Real-rooted atlas

narayana → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-2 - 13\,x - 28\,x^{2} - 25\,x^{3} - 8\,x^{4} + 4\,n + 17\,n x + 26\,n x^{2} + 17\,n x^{3} + 4\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 10\,x^{2} - 18\,x^{3} - 14\,x^{4} - 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2235
35151131
414639345215
5422525605461971597
613299029404320314090312425
7429386114355282153091517901427999955
81430150156706716516524216521136510167520793824675
94862583443043049009001655654193393214020165767081030496939769
1016796226746135252046781281033905615140034146884929103536327043851885959334605
11587868817905920590234304206051864010659448812965316010752651058188585185532752646723513972967
1220801234321982561713011415789033746394069479625610165903441056991320765316860367494435105324681136488694501041935
137429001337220010983033654424854018121365904270848120730525224091372358888293857660532790676022991601605984077507103937339784038517
1426744405215158046743268025496328009442140136250692645064909151676071786574288783856890966312170264436435419592142757372163402883848372693519354455513105
15969484520359174519774140751177321918548002149195141775518885312798513027523691390079668552103219647614352385468563894799245538217197880917074311936798863319688015193179928556219
Technical details
Node id
n-c0c21fffce187dd8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ca6d936c14642db2

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,      3;
   3:     5,     15,      11;
   4:    14,     63,      93,      45;
   5:    42,    252,     560,     546,      197;
   6:   132,    990,    2940,    4320,     3140,      903;
   7:   429,   3861,   14355,   28215,    30915,    17901,     4279;
   8:  1430,  15015,   67067,  165165,   242165,   211365,   101675,   20793;
   9:  4862,  58344,  304304,  900900,  1655654,  1933932,  1402016,  576708,  103049;
  10: 16796, 226746, 1352520, 4678128, 10339056, 15140034, 14688492, 9103536, 3270438, 518859;
%F Formula
The row polynomials satisfy (n+1+(2*n+2)*x+(n+1)*x^2)*P(n,x) = (4*n-2+(17*n-13)*x+(26*n-28)*x^2+(17*n-25)*x^3+(4*n-8)*x^4)*P(n-1,x) - (2*x+10*x^2+18*x^3+14*x^4+4*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 + (17*n - 13)*x + (26*n - 28)*x^2 + (17*n - 25)*x^3 + (4*n - 8)*x^4)*P[n - 1, x] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*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 + (17*n - 13)*x + (26*n - 28)*x^2 + (17*n - 25)*x^3 + (4*n - 8)*x^4)*v[i - 1] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*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