Real-rooted atlas

A094816 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 7\,n - n^{2}\right)\,P_{n-4}(x) + \left(9 - 48\,x - 9\,n + 16\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(48\,x^{2} - 16\,n x^{2}\right)\,P_{n-3}'(x) + \left(4 + 16\,x + 144\,x^{2} - 16\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-16\,x^{2} + 64\,x^{3} + 16\,n x^{2}\right)\,P_{n-2}'(x) + \left(-1 + 16\,x + 2\,n\right)\,P_{n-1}(x) + \left(-16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 16\,x\)
\(P_{3}(x) = 5 + 80\,x + 144\,x^{2}\)
\(P_{4}(x) = 16 + 368\,x + 1296\,x^{2} + 1024\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
221618
3580144229
416368129610242704
5651856936014336640032017
6326107046768014848012800036864392054
7195770864523296144896017920009953282007045033109
81370053400044180641442201622624000180633607024640104857668148356

All 15 rows

Technical details
Node id
n-c56a959bf9d0484a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8bbf265e2eb9b49a

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,       16;
   3:      5,       80,       144;
   4:     16,      368,      1296,       1024;
   5:     65,     1856,      9360,      14336,       6400;
   6:    326,    10704,     67680,     148480,     128000,      36864;
   7:   1957,    70864,    523296,    1448960,    1792000,     995328,     200704;
   8:  13700,   534000,   4418064,   14422016,   22624000,   18063360,    7024640,    1048576;
   9: 109601,  4529152,  40905504,  151072768,  281209600,  285917184,  160161792,   46137344,   5308416;
  10: 986410, 42773136, 414472896, 1686235136, 3575308800, 4334432256, 3102081024, 1289748480, 286654464, 26214400;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(16*n-48)*x)*P(n-3,x) - (16*n-48)*x^2*d/dx P(n-3,x) - (n^2-4+(16*n-16)*x-144*x^2)*P(n-2,x) + ((16*n-16)*x^2+64*x^3)*d/dx P(n-2,x) + (2*n-1+16*x)*P(n-1,x) - 16*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + 16*x, P(3,x) = 5 + 80*x + 144*x^2, P(4,x) = 16 + 368*x + 1296*x^2 + 1024*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 16*x;
P[3, x_] := 5 + 80*x + 144*x^2;
P[4, x_] := 16 + 368*x + 1296*x^2 + 1024*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (16*n - 48)*x)*P[n - 3, x] - (16*n - 48)*x^2*D[P[n - 3, x], x] - (n^2 - 4 + (16*n - 16)*x - 144*x^2)*P[n - 2, x] + ((16*n - 16)*x^2 + 64*x^3)*D[P[n - 2, x], x] + (2*n - 1 + 16*x)*P[n - 1, x] - 16*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 16*x; v[3] = 5 + 80*x + 144*x^2; v[4] = 16 + 368*x + 1296*x^2 + 1024*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (16*n - 48)*x)*v[i - 3] - (16*n - 48)*x^2*deriv(v[i - 3], x) - (n^2 - 4 + (16*n - 16)*x - 144*x^2)*v[i - 2] + ((16*n - 16)*x^2 + 64*x^3)*deriv(v[i - 2], x) + (2*n - 1 + 16*x)*v[i - 1] - 16*x^2*deriv(v[i - 1], x)); 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