Real-rooted atlas

narayana → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2224
3410519
48364214100
51611222416842562
63232096012006601323304
764864360066005940257442920071
8128224012320308004004028028100101430124996
9256563239424128128224224224224128128388964862793774
1051213824119808489216110073614676481188096572832151164167965120632
111024332803494401747200489216083166728910720604656025194005878605878633463102
1220487884898560059136002010624042223104573027845116320029845200109432402288132208012221060008
134096184320270336019148800775526401964666883274444803683750402813976001438254404707014489148007429001473830308
14819242598472417285974425628378521685135564817027112962341228032224367686414957845766799020802008801603476772026744409904186192
1516384974848190095361805905929932482563476368896819429811213462061184157057380481308811504077338861603163862520851809140135727830969484567015401391
Technical details
Node id
n-7a07750c70fa8b62 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-665cb75b242a18c2

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,     2;
   3:   4,    10,      5;
   4:   8,    36,     42,     14;
   5:  16,   112,    224,    168,      42;
   6:  32,   320,    960,   1200,     660,     132;
   7:  64,   864,   3600,   6600,    5940,    2574,     429;
   8: 128,  2240,  12320,  30800,   40040,   28028,   10010,   1430;
   9: 256,  5632,  39424, 128128,  224224,  224224,  128128,  38896,   4862;
  10: 512, 13824, 119808, 489216, 1100736, 1467648, 1188096, 572832, 151164, 16796;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (2*n+2+(2*n+2)*x)*P(n-1,x) + (4*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[((2*n + 2 + (2*n + 2)*x)*P[n - 1, x] + (4*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] = ((2*n + 2 + (2*n + 2)*x)*v[i - 1] + (4*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

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