Real-rooted atlas

A063007 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-12 - 15\,x - 4\,x^{2} + 8\,n + 9\,n x + 2\,n x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 6\,x^{2} - 2\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2437
324361373
416036026463847
5112033603720180032110321
68064302404480032760118201683129367
7591362661124939204838402637607585289891651609
8439296230630451448326320160461664020049124792484863921360031
932947201976832051507456761080326974352040582080146412002994192265729278905249
10248934401680307205007974408648640009536567046962155203364838401038074401854882014625633668760487
11189190144141892608047608704009409720320121311590401065858393664629364802670370560719496360114148980809745348543499753
1214447247361191897907244459683840989700940801460747059201500850391041095309895685677399728020482381920489806460069873487245046719645382441711
131107622297699686006784409218281472101311322112016846199884801981955948544169160633241610553001338884775577206401528776849603286130047242584471282515959698614382884849
148520171520083071672320037216109199361014305119027218762459555840248734194508802431187945472017738420603904966068540236838787161212801115859064320217820421000258583317721409933619115367108888311
156572703744006901338931200335055745024009968600678400020304378148454429949414663168032989499262720027566044526592017558394357504084835568353536306057524572807994412840960142921121616015653127798079238482531549456900170553
Technical details
Node id
n-89174466dc5ea2fd · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-79e2479186e12230

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:        4,         3;
   3:       24,        36,        13;
   4:      160,       360,       264,        63;
   5:     1120,      3360,      3720,      1800,       321;
   6:     8064,     30240,     44800,     32760,     11820,      1683;
   7:    59136,    266112,    493920,    483840,    263760,     75852,      8989;
   8:   439296,   2306304,   5144832,   6320160,   4616640,   2004912,    479248,     48639;
   9:  3294720,  19768320,  51507456,  76108032,  69743520,  40582080,  14641200,   2994192,   265729;
  10: 24893440, 168030720, 500797440, 864864000, 953656704, 696215520, 336483840, 103807440, 18548820, 1462563;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (8*n-12+(9*n-15)*x+(2*n-4)*x^2)*P(n-1,x) - (4*x+6*x^2+2*x^3)*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[((8*n - 12 + (9*n - 15)*x + (2*n - 4)*x^2)*P[n - 1, x] - (4*x + 6*x^2 + 2*x^3)*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] = ((8*n - 12 + (9*n - 15)*x + (2*n - 4)*x^2)*v[i - 1] - (4*x + 6*x^2 + 2*x^3)*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

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