Real-rooted atlas

narayana → basis A013611 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(32 + 256\,x + 512\,x^{2} - 16\,n - 128\,n x - 256\,n x^{2}\right)\,P_{n-2}(x) + \left(-6 - 16\,x + 12\,n + 32\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
261622
341208256505
430622565376409612034
52426233608192012288065536296122
62007623824011008002457600262144010485767486732
717148124171361385088041062400658636805347737616777216193620169
81500666244898081674677766191718401328087040163682713610569646082684354565102944330
91338620624817753619718594568736587776234890199043896088985638520487936204010946564294967296136636470622
101212674762517297120227864678401176836505603806063493127949695057921064011038720869730877440386547056640687194767363707692987636
11111267402625563902880259684723200153191178240057923312025601460425614950424718878965760275012124672001900523028480072155450572801099511627776101755238837386
121031893995625994559465629281221657601942518927360084093830430720248563372130304509493582495744719653982699520680322819686400404414120591360133040906960896175921860444162819798377013332
13965721689162646680231808327403418664962412704816332801176892572303360399158735693414496160618716856321649825357915750419873930370088960162370379631820808417861022253056242772167412940828147497671065678797575462345300
1491089699285626981461905216363608371316736294692405713305615993809821237248612325513746186241696143604596080643428342281210429445030431654611517445255025115246100483760945493505474561719812107702108164391009636686233645035996273704962218048507664607640
15865056660140127538906243128040157372338137603550042058594304021219175482477772890525275767544217628353726153700147206602340782718320640114384667035461222401459348461472815513613397739399962165248849271577387204608034579200888904089607881299347898368007205759403792793662835472217864007145
Technical details
Node id
n-89ebe2b84861e9f6 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-935f0bc54b4916f9

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:         6,         16;
   3:        41,        208,         256;
   4:       306,       2256,        5376,         4096;
   5:      2426,      23360,       81920,       122880,        65536;
   6:     20076,     238240,     1100800,      2457600,      2621440,      1048576;
   7:    171481,    2417136,    13850880,     41062400,     65863680,     53477376,      16777216;
   8:   1500666,   24489808,   167467776,    619171840,   1328087040,   1636827136,    1056964608,    268435456;
   9:  13386206,  248177536,  1971859456,   8736587776,  23489019904,  38960889856,   38520487936,  20401094656,   4294967296;
  10: 121267476, 2517297120, 22786467840, 117683650560, 380606349312, 794969505792, 1064011038720, 869730877440, 386547056640, 68719476736;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(16*n-32+(128*n-256)*x+(256*n-512)*x^2)*P(n-2,x) + (12*n-6+(32*n-16)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 16*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n - 32 + (128*n - 256)*x + (256*n - 512)*x^2)*P[n - 2, x] + (12*n - 6 + (32*n - 16)*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] = 6 + 16*x; for(n = 3, N, my(i = n); v[i] = (-(16*n - 32 + (128*n - 256)*x + (256*n - 512)*x^2)*v[i - 2] + (12*n - 6 + (32*n - 16)*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