Real-rooted atlas

narayana → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 15\,x + 4\,x^{2} - 4\,n - 7\,n x + 4\,n x^{2} - 6\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(-4 + 10\,x + 124\,x^{2} - 48\,x^{3} + 20\,n + 32\,n x - 212\,n x^{2} + 48\,n x^{3} - 32\,n^{2} - 104\,n^{2} x + 96\,n^{2} x^{2} + 16\,n^{3} + 48\,n^{3} x\right)\,P_{n-1}(x) + \left(-2\,x + 6\,x^{2} + 48\,x^{3} - 32\,x^{4} + 8\,n x + 16\,n x^{2} - 64\,n x^{3} - 8\,n^{2} x - 32\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
26410
325521693
49837233664870
53782128358419202568266
6145210920278402816010240102479636
7557752668182160283200195840522244096775765
8214502442441073072229152024640001254400258048163847623118
98265411028165893888161441282408806419238912756940812451846553675430590
10319124488498430805632103434240199600128223082496138657792436469765898240262144750591756
111234506213309201551950406186585601473239040214472294418785894409392947202428108802752512010485767503649746
124784276921221847600577603513206400998705664018047361024207174696961468465152060555264001312030720126877696419430475305338620
13185725003943907523640091136191600499206344236800013752668160019699059916818397554278410809114624037486592000692060160057881395216777216758242226868
1472209880167657672017121170560101169429504382920851456970829223936167546791526419615234129921525629321216757079212032224319766528357690245122617245696671088647656262469160
152811505057085928780793399370405202796144002217840584960644982309171213067174055936185050663649281811684818944011950763016192508543107072013042062458881816804720641174405120026843545677497832209221
Technical details
Node id
n-780e1545553d103c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1f944cc0b901d444

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,       4;
   3:     25,      52,       16;
   4:     98,     372,      336,        64;
   5:    378,    2128,     3584,      1920,       256;
   6:   1452,   10920,    27840,     28160,     10240,      1024;
   7:   5577,   52668,   182160,    283200,    195840,     52224,      4096;
   8:  21450,  244244,  1073072,   2291520,   2464000,   1254400,    258048,    16384;
   9:  82654, 1102816,  5893888,  16144128,  24088064,  19238912,   7569408,  1245184,   65536;
  10: 319124, 4884984, 30805632, 103434240, 199600128, 223082496, 138657792, 43646976, 5898240, 262144;
%F Formula
The row polynomials satisfy (4*n^3-6*n^2-4*n+6+(8*n^2-7*n-15)*x+(4*n+4)*x^2)*P(n,x) = (16*n^3-32*n^2+20*n-4+(48*n^3-104*n^2+32*n+10)*x+(96*n^2-212*n+124)*x^2+(48*n-48)*x^3)*P(n-1,x) - ((8*n^2-8*n+2)*x+(32*n^2-16*n-6)*x^2+(64*n-48)*x^3+32*x^4)*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[((16*n^3 - 32*n^2 + 20*n - 4 + (48*n^3 - 104*n^2 + 32*n + 10)*x + (96*n^2 - 212*n + 124)*x^2 + (48*n - 48)*x^3)*P[n - 1, x] - ((8*n^2 - 8*n + 2)*x + (32*n^2 - 16*n - 6)*x^2 + (64*n - 48)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(4*n^3 - 6*n^2 - 4*n + 6 + (8*n^2 - 7*n - 15)*x + (4*n + 4)*x^2)]];
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] = ((16*n^3 - 32*n^2 + 20*n - 4 + (48*n^3 - 104*n^2 + 32*n + 10)*x + (96*n^2 - 212*n + 124)*x^2 + (48*n - 48)*x^3)*v[i - 1] - ((8*n^2 - 8*n + 2)*x + (32*n^2 - 16*n - 6)*x^2 + (64*n - 48)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(4*n^3 - 6*n^2 - 4*n + 6 + (8*n^2 - 7*n - 15)*x + (4*n + 4)*x^2)); 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