Real-rooted atlas

A063007 → x ↦ x+1 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−3, −2)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-5 - 2\,x + 5\,n + 2\,n x\right)\,P_{n-1}(x) + \left(12 + 10\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2527
33730673
430537215020847
52641430026107007010321
6235254791038850156803150252129367
721344552185052962028560086310138609241651609
81961825559748068229004605720185955044906460060343221360031

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(1 - n\right)\,P_{n}(x) = \left(-2 + n\right)\,P_{n-2}(x) + \left(15 + 6\,x - 10\,n - 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 2\,x\)
Technical details
Node id
n-abad200ec25a15c5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-7a3496783bbc03d9)
Fingerprint
f6-5d0768679fdfa04b

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:         5,         2;
   3:        37,        30,          6;
   4:       305,       372,        150,        20;
   5:      2641,      4300,       2610,       700,        70;
   6:     23525,     47910,      38850,     15680,      3150,       252;
   7:    213445,    521850,     529620,    285600,     86310,     13860,      924;
   8:   1961825,   5597480,    6822900,   4605720,   1859550,    449064,    60060,     3432;
   9:  18205345,  59376600,   84491820,  68514600,  34629210,  11171160,  2246244,   257400,   12870;
  10: 170195525, 624579210, 1016225100, 962179680, 584233650, 235927692, 63363300, 10913760, 1093950, 48620;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n-5+(2*n-2)*x)*P(n-1,x) + (12+10*x+2*x^2)*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[((5*n - 5 + (2*n - 2)*x)*P[n - 1, x] + (12 + 10*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];
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] = ((5*n - 5 + (2*n - 2)*x)*v[i - 1] + (12 + 10*x + 2*x^2)*deriv(v[i - 1], x))/(n - 1)); 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

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