Real-rooted atlas

narayana → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n x + 4\,n^{2} - 8\,n^{2} x + n^{2} x^{2} + 4\,n^{3} - 3\,n^{3} x - n^{3} x^{2} + 6\,n^{4} x - n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(4\,x^{2} - 2\,x^{3} - 2\,n x + 14\,n x^{2} + 3\,n x^{3} - 8\,n^{2} - 26\,n^{2} x - 12\,n^{2} x^{2} + 3\,n^{2} x^{3} + 16\,n^{3} - 30\,n^{3} x^{2} - 6\,n^{3} x^{3} + 24\,n^{4} x + 20\,n^{4} x^{2} - n^{4} x^{3} + 4\,n^{5} x^{2} + 3\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(x^{2} - x^{4} + 5\,n x^{2} + 6\,n x^{3} + n x^{4} - 8\,n^{2} x - 8\,n^{2} x^{2} + 2\,n^{2} x^{3} + 2\,n^{2} x^{4} - 12\,n^{3} x^{2} - 14\,n^{3} x^{3} - 2\,n^{3} x^{4} - 2\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2246
3520934
414848116195
542336504224251131
6132132027001920500366608
74295148133651320056259724938788
814302002063063800804812513860171564228357
9486277792288288448448350350144144301842816811346965
101679630232812888722376192229320012383283745565990443741007954650
115878611757205668650120931201392300093562563745560873600110565650012147011878
12208012457626424622290596904007994250064465632323270641005312018711001925009317144277958343
137429001782960010590782428765088043968375041442192025069968098233344245381403740000319440129601691643780607
142674440695354404519803601359804160233716340025241364721792502712851355648269374248554268007059624509184175761969721540260
1596948452714556601916570565632772504012084197125147241294201202470569367310305922592727137678978300117366249126977767843292332422557492086280
Technical details
Node id
n-68b0acc691b7061a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-097fa7332775e3f8

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,      4;
   3:     5,     20,       9;
   4:    14,     84,      81,      16;
   5:    42,    336,     504,     224,      25;
   6:   132,   1320,    2700,    1920,     500,      36;
   7:   429,   5148,   13365,   13200,    5625,     972,     49;
   8:  1430,  20020,   63063,   80080,   48125,   13860,   1715,    64;
   9:  4862,  77792,  288288,  448448,  350350,  144144,  30184,  2816,   81;
  10: 16796, 302328, 1288872, 2376192, 2293200, 1238328, 374556, 59904, 4374, 100;
%F Formula
The row polynomials satisfy (4*n^3+4*n^2+(6*n^4-3*n^3-8*n^2+n)*x+(n^5-n^4-n^3+n^2)*x^2)*P(n,x) = (16*n^3-8*n^2+(24*n^4-26*n^2-2*n)*x+(4*n^5+20*n^4-30*n^3-12*n^2+14*n+4)*x^2+(3*n^5-n^4-6*n^3+3*n^2+3*n-2)*x^3)*P(n-1,x) - (8*n^2*x+(12*n^3+8*n^2-5*n-1)*x^2+(2*n^4+14*n^3-2*n^2-6*n)*x^3+(2*n^4+2*n^3-2*n^2-n+1)*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 - 8*n^2 + (24*n^4 - 26*n^2 - 2*n)*x + (4*n^5 + 20*n^4 - 30*n^3 - 12*n^2 + 14*n + 4)*x^2 + (3*n^5 - n^4 - 6*n^3 + 3*n^2 + 3*n - 2)*x^3)*P[n - 1, x] - (8*n^2*x + (12*n^3 + 8*n^2 - 5*n - 1)*x^2 + (2*n^4 + 14*n^3 - 2*n^2 - 6*n)*x^3 + (2*n^4 + 2*n^3 - 2*n^2 - n + 1)*x^4)*D[P[n - 1, x], x])/(4*n^3 + 4*n^2 + (6*n^4 - 3*n^3 - 8*n^2 + n)*x + (n^5 - n^4 - n^3 + 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] = ((16*n^3 - 8*n^2 + (24*n^4 - 26*n^2 - 2*n)*x + (4*n^5 + 20*n^4 - 30*n^3 - 12*n^2 + 14*n + 4)*x^2 + (3*n^5 - n^4 - 6*n^3 + 3*n^2 + 3*n - 2)*x^3)*v[i - 1] - (8*n^2*x + (12*n^3 + 8*n^2 - 5*n - 1)*x^2 + (2*n^4 + 14*n^3 - 2*n^2 - 6*n)*x^3 + (2*n^4 + 2*n^3 - 2*n^2 - n + 1)*x^4)*deriv(v[i - 1], x))/(4*n^3 + 4*n^2 + (6*n^4 - 3*n^3 - 8*n^2 + n)*x + (n^5 - n^4 - n^3 + 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