Real-rooted atlas

narayana → basis A007318 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(2 + 16\,x + 32\,x^{2} - n - 8\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 4\,x + 6\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2347
311281655
44516819264469
5197968166411522564237
6903550012640134406400102439907
74279310928952012896096000337924096387739
82079317550460748811065601128960630784172032163843858505
91030499907364006912883456011490304888832038993928519686553639130777
105188595597172259032966708172810625126410554163264684032230031364128768262144402972031
1126467233165582016498632049094592091768320011162296328842444804428595201307443201966080010485764202705311
121364886917925151210389667203493490880753181440010858315776105858170886893568000288718848072089600092274688419430444299426717
137103937310162589046484376448243155616005942141568099150999552115021479936926041374725070422016018080071680387553689642781900816777216471189693925
1437269351957685275964017960835216625294400045426512102486164264243211606340894721115966472192758625533952355410378752109460848640203927060481962934272671088645051001609115
1519688015193278093053224750069489611201863370883384772495104719570230579211047442165760123708213657601007737995264058844577792002392094408704643896967168105344139264892547891226843545654513542257795
Technical details
Node id
n-5a3d2864ab053941 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-f60b2631b48d7d2e

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:      3,       4;
   3:     11,      28,       16;
   4:     45,     168,      192,       64;
   5:    197,     968,     1664,     1152,       256;
   6:    903,    5500,    12640,    13440,      6400,      1024;
   7:   4279,   31092,    89520,   128960,     96000,     33792,     4096;
   8:  20793,  175504,   607488,  1106560,   1128960,    630784,   172032,    16384;
   9: 103049,  990736,  4006912,  8834560,  11490304,   8888320,  3899392,   851968,   65536;
  10: 518859, 5597172, 25903296, 67081728, 106251264, 105541632, 64684032, 23003136, 4128768, 262144;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(n-2+(8*n-16)*x+(16*n-32)*x^2)*P(n-2,x) + (6*n-3+(8*n-4)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (8*n - 16)*x + (16*n - 32)*x^2)*P[n - 2, x] + (6*n - 3 + (8*n - 4)*x)*P[n - 1, 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; v[2] = 3 + 4*x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 + (8*n - 16)*x + (16*n - 32)*x^2)*v[i - 2] + (6*n - 3 + (8*n - 4)*x)*v[i - 1])/(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