Real-rooted atlas

A091042 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-12 + 33\,x - 3\,x^{2} + 32\,n - 127\,n x + 20\,n x^{2} - 28\,n^{2} + 178\,n^{2} x - 51\,n^{2} x^{2} + 8\,n^{3} - 108\,n^{3} x + 62\,n^{3} x^{2} + 24\,n^{4} x - 36\,n^{4} x^{2} + 8\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-48 - 156\,x - 132\,x^{2} - 24\,x^{3} + 128\,n + 260\,n x - 8\,n x^{2} - 20\,n x^{3} - 112\,n^{2} + 40\,n^{2} x + 544\,n^{2} x^{2} + 126\,n^{2} x^{3} + 32\,n^{3} - 240\,n^{3} x - 536\,n^{3} x^{2} - 44\,n^{3} x^{3} + 96\,n^{4} x + 96\,n^{4} x^{2} - 88\,n^{4} x^{3} + 32\,n^{5} x^{2} + 48\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-16\,x - 54\,x^{2} - 50\,x^{3} - 12\,x^{4} + 32\,n x + 60\,n x^{2} - 16\,n x^{3} - 16\,n x^{4} - 16\,n^{2} x + 40\,n^{2} x^{2} + 172\,n^{2} x^{3} + 56\,n^{2} x^{4} - 48\,n^{3} x^{2} - 96\,n^{3} x^{3} - 16\,n^{4} x^{3} - 32\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
242428
316160180356
46489620168963872
525646081555215360360039376
61024225281013761576968800012672383296
7409610649659904012779521164800419328407683612480
81638449152033177609011200115200006967296175616012288033203200

All 15 rows

Technical details
Node id
n-5c5729364af4d88b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e0a1e5842292e9c5

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:      4,      24;
   3:     16,     160,      180;
   4:     64,     896,     2016,       896;
   5:    256,    4608,    15552,     15360,      3600;
   6:   1024,   22528,   101376,    157696,     88000,     12672;
   7:   4096,  106496,   599040,   1277952,   1164800,    419328,     40768;
   8:  16384,  491520,  3317760,   9011200,  11520000,   6967296,   1756160,    122880;
   9:  65536, 2228224, 17547264,  57933824,  95744000,  82722816,  35825664,   6684672,   352512;
  10: 262144, 9961472, 89653248, 348651520, 708198400, 801275904, 503365632, 164364288, 23639040, 972800;
%F Formula
The row polynomials satisfy (8*n^3-28*n^2+32*n-12+(24*n^4-108*n^3+178*n^2-127*n+33)*x+(8*n^5-36*n^4+62*n^3-51*n^2+20*n-3)*x^2)*P(n,x) = (32*n^3-112*n^2+128*n-48+(96*n^4-240*n^3+40*n^2+260*n-156)*x+(32*n^5+96*n^4-536*n^3+544*n^2-8*n-132)*x^2+(48*n^5-88*n^4-44*n^3+126*n^2-20*n-24)*x^3)*P(n-1,x) - ((16*n^2-32*n+16)*x+(48*n^3-40*n^2-60*n+54)*x^2+(16*n^4+96*n^3-172*n^2+16*n+50)*x^3+(32*n^4-56*n^2+16*n+12)*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[((32*n^3 - 112*n^2 + 128*n - 48 + (96*n^4 - 240*n^3 + 40*n^2 + 260*n - 156)*x + (32*n^5 + 96*n^4 - 536*n^3 + 544*n^2 - 8*n - 132)*x^2 + (48*n^5 - 88*n^4 - 44*n^3 + 126*n^2 - 20*n - 24)*x^3)*P[n - 1, x] - ((16*n^2 - 32*n + 16)*x + (48*n^3 - 40*n^2 - 60*n + 54)*x^2 + (16*n^4 + 96*n^3 - 172*n^2 + 16*n + 50)*x^3 + (32*n^4 - 56*n^2 + 16*n + 12)*x^4)*D[P[n - 1, x], x])/(8*n^3 - 28*n^2 + 32*n - 12 + (24*n^4 - 108*n^3 + 178*n^2 - 127*n + 33)*x + (8*n^5 - 36*n^4 + 62*n^3 - 51*n^2 + 20*n - 3)*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] = ((32*n^3 - 112*n^2 + 128*n - 48 + (96*n^4 - 240*n^3 + 40*n^2 + 260*n - 156)*x + (32*n^5 + 96*n^4 - 536*n^3 + 544*n^2 - 8*n - 132)*x^2 + (48*n^5 - 88*n^4 - 44*n^3 + 126*n^2 - 20*n - 24)*x^3)*v[i - 1] - ((16*n^2 - 32*n + 16)*x + (48*n^3 - 40*n^2 - 60*n + 54)*x^2 + (16*n^4 + 96*n^3 - 172*n^2 + 16*n + 50)*x^3 + (32*n^4 - 56*n^2 + 16*n + 12)*x^4)*deriv(v[i - 1], x))/(8*n^3 - 28*n^2 + 32*n - 12 + (24*n^4 - 108*n^3 + 178*n^2 - 127*n + 33)*x + (8*n^5 - 36*n^4 + 62*n^3 - 51*n^2 + 20*n - 3)*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

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