Real-rooted atlas

A056939 → θ+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(24\,n + 50\,n^{2} + 35\,n^{3} + 10\,n^{4} + n^{5}\right)\,P_{n}(x) = \left(24 - 12\,x + 74\,n - 25\,n x + 85\,n^{2} + 22\,n^{2} x + 45\,n^{3} + 75\,n^{3} x + 11\,n^{4} + 50\,n^{4} x + n^{5} + 10\,n^{5} x\right)\,P_{n-1}(x) + \left(48\,x + 12\,x^{2} + 148\,n x + n x^{2} + 125\,n^{2} x - 70\,n^{2} x^{2} + 37\,n^{3} x - 65\,n^{3} x^{2} + 3\,n^{4} x - 15\,n^{4} x^{2}\right)\,P_{n-1}'(x) + \left(12\,x^{2} + 12\,x^{3} + 37\,n x^{2} + 37\,n x^{3} + 28\,n^{2} x^{2} + 28\,n^{2} x^{3} + 6\,n^{3} x^{2} + 6\,n^{3} x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
238112
343020155
4580150401276
561757005257011477
6733624503920147011218296
78588705620580164643528168148393

All 15 rows

Technical details
Node id
n-ddd727485ef0b1d6 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f7f5c9aa0a83f4e6

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:  2,    1;
  2:  3,    8,     1;
  3:  4,   30,    20,      1;
  4:  5,   80,   150,     40,      1;
  5:  6,  175,   700,    525,     70,      1;
  6:  7,  336,  2450,   3920,   1470,    112,      1;
  7:  8,  588,  7056,  20580,  16464,   3528,    168,     1;
  8:  9,  960, 17640,  84672, 123480,  56448,   7560,   240,   1;
  9: 10, 1485, 39600, 291060, 698544, 582120, 166320, 14850, 330, 1;
%F Formula
The row polynomials satisfy (n^5+10*n^4+35*n^3+50*n^2+24*n)*P(n,x) = (n^5+11*n^4+45*n^3+85*n^2+74*n+24+(10*n^5+50*n^4+75*n^3+22*n^2-25*n-12)*x)*P(n-1,x) + ((3*n^4+37*n^3+125*n^2+148*n+48)*x-(15*n^4+65*n^3+70*n^2-n-12)*x^2)*d/dx P(n-1,x) + ((6*n^3+28*n^2+37*n+12)*x^2+(6*n^3+28*n^2+37*n+12)*x^3)*d^2/dx^2 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[((n^5 + 11*n^4 + 45*n^3 + 85*n^2 + 74*n + 24 + (10*n^5 + 50*n^4 + 75*n^3 + 22*n^2 - 25*n - 12)*x)*P[n - 1, x] + ((3*n^4 + 37*n^3 + 125*n^2 + 148*n + 48)*x - (15*n^4 + 65*n^3 + 70*n^2 - n - 12)*x^2)*D[P[n - 1, x], x] + ((6*n^3 + 28*n^2 + 37*n + 12)*x^2 + (6*n^3 + 28*n^2 + 37*n + 12)*x^3)*D[P[n - 1, x], {x, 2}])/(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)]];
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] = ((n^5 + 11*n^4 + 45*n^3 + 85*n^2 + 74*n + 24 + (10*n^5 + 50*n^4 + 75*n^3 + 22*n^2 - 25*n - 12)*x)*v[i - 1] + ((3*n^4 + 37*n^3 + 125*n^2 + 148*n + 48)*x - (15*n^4 + 65*n^3 + 70*n^2 - n - 12)*x^2)*deriv(v[i - 1], x) + ((6*n^3 + 28*n^2 + 37*n + 12)*x^2 + (6*n^3 + 28*n^2 + 37*n + 12)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*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

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