Real-rooted atlas

narayana → basis A013610 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(18 + 72\,x + 72\,x^{2} - 9\,n - 36\,n x - 36\,n x^{2}\right)\,P_{n-2}(x) + \left(-5 - 6\,x + 10\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
25611
3296636131
41856126482161661
51257543684965616129622101
68925476109828097200453607776305151
76544541499010643401402920991440349920466564335711
849182536128401107288018249840173275209362304261273627993663009881
937682093146210411215814422194950426501126419377792083607552190356481679616932449961
1029324405274245966111475569625752504963707798976342725644820144194567155164161360488961007769614004694451
112311531332393447610109254090602886247080048673638720544644414724066070400019794274560592064640095738112060466176212944033051
121841801065209158988281059311127603150059576406091065604808007709248007263197061124510860710401859586854404766750208066512793603627970563271618296661
1314810069497183019254228101841546628833666971263207343894503440110975764711681185982232371289788945720324741530773760168388222464037513215590445712429056217678233650711564152381
141200296578051603483696386972422137346435377924344048859468246322401468122576972481807582533419521616681023626241043976494246404767337737676814785794017280289584610099231127987404813060694016792088104593511
159794701406611406537069779892329970579028366602408493048981661253581008187072197014102426085669966195842693701101446784206003939812070411537285898378244617577531607041264786724597762198767837593621027717365767836416409612454801769554551
Technical details
Node id
n-5d670dabde30c681 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-45c3a4579ee54f12

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,         6;
   3:       29,        66,         36;
   4:      185,       612,        648,        216;
   5:     1257,      5436,       8496,       5616,       1296;
   6:     8925,     47610,      98280,      97200,      45360,       7776;
   7:    65445,    414990,    1064340,    1402920,     991440,     349920,      46656;
   8:   491825,   3612840,   11072880,   18249840,   17327520,    9362304,    2612736,    279936;
   9:  3768209,  31462104,  112158144,  221949504,  265011264,  193777920,   83607552,  19035648,   1679616;
  10: 29324405, 274245966, 1114755696, 2575250496, 3707798976, 3427256448, 2014419456, 715516416, 136048896, 10077696;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(9*n-18+(36*n-72)*x+(36*n-72)*x^2)*P(n-2,x) + (10*n-5+(12*n-6)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 6*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n - 18 + (36*n - 72)*x + (36*n - 72)*x^2)*P[n - 2, x] + (10*n - 5 + (12*n - 6)*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] = 5 + 6*x; for(n = 3, N, my(i = n); v[i] = (-(9*n - 18 + (36*n - 72)*x + (36*n - 72)*x^2)*v[i - 2] + (10*n - 5 + (12*n - 6)*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