Real-rooted atlas

A109983 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(1 + 3\,x + x^{2} - n - 3\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(3\,x + 2\,x^{2} - 3\,n x - 2\,n x^{2}\right)\,P_{n-2}'(x) + \left(x^{2} - n x^{2}\right)\,P_{n-2}''(x) + \left(-3 - 7\,x - 2\,x^{2} + 6\,n + 14\,n x + 4\,n x^{2}\right)\,P_{n-1}(x) + \left(-7\,x - 4\,x^{2} + 14\,n x + 8\,n x^{2}\right)\,P_{n-1}'(x) + \left(-2\,x^{2} + 4\,n x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 7\,x + 2\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
137212
213135187666407
36323779026907132624502024269
432141935375905814850646471224370363502660702142932
516837511171496613262934125930839856114998520093754348663031997014490252256802123
689891365149158835549545548825841161315688412886911555723182615522347614943976404184123749024323467484492439564940669
7486392510649612310610461832001350705113466156536982376908518250209792213856410281471083712983095578305317424973059123521804071582901672532437237234327550784563310

All 15 rows

Technical details
Node id
n-1b547e4b3d19bf6d · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-2155513c91a20d32

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,           7,              2;
  2:      13,         135,            187,               66,                 6;
  3:      63,        2377,           9026,             9071,              3262,               450,                 20;
  4:     321,       41935,         375905,           814850,            646471,            224370,              36350,               2660,                70;
  5:    1683,      751117,       14966132,         62934125,          93083985,          61149985,           20093754,            3486630,            319970,             14490,              252;
  6:    8989,    13651491,      588355495,       4554882584,       11613156884,       12886911555,         7231826155,         2234761494,         397640418,          41237490,          2432346,            74844,             924;
  7:   48639,   251064961,    23106104618,     320013507051,     1346615653698,     2376908518250,      2097922138564,      1028147108371,      298309557830,       53174249730,       5912352180,        407158290,        16725324,         372372,         3432;
  8:  265729,  4660302591,   910288233889,   22184328433290,   150066446050371,   406374021639978,    537942408789954,    392801303472516,   170917111295971,    46594624435266,    8209143592854,     949361546364,     72109283862,     3535023492,    106905084,     1801800,     12870;
  9: 1462563, 87138646237, 36018966955292, 1529648141313605, 16357340665435285, 66386287592022213, 127863864211009526, 133852422505806930, 83168104891674705, 32546665634639201, 8351370659086666, 1442564948310750, 170259121838410, 13787972368170, 760244725812, 27872807820, 646352850, 8532810, 48620;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n-1+(3*n-3)*x+(n-1)*x^2)*P(n-2,x) - ((3*n-3)*x+(2*n-2)*x^2)*d/dx P(n-2,x) - (n-1)*x^2*d^2/dx^2 P(n-2,x) + (6*n-3+(14*n-7)*x+(4*n-2)*x^2)*P(n-1,x) + ((14*n-7)*x+(8*n-4)*x^2)*d/dx P(n-1,x) + (4*n-2)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 7*x + 2*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 7*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (3*n - 3)*x + (n - 1)*x^2)*P[n - 2, x] - ((3*n - 3)*x + (2*n - 2)*x^2)*D[P[n - 2, x], x] - (n - 1)*x^2*D[P[n - 2, x], {x, 2}] + (6*n - 3 + (14*n - 7)*x + (4*n - 2)*x^2)*P[n - 1, x] + ((14*n - 7)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x] + (4*n - 2)*x^2*D[P[n - 1, x], {x, 2}])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 7*x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (3*n - 3)*x + (n - 1)*x^2)*v[i - 2] - ((3*n - 3)*x + (2*n - 2)*x^2)*deriv(v[i - 2], x) - (n - 1)*x^2*deriv(deriv(v[i - 2], x), x) + (6*n - 3 + (14*n - 7)*x + (4*n - 2)*x^2)*v[i - 1] + ((14*n - 7)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x) + (4*n - 2)*x^2*deriv(deriv(v[i - 1], x), x))/(n)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.