Real-rooted atlas

narayana → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n x + 2\,n^{2} - 8\,n^{2} x + 2\,n^{2} x^{2} + 2\,n^{3} - 3\,n^{3} x - 2\,n^{3} x^{2} + 6\,n^{4} x - 2\,n^{4} x^{2} + 2\,n^{5} x^{2}\right)\,P_{n}(x) = \left(8\,x^{2} - 8\,x^{3} - 2\,n x + 28\,n x^{2} + 12\,n x^{3} - 4\,n^{2} - 26\,n^{2} x - 24\,n^{2} x^{2} + 12\,n^{2} x^{3} + 8\,n^{3} - 60\,n^{3} x^{2} - 24\,n^{3} x^{3} + 24\,n^{4} x + 40\,n^{4} x^{2} - 4\,n^{4} x^{3} + 8\,n^{5} x^{2} + 12\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(x^{2} - 4\,x^{4} + 5\,n x^{2} + 12\,n x^{3} + 4\,n x^{4} - 4\,n^{2} x - 8\,n^{2} x^{2} + 4\,n^{2} x^{3} + 8\,n^{2} x^{4} - 12\,n^{3} x^{2} - 28\,n^{3} x^{3} - 8\,n^{3} x^{4} - 4\,n^{4} x^{3} - 8\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
22810
35403681
414168324128634
542672201617924004922
6132264010800153608000115238084
7429102965346010560090000311043136294025
814304004025225264064077000044352010976081922265834
94862155584115315235875845605600461260819317763604482073617432350
10167966046565155488190095363669120039626496239715847667712111974451200133914412
1158786235144022674600967449602227680002994001922397158401118208002830464033280001239041027291162
1220801291525289848916047752320012790800002062900224206893209612867993604790016009856000095406082949127870481700
137429003565920042363129623012070407034940000132615014401604477952012573868032628176384019148800003271065602654208069222460227314132
1426744401390708801807921440108784332803739461440080772367104114720173568108973522944689598074882837852160072290549761042808832719912961605632460372567880
15969484554291132076662822605062180032019334715400047117214144076958116435286157191577666373814707234763688960012018303897626005045248321261158419107020836864003515483553401
Technical details
Node id
n-842205b36b684b27 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9d61137180e532ac

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,      8;
   3:     5,     40,      36;
   4:    14,    168,     324,      128;
   5:    42,    672,    2016,     1792,      400;
   6:   132,   2640,   10800,    15360,     8000,     1152;
   7:   429,  10296,   53460,   105600,    90000,    31104,     3136;
   8:  1430,  40040,  252252,   640640,   770000,   443520,   109760,    8192;
   9:  4862, 155584, 1153152,  3587584,  5605600,  4612608,  1931776,  360448,   20736;
  10: 16796, 604656, 5155488, 19009536, 36691200, 39626496, 23971584, 7667712, 1119744, 51200;
%F Formula
The row polynomials satisfy (2*n^3+2*n^2+(6*n^4-3*n^3-8*n^2+n)*x+(2*n^5-2*n^4-2*n^3+2*n^2)*x^2)*P(n,x) = (8*n^3-4*n^2+(24*n^4-26*n^2-2*n)*x+(8*n^5+40*n^4-60*n^3-24*n^2+28*n+8)*x^2+(12*n^5-4*n^4-24*n^3+12*n^2+12*n-8)*x^3)*P(n-1,x) - (4*n^2*x+(12*n^3+8*n^2-5*n-1)*x^2+(4*n^4+28*n^3-4*n^2-12*n)*x^3+(8*n^4+8*n^3-8*n^2-4*n+4)*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[((8*n^3 - 4*n^2 + (24*n^4 - 26*n^2 - 2*n)*x + (8*n^5 + 40*n^4 - 60*n^3 - 24*n^2 + 28*n + 8)*x^2 + (12*n^5 - 4*n^4 - 24*n^3 + 12*n^2 + 12*n - 8)*x^3)*P[n - 1, x] - (4*n^2*x + (12*n^3 + 8*n^2 - 5*n - 1)*x^2 + (4*n^4 + 28*n^3 - 4*n^2 - 12*n)*x^3 + (8*n^4 + 8*n^3 - 8*n^2 - 4*n + 4)*x^4)*D[P[n - 1, x], x])/(2*n^3 + 2*n^2 + (6*n^4 - 3*n^3 - 8*n^2 + n)*x + (2*n^5 - 2*n^4 - 2*n^3 + 2*n^2)*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] = ((8*n^3 - 4*n^2 + (24*n^4 - 26*n^2 - 2*n)*x + (8*n^5 + 40*n^4 - 60*n^3 - 24*n^2 + 28*n + 8)*x^2 + (12*n^5 - 4*n^4 - 24*n^3 + 12*n^2 + 12*n - 8)*x^3)*v[i - 1] - (4*n^2*x + (12*n^3 + 8*n^2 - 5*n - 1)*x^2 + (4*n^4 + 28*n^3 - 4*n^2 - 12*n)*x^3 + (8*n^4 + 8*n^3 - 8*n^2 - 4*n + 4)*x^4)*deriv(v[i - 1], x))/(2*n^3 + 2*n^2 + (6*n^4 - 3*n^3 - 8*n^2 + n)*x + (2*n^5 - 2*n^4 - 2*n^3 + 2*n^2)*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