Real-rooted atlas

narayana → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-2 - 14\,x - 8\,x^{2} - n + 9\,n x + 20\,n x^{2} + n^{2} + 11\,n^{2} x - 10\,n^{2} x^{2} - 10\,n^{3} x - 13\,n^{3} x^{2} + 2\,n^{4} x + 17\,n^{4} x^{2} - 7\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(8\,x + 14\,x^{2} + 2\,x^{3} + n - 21\,n x - 47\,n x^{2} - 9\,n x^{3} + n^{2} + 11\,n^{2} x + 74\,n^{2} x^{2} + 14\,n^{2} x^{3} - 6\,n^{3} x - 47\,n^{3} x^{2} - 29\,n^{3} x^{3} + 2\,n^{4} x + 16\,n^{4} x^{2} + 34\,n^{4} x^{3} - 5\,n^{5} x^{2} - 17\,n^{5} x^{3} + n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-x - 3\,x^{2} + 3\,x^{3} + x^{4} + 2\,n x + 8\,n x^{2} - 6\,n x^{3} - 4\,n x^{4} - 12\,n^{2} x^{2} + 7\,n^{2} x^{3} + 5\,n^{2} x^{4} + 4\,n^{3} x^{2} + 4\,n^{3} x^{3} - 8\,n^{3} x^{4} - 7\,n^{4} x^{3} + 7\,n^{4} x^{4} + 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
33811
46482781
51016027064504
61540013509601252850
72184047256720262521615147
8281568132303136024500604834377077
9362688317521128961470007257612348512379808
1045432068040338688661500544320185220230407291825902
115566001336508870402425500299376016978504224004009510008607950
12669680245025209088076230001317254411205810464640088209066000133139942826
1378137284247104530240212355004892659258270212362419201146717017160001038181728182931696
1491189287027029161152536786251590114242525042522198676481043512472602600031491461572482197828630660
1510525480111793517489472125250125463783320946890945109933824073045872927327300055110055550368023068527443718474515
Technical details
Node id
n-3cfdab3970f8c02c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2751883146f8224a

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  0;
   2:  1;
   3:  3,    8;
   4:  6,   48,    27;
   5: 10,  160,   270,     64;
   6: 15,  400,  1350,    960,    125;
   7: 21,  840,  4725,   6720,   2625,    216;
   8: 28, 1568, 13230,  31360,  24500,   6048,    343;
   9: 36, 2688, 31752, 112896, 147000,  72576,  12348,   512;
  10: 45, 4320, 68040, 338688, 661500, 544320, 185220, 23040, 729;
%F Formula
The row polynomials satisfy (n^2-n-2+(2*n^4-10*n^3+11*n^2+9*n-14)*x+(n^6-7*n^5+17*n^4-13*n^3-10*n^2+20*n-8)*x^2)*P(n,x) = (n^2+n+(2*n^4-6*n^3+11*n^2-21*n+8)*x+(n^6-5*n^5+16*n^4-47*n^3+74*n^2-47*n+14)*x^2+(3*n^6-17*n^5+34*n^4-29*n^3+14*n^2-9*n+2)*x^3)*P(n-1,x) + ((2*n-1)*x+(4*n^3-12*n^2+8*n-3)*x^2+(2*n^5-7*n^4+4*n^3+7*n^2-6*n+3)*x^3-(2*n^5-7*n^4+8*n^3-5*n^2+4*n-1)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + (2*n^4 - 6*n^3 + 11*n^2 - 21*n + 8)*x + (n^6 - 5*n^5 + 16*n^4 - 47*n^3 + 74*n^2 - 47*n + 14)*x^2 + (3*n^6 - 17*n^5 + 34*n^4 - 29*n^3 + 14*n^2 - 9*n + 2)*x^3)*P[n - 1, x] + ((2*n - 1)*x + (4*n^3 - 12*n^2 + 8*n - 3)*x^2 + (2*n^5 - 7*n^4 + 4*n^3 + 7*n^2 - 6*n + 3)*x^3 - (2*n^5 - 7*n^4 + 8*n^3 - 5*n^2 + 4*n - 1)*x^4)*D[P[n - 1, x], x])/(n^2 - n - 2 + (2*n^4 - 10*n^3 + 11*n^2 + 9*n - 14)*x + (n^6 - 7*n^5 + 17*n^4 - 13*n^3 - 10*n^2 + 20*n - 8)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n^2 + n + (2*n^4 - 6*n^3 + 11*n^2 - 21*n + 8)*x + (n^6 - 5*n^5 + 16*n^4 - 47*n^3 + 74*n^2 - 47*n + 14)*x^2 + (3*n^6 - 17*n^5 + 34*n^4 - 29*n^3 + 14*n^2 - 9*n + 2)*x^3)*v[i - 1] + ((2*n - 1)*x + (4*n^3 - 12*n^2 + 8*n - 3)*x^2 + (2*n^5 - 7*n^4 + 4*n^3 + 7*n^2 - 6*n + 3)*x^3 - (2*n^5 - 7*n^4 + 8*n^3 - 5*n^2 + 4*n - 1)*x^4)*deriv(v[i - 1], x))/(n^2 - n - 2 + (2*n^4 - 10*n^3 + 11*n^2 + 9*n - 14)*x + (n^6 - 7*n^5 + 17*n^4 - 13*n^3 - 10*n^2 + 20*n - 8)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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