Real-rooted atlas

narayana → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-n - 2\,n x - n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(-n - 3\,n x - 3\,n x^{2} - n x^{3} + n^{2} + 4\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-x - 3\,x^{2} - 3\,x^{3} - x^{4} + 2\,n x + 6\,n x^{2} + 6\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2134
3181019
4115453596
5124126224126501
613528084010504622668
7148540240049504752171614407
81639455775173252702721021643578592
9180154012320500501121121401409152024310432073
10199237624024126126378378672672700128393822923782390004
1111203510436802866501100736259896038188803401190167960035271613286043
121143500575075600600285885685765681662804020785050161661507113106135207874160672
131168693012320011781006785856250699686139584010072755010943240075508356299537285200300415382397
141195936019448021879001496523666512160199536480411543990584290850560919216347677200125550100200583002333445468
15122412376297024387987631039008162954792585307008147241294226176230083272028760281232224015819312605241902407755876013141557519
Technical details
Node id
n-13efab40bf59cd4d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-883c9e310e304e0b

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: 1,  3;
   3: 1,  8,   10;
   4: 1, 15,   45,    35;
   5: 1, 24,  126,   224,    126;
   6: 1, 35,  280,   840,   1050,    462;
   7: 1, 48,  540,  2400,   4950,   4752,   1716;
   8: 1, 63,  945,  5775,  17325,  27027,  21021,   6435;
   9: 1, 80, 1540, 12320,  50050, 112112, 140140,  91520,  24310;
  10: 1, 99, 2376, 24024, 126126, 378378, 672672, 700128, 393822, 92378;
%F Formula
The row polynomials satisfy (n^2-n+(2*n^2-2*n)*x+(n^2-n)*x^2)*P(n,x) = (n^2-n+(4*n^2-3*n)*x+(5*n^2-3*n)*x^2+(2*n^2-n)*x^3)*P(n-1,x) + ((2*n-1)*x+(6*n-3)*x^2+(6*n-3)*x^3+(2*n-1)*x^4)*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[((n^2 - n + (4*n^2 - 3*n)*x + (5*n^2 - 3*n)*x^2 + (2*n^2 - n)*x^3)*P[n - 1, x] + ((2*n - 1)*x + (6*n - 3)*x^2 + (6*n - 3)*x^3 + (2*n - 1)*x^4)*D[P[n - 1, x], x])/(n^2 - n + (2*n^2 - 2*n)*x + (n^2 - n)*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] = ((n^2 - n + (4*n^2 - 3*n)*x + (5*n^2 - 3*n)*x^2 + (2*n^2 - n)*x^3)*v[i - 1] + ((2*n - 1)*x + (6*n - 3)*x^2 + (6*n - 3)*x^3 + (2*n - 1)*x^4)*deriv(v[i - 1], x))/(n^2 - n + (2*n^2 - 2*n)*x + (n^2 - n)*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