Real-rooted atlas

A207606 → x ↦ x+1 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(18 + 24\,n + 8\,n^{2}\right)\,P_{n}(x) = \left(18 + 69\,x + 57\,x^{2} + 24\,n + 100\,n x + 95\,n x^{2} + 15\,n x^{3} + 8\,n^{2} + 36\,n^{2} x + 38\,n^{2} x^{2} + 15\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-6\,x - 45\,x^{2} - 57\,x^{3} - 4\,n x - 30\,n x^{2} - 56\,n x^{3} - 30\,n x^{4}\right)\,P_{n-1}'(x) + \left(3\,x^{3} + 18\,x^{4} + 15\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 2 + 11\,x + 13\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02111326
1217443497
22239315589362
32291604175102331351
4235245874166816086105042
52413481580413061954924159718817
624746925898625177112184114761418170226
72536083955160444200670844740894354010946262087

All 15 rows

Technical details
Node id
n-4a8b92b99ec82596 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-797bb38c8aa87a31

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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: 2, 11,  13;
  1: 2, 17,  44,   34;
  2: 2, 23,  93,  155,    89;
  3: 2, 29, 160,  417,   510,    233;
  4: 2, 35, 245,  874,  1668,   1608,    610;
  5: 2, 41, 348, 1580,  4130,   6195,   4924,   1597;
  6: 2, 47, 469, 2589,  8625,  17711,  21841,  14761,   4181;
  7: 2, 53, 608, 3955, 16044,  42006,  70844,  74089,  43540,  10946;
  8: 2, 59, 765, 5732, 27440,  87549, 188237, 268910, 243966, 126805,  28657;
  9: 2, 65, 940, 7974, 44028, 165914, 434840, 791615, 979852, 784614, 365532, 75025;
%F Formula
The row polynomials satisfy (8*n^2+24*n+18)*P(n,x) = (8*n^2+24*n+18+(36*n^2+100*n+69)*x+(38*n^2+95*n+57)*x^2+(15*n^2+15*n)*x^3)*P(n-1,x) - ((4*n+6)*x+(30*n+45)*x^2+(56*n+57)*x^3+30*n*x^4)*d/dx P(n-1,x) + (3*x^3+18*x^4+15*x^5)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 2 + 11*x + 13*x^2.
%t Mathematica
P[0, x_] := 2 + 11*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 + 24*n + 18 + (36*n^2 + 100*n + 69)*x + (38*n^2 + 95*n + 57)*x^2 + (15*n^2 + 15*n)*x^3)*P[n - 1, x] - ((4*n + 6)*x + (30*n + 45)*x^2 + (56*n + 57)*x^3 + 30*n*x^4)*D[P[n - 1, x], x] + (3*x^3 + 18*x^4 + 15*x^5)*D[P[n - 1, x], {x, 2}])/(8*n^2 + 24*n + 18)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2 + 11*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^2 + 24*n + 18 + (36*n^2 + 100*n + 69)*x + (38*n^2 + 95*n + 57)*x^2 + (15*n^2 + 15*n)*x^3)*v[i - 1] - ((4*n + 6)*x + (30*n + 45)*x^2 + (56*n + 57)*x^3 + 30*n*x^4)*deriv(v[i - 1], x) + (3*x^3 + 18*x^4 + 15*x^5)*deriv(deriv(v[i - 1], x), x))/(8*n^2 + 24*n + 18)); 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

Candidate equalities ?

Same fingerprint ?

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023