Real-rooted atlas

A056939 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-12 + 5\,n + 6\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(24 + 16\,n - 10\,n x + 9\,n^{2} + 11\,n^{3} + 10\,n^{3} x\right)\,P_{n-1}(x) + \left(36 + 48\,x + 12\,x^{2} + 36\,n + 39\,n x + 3\,n x^{2} - 12\,n^{2} - 27\,n^{2} x - 15\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(8 + 20\,x + 16\,x^{2} + 4\,x^{3} + 12\,n + 30\,n x + 24\,n x^{2} + 6\,n x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2628
33326362
4184232724492
51055184097516054035
6622213890105603140310634128
7376391025221021654516485055467296548
82328087490009260165493041590402032889682637400
914680805449560805707060286802396100485604441421392923930637
109414800396156006822720061807200314958008954820132678088800207010220933080
11612730802882466405670205206040848003772692001397246402964192033132001678052970112070744786
124039799522101238128464983728057013587844227318480194194440054489388888868736767764830096041361219667422404
13269424634715354029288377606121755240785378445061848975248073780248735441715191191228824521339716699540516373561613188995757535
1418153391102112490094626304428598656471988168652462212680930297301594590126618431940351738987606137381250630279650343983648528527462141835169778848
Technical details
Node id
n-c4bf52064cb30a51 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a02029c15ad60f22

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:       0;
  1:       1;
  2:       6,       2;
  3:      33,      26,       3;
  4:     184,     232,      72,       4;
  5:    1055,    1840,     975,     160,       5;
  6:    6222,   13890,   10560,    3140,     310,      6;
  7:   37639,  102522,  102165,   45164,    8505,    546,     7;
  8:  232808,  749000,  926016,  549304,  159040,  20328,   896,    8;
  9: 1468080, 5449560, 8057070, 6028680, 2396100, 485604, 44142, 1392, 9;
%F Formula
The row polynomials satisfy (n^3+6*n^2+5*n-12)*P(n,x) = (11*n^3+9*n^2+16*n+24+(10*n^3-10*n)*x)*P(n-1,x) - (12*n^2-36*n-36+(27*n^2-39*n-48)*x+(15*n^2-3*n-12)*x^2)*d/dx P(n-1,x) + (12*n+8+(30*n+20)*x+(24*n+16)*x^2+(6*n+4)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((11*n^3 + 9*n^2 + 16*n + 24 + (10*n^3 - 10*n)*x)*P[n - 1, x] - (12*n^2 - 36*n - 36 + (27*n^2 - 39*n - 48)*x + (15*n^2 - 3*n - 12)*x^2)*D[P[n - 1, x], x] + (12*n + 8 + (30*n + 20)*x + (24*n + 16)*x^2 + (6*n + 4)*x^3)*D[P[n - 1, x], {x, 2}])/(n^3 + 6*n^2 + 5*n - 12)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((11*n^3 + 9*n^2 + 16*n + 24 + (10*n^3 - 10*n)*x)*v[i - 1] - (12*n^2 - 36*n - 36 + (27*n^2 - 39*n - 48)*x + (15*n^2 - 3*n - 12)*x^2)*deriv(v[i - 1], x) + (12*n + 8 + (30*n + 20)*x + (24*n + 16)*x^2 + (6*n + 4)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^3 + 6*n^2 + 5*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

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