461141416
41056301600222618308902403017552
5945693022225407964706235216169905060841621176128
610395900903490207957181182195119653283697640143612795325410277212615018624
713513513513506136515167167003039333538808946356430992369853611315829378978685136111852488212541168968192

All 15 rows

Technical details
Node id
n-be6279abbb83a349 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-5870953131fc996b

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:        1,         2,          1;
  2:        3,        10,         12,          6,           1;
  3:       15,        70,        131,        124,          61,          14,           1;
  4:      105,       630,       1600,       2226,        1830,         890,         240,          30,           1;
  5:      945,      6930,      22225,      40796,       47062,       35216,       16990,        5060,         841,          62,           1;
  6:    10395,     90090,     349020,     795718,     1182195,     1196532,      836976,      401436,      127953,       25410,        2772,        126,          1;
  7:   135135,   1351350,    6136515,   16716700,    30393335,    38808946,    35643099,    23698536,    11315829,     3789786,      851361,     118524,       8821,       254,        1;
  8:  2027025,  22972950,  119639520,  379068690,   815302180,  1257343230,  1431293496,  1219253530,   779175270,   370748610,   128945040,   31726422,    5231460,    527530,    27480,     510,     1;
  9: 34459425, 436486050, 2564186625, 9263714460, 23007444460, 41592937100, 56519394316, 58769110972, 47141876486, 29183679040, 13847170470, 4961577972, 1309489644, 244783252, 30459980, 2276020, 84481, 1022, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n-1,x) - (x+2*x^2+x^3)*d/dx 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[(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)*P[n - 1, x] - (x + 2*x^2 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)*v[i - 1] - (x + 2*x^2 + 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

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.