Real-rooted atlas

A008459 → x ↦ x+1 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(1 + 4\,x + 4\,x^{2} - n - 4\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 2\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21316433
363114608245
4321768624192161921
516835010552027205603215525
6898932016445803040010320153664127905
7486392016983396122965201416803561640321281067925
82657291257472248416026467841637440587776114688102402569004545
914625637777314176353922219481616890048792691222364163502082534451276499525
10809745347800080122356260177672960160668480934133763475584079564801025280614401024653808673
11450467192922929468336453401372173000143838816099913228846344883214099712026822400290048014643220485614995765
12251595969177985612855967994721029886176012283341840993261772855313448962110040064537166080865075207974912344064409648416454529
13140993361910799942322371197476007551873014410100413752093247453728606861066242791726080089580142081942786560268849152214056967987208192418895174885
14792384825365336473104243678752772543143729856805220624208835990611456623128554048336199378944130628289792358799400966725182464809582592562831361835008163843634723102113
Technical details
Node id
n-9a53de4e8b612343 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-e7d8403424fc1d9b

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       3,       2;
  2:      13,      16,        4;
  3:      63,     114,       60,        8;
  4:     321,     768,      624,      192,       16;
  5:    1683,    5010,     5520,     2720,      560,      32;
  6:    8989,   32016,    44580,    30400,    10320,    1536,      64;
  7:   48639,  201698,   339612,   296520,   141680,   35616,    4032,    128;
  8:  265729, 1257472,  2484160,  2646784,  1637440,  587776,  114688,  10240,   256;
  9: 1462563, 7777314, 17635392, 22194816, 16890048, 7926912, 2236416, 350208, 25344, 512;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n-1+(4*n-4)*x+(4*n-4)*x^2)*P(n-2,x) + (6*n-3+(4*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (4*n - 4)*x + (4*n - 4)*x^2)*P[n - 2, x] + (6*n - 3 + (4*n - 2)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (4*n - 4)*x + (4*n - 4)*x^2)*v[i - 2] + (6*n - 3 + (4*n - 2)*x)*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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