Real-rooted atlas

narayana → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-2 - 19\,x - 48\,x^{2} - 47\,x^{3} - 16\,x^{4} + 4\,n + 23\,n x + 42\,n x^{2} + 31\,n x^{3} + 8\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 14\,x^{2} - 30\,x^{3} - 26\,x^{4} - 8\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2257
35252959
414105249185553
5424201512231012575541
6132165079801860020820892558107
742964353910512292521028518517565445629799
81430250251831837257251677445225802516363554918256999013

All 15 rows

Technical details
Node id
n-c5893904d5bd8161 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-68cd55a004f0db2e

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     2,      5;
   3:     5,     25,      29;
   4:    14,    105,     249,      185;
   5:    42,    420,    1512,     2310,     1257;
   6:   132,   1650,    7980,    18600,    20820,      8925;
   7:   429,   6435,   39105,   122925,   210285,    185175,     65445;
   8:  1430,  25025,  183183,   725725,  1677445,   2258025,   1636355,    491825;
   9:  4862,  97240,  832832,  3983980, 11625614,  21161140,  23432024,  14414620,   3768209;
  10: 16796, 377910, 3707496, 20791680, 73369296, 168754950, 252722652, 237383640, 126796662, 29324405;
%F Formula
The row polynomials satisfy (n+1+(2*n+2)*x+(n+1)*x^2)*P(n,x) = (4*n-2+(23*n-19)*x+(42*n-48)*x^2+(31*n-47)*x^3+(8*n-16)*x^4)*P(n-1,x) - (2*x+14*x^2+30*x^3+26*x^4+8*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 2 + (23*n - 19)*x + (42*n - 48)*x^2 + (31*n - 47)*x^3 + (8*n - 16)*x^4)*P[n - 1, x] - (2*x + 14*x^2 + 30*x^3 + 26*x^4 + 8*x^5)*D[P[n - 1, x], x])/(n + 1 + (2*n + 2)*x + (n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n - 2 + (23*n - 19)*x + (42*n - 48)*x^2 + (31*n - 47)*x^3 + (8*n - 16)*x^4)*v[i - 1] - (2*x + 14*x^2 + 30*x^3 + 26*x^4 + 8*x^5)*deriv(v[i - 1], x))/(n + 1 + (2*n + 2)*x + (n + 1)*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