Real-rooted atlas

A069466 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n^{3} - 3\,n^{3} x + 6\,n^{4} x + n^{5} x^{2}\right)\,P_{n}(x) = \left(-4\,x^{2} + 16\,n + 52\,n x + 52\,n x^{2} + 8\,n x^{3} - 64\,n^{2} - 152\,n^{2} x - 92\,n^{2} x^{2} - 14\,n^{2} x^{3} + 64\,n^{3} + 48\,n^{3} x - 48\,n^{3} x^{2} - 16\,n^{3} x^{3} + 96\,n^{4} x + 104\,n^{4} x^{2} + 18\,n^{4} x^{3} + 16\,n^{5} x^{2} + 12\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 2\,x^{3} + 16\,n x + 44\,n x^{2} + 36\,n x^{3} + 8\,n x^{4} - 32\,n^{2} x - 56\,n^{2} x^{2} - 28\,n^{2} x^{3} - 4\,n^{2} x^{4} - 48\,n^{3} x^{2} - 68\,n^{3} x^{3} - 20\,n^{3} x^{4} - 8\,n^{4} x^{3} - 8\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
23614454234
3400240021603205280
449003920056700224001750124950
563504635040127008084672018900090723013416
6853776102453122619540024837120970200013970884527673275972
7117786241649007365137292166342336003603600009340531294174082196481788044544

All 15 rows

Technical details
Node id
n-c017ad02383332ad · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e884231a3e2c9072

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:          4,           8;
  2:         36,         144,           54;
  3:        400,        2400,         2160,          320;
  4:       4900,       39200,        56700,        22400,         1750;
  5:      63504,      635040,      1270080,       846720,       189000,         9072;
  6:     853776,    10245312,     26195400,     24837120,      9702000,      1397088,       45276;
  7:   11778624,   164900736,    513729216,    634233600,    360360000,     93405312,     9417408,     219648;
  8:  165636900,  2650190400,   9739449720,  14841066240,  11148637500,   4281076800,   794593800,   59304960,   1042470;
  9: 2363904400, 42550279200, 180212947200, 327053126400, 306612306000, 157686328800, 44026382400, 6161126400, 354439800, 4862000;
%F Formula
The row polynomials satisfy (4*n^3+(6*n^4-3*n^3)*x+n^5*x^2)*P(n,x) = (64*n^3-64*n^2+16*n+(96*n^4+48*n^3-152*n^2+52*n)*x+(16*n^5+104*n^4-48*n^3-92*n^2+52*n-4)*x^2+(12*n^5+18*n^4-16*n^3-14*n^2+8*n)*x^3)*P(n-1,x) - ((32*n^2-16*n)*x+(48*n^3+56*n^2-44*n+2)*x^2+(8*n^4+68*n^3+28*n^2-36*n+2)*x^3+(8*n^4+20*n^3+4*n^2-8*n)*x^4)*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[Together[((64*n^3 - 64*n^2 + 16*n + (96*n^4 + 48*n^3 - 152*n^2 + 52*n)*x + (16*n^5 + 104*n^4 - 48*n^3 - 92*n^2 + 52*n - 4)*x^2 + (12*n^5 + 18*n^4 - 16*n^3 - 14*n^2 + 8*n)*x^3)*P[n - 1, x] - ((32*n^2 - 16*n)*x + (48*n^3 + 56*n^2 - 44*n + 2)*x^2 + (8*n^4 + 68*n^3 + 28*n^2 - 36*n + 2)*x^3 + (8*n^4 + 20*n^3 + 4*n^2 - 8*n)*x^4)*D[P[n - 1, x], x])/(4*n^3 + (6*n^4 - 3*n^3)*x + n^5*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] = ((64*n^3 - 64*n^2 + 16*n + (96*n^4 + 48*n^3 - 152*n^2 + 52*n)*x + (16*n^5 + 104*n^4 - 48*n^3 - 92*n^2 + 52*n - 4)*x^2 + (12*n^5 + 18*n^4 - 16*n^3 - 14*n^2 + 8*n)*x^3)*v[i - 1] - ((32*n^2 - 16*n)*x + (48*n^3 + 56*n^2 - 44*n + 2)*x^2 + (8*n^4 + 68*n^3 + 28*n^2 - 36*n + 2)*x^3 + (8*n^4 + 20*n^3 + 4*n^2 - 8*n)*x^4)*deriv(v[i - 1], x))/(4*n^3 + (6*n^4 - 3*n^3)*x + n^5*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

Same fingerprint ?

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023