Real-rooted atlas

A094816 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 7\,n - n^{2}\right)\,P_{n-4}(x) + \left(9 - 24\,x - 9\,n + 8\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(24\,x^{2} - 8\,n x^{2}\right)\,P_{n-3}'(x) + \left(4 + 8\,x + 36\,x^{2} - 8\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-8\,x^{2} + 16\,x^{3} + 8\,n x^{2}\right)\,P_{n-2}'(x) + \left(-1 + 8\,x + 2\,n\right)\,P_{n-1}(x) + \left(-8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 8\,x\)
\(P_{3}(x) = 5 + 40\,x + 36\,x^{2}\)
\(P_{4}(x) = 16 + 184\,x + 324\,x^{2} + 128\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
22810
35403681
416184324128652
565928234017924005525
6326535216920185608000115250310
7195735432130824181120112000311043136495573
81370026700011045161802752141400056448010976081925284400
91096012264576102263761888409617575600893491225025283604482073660878873
10986410213865681036182242107793922234568001354510084847001610076160111974451200755395522
119864101222762344114400263625170368002958530000205310246487937516823420928037635840332800012390410059970537
1210850511225381952561369427994032164212608411335320003180328588815656803008500689305610332748801323520009540608294912143281169268
13130206134531414193440176863488876439138132224602705250400509323161600280440923840103412711424256922772484235264000442956800265420806922242174997655501
141692679748641975685584824531213554646390811260288931657778480084846759528965124782126464212027561574460905600870412114319360016336246784142236057671991296160563235074959155582
152369751648056023382702888363865008278889887558244928015188478370520014748068221286496323173314368438469549506561416878817868832659314688005324517918725987332915244094668801910702083686400599089684319949
Technical details
Node id
n-c61130f531ece40f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e6b536d657c17a12

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:     16,      184,       324,       128;
   5:     65,      928,      2340,      1792,       400;
   6:    326,     5352,     16920,     18560,      8000,      1152;
   7:   1957,    35432,    130824,    181120,    112000,     31104,     3136;
   8:  13700,   267000,   1104516,   1802752,   1414000,    564480,   109760,     8192;
   9: 109601,  2264576,  10226376,  18884096,  17575600,   8934912,  2502528,   360448,   20736;
  10: 986410, 21386568, 103618224, 210779392, 223456800, 135451008, 48470016, 10076160, 1119744, 51200;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(8*n-24)*x)*P(n-3,x) - (8*n-24)*x^2*d/dx P(n-3,x) - (n^2-4+(8*n-8)*x-36*x^2)*P(n-2,x) + ((8*n-8)*x^2+16*x^3)*d/dx P(n-2,x) + (2*n-1+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + 8*x, P(3,x) = 5 + 40*x + 36*x^2, P(4,x) = 16 + 184*x + 324*x^2 + 128*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 8*x;
P[3, x_] := 5 + 40*x + 36*x^2;
P[4, x_] := 16 + 184*x + 324*x^2 + 128*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (8*n - 24)*x)*P[n - 3, x] - (8*n - 24)*x^2*D[P[n - 3, x], x] - (n^2 - 4 + (8*n - 8)*x - 36*x^2)*P[n - 2, x] + ((8*n - 8)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (2*n - 1 + 8*x)*P[n - 1, x] - 8*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 + 8*x; v[3] = 5 + 40*x + 36*x^2; v[4] = 16 + 184*x + 324*x^2 + 128*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 + (8*n - 24)*x)*v[i - 3] - (8*n - 24)*x^2*deriv(v[i - 3], x) - (n^2 - 4 + (8*n - 8)*x - 36*x^2)*v[i - 2] + ((8*n - 8)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (2*n - 1 + 8*x)*v[i - 1] - 8*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