Real-rooted atlas

A056939 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(12 + 7\,n + n^{2}\right)\,P_{n}(x) = \left(12 + 36\,x + 36\,x^{2} + 24\,x^{3} + 7\,n - 4\,n x - 54\,n x^{2} - 36\,n x^{3} + n^{2} + 8\,n^{2} x + 18\,n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(24\,x + 72\,x^{2} + 96\,x^{3} + 48\,x^{4} + 3\,n x - 21\,n x^{2} - 48\,n x^{3} - 24\,n x^{4}\right)\,P_{n-1}'(x) + \left(6\,x^{2} + 24\,x^{3} + 30\,x^{4} + 12\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
216613
3113332269
412411618492417
514032592010554222763
6162785352069456222207419609
719117011129134055512613763910754146793

All 15 rows

Technical details
Node id
n-d3f0e02b70bbc824 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-bb0a136ce054ce63

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: 1,   2;
  2: 1,   6,    6;
  3: 1,  13,   33,    22;
  4: 1,  24,  116,   184,     92;
  5: 1,  40,  325,   920,   1055,     422;
  6: 1,  62,  785,  3520,   6945,    6222,    2074;
  7: 1,  91, 1701, 11291,  34055,   51261,   37639,   10754;
  8: 1, 128, 3388, 31808, 137326,  308672,  374500,  232808,   58202;
  9: 1, 174, 6306, 80934, 479220, 1507170, 2685690, 2724780, 1468080, 326240;
%F Formula
The row polynomials satisfy (n^2+7*n+12)*P(n,x) = (n^2+7*n+12+(8*n^2-4*n+36)*x+(18*n^2-54*n+36)*x^2+(12*n^2-36*n+24)*x^3)*P(n-1,x) + ((3*n+24)*x-(21*n-72)*x^2-(48*n-96)*x^3-(24*n-48)*x^4)*d/dx P(n-1,x) + (6*x^2+24*x^3+30*x^4+12*x^5)*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^2 + 7*n + 12 + (8*n^2 - 4*n + 36)*x + (18*n^2 - 54*n + 36)*x^2 + (12*n^2 - 36*n + 24)*x^3)*P[n - 1, x] + ((3*n + 24)*x - (21*n - 72)*x^2 - (48*n - 96)*x^3 - (24*n - 48)*x^4)*D[P[n - 1, x], x] + (6*x^2 + 24*x^3 + 30*x^4 + 12*x^5)*D[P[n - 1, x], {x, 2}])/(n^2 + 7*n + 12)]];
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^2 + 7*n + 12 + (8*n^2 - 4*n + 36)*x + (18*n^2 - 54*n + 36)*x^2 + (12*n^2 - 36*n + 24)*x^3)*v[i - 1] + ((3*n + 24)*x - (21*n - 72)*x^2 - (48*n - 96)*x^3 - (24*n - 48)*x^4)*deriv(v[i - 1], x) + (6*x^2 + 24*x^3 + 30*x^4 + 12*x^5)*deriv(deriv(v[i - 1], x), x))/(n^2 + 7*n + 12)); 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. [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