Real-rooted atlas

A207607 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 + 580\,x + 2336\,x^{2} + 1808\,x^{3} - 1296\,x^{4} + 840\,n x^{2} + 3840\,n x^{3} + 4320\,n x^{4} - 140\,n^{2} x^{2} - 1040\,n^{2} x^{3} - 1056\,n^{2} x^{4} + 576\,n^{2} x^{5} - 192\,n^{3} x^{4} - 384\,n^{3} x^{5} + 16\,n^{4} x^{4} + 64\,n^{4} x^{5}\right)\,P_{n}(x) = \left(-25 - 580\,x - 3036\,x^{2} - 4608\,x^{3} - 1792\,x^{4} - 256\,x^{5} - 560\,n x^{2} - 1760\,n x^{3} - 1696\,n x^{4} - 256\,n x^{5} + 140\,n^{2} x^{2} + 1040\,n^{2} x^{3} + 1536\,n^{2} x^{4} + 192\,n^{2} x^{5} + 128\,n^{3} x^{4} + 128\,n^{3} x^{5} - 16\,n^{4} x^{4} - 64\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(75 + 1940\,x + 12068\,x^{2} + 27872\,x^{3} + 26144\,x^{4} + 8832\,x^{5} + 1152\,x^{6} + 2100\,n x^{2} + 10400\,n x^{3} + 17744\,n x^{4} + 10752\,n x^{5} + 1536\,n x^{6} - 420\,n^{2} x^{2} - 3520\,n^{2} x^{3} - 6976\,n^{2} x^{4} - 4000\,n^{2} x^{5} - 256\,n^{2} x^{6} - 480\,n^{3} x^{4} - 1088\,n^{3} x^{5} - 512\,n^{3} x^{6} + 48\,n^{4} x^{4} + 224\,n^{4} x^{5} + 128\,n^{4} x^{6}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131619
287272152
3212645402561081
455888273626888007167
51442840116641753610800230445288
63778784450369139286400380166272276277
798726528162972416768534000360576122304163841640519

All 15 rows

Technical details
Node id
n-e3e3a7704a5fb54e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-fc3bcf52c051b0a7

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    1;
  1:    3,     16;
  2:    8,     72,      72;
  3:   21,    264,     540,     256;
  4:   55,    888,    2736,    2688,      800;
  5:  144,   2840,   11664,   17536,    10800,     2304;
  6:  377,   8784,   45036,   91392,    86400,    38016,     6272;
  7:  987,  26528,  162972,  416768,   534000,   360576,   122304,   16384;
  8: 2584,  78696,  563256, 1738368,  2818000,  2581632,  1342208,  368640,   41472;
  9: 6765, 230232, 1880928, 6801024, 13352400, 15500160, 10932096, 4595712, 1057536, 102400;
%F Formula
The row polynomials satisfy (25+580*x-(140*n^2-840*n-2336)*x^2-(1040*n^2-3840*n-1808)*x^3+(16*n^4-192*n^3-1056*n^2+4320*n-1296)*x^4+(64*n^4-384*n^3+576*n^2)*x^5)*P(n,x) = -(25+580*x-(140*n^2-560*n-3036)*x^2-(1040*n^2-1760*n-4608)*x^3+(16*n^4-128*n^3-1536*n^2+1696*n+1792)*x^4+(64*n^4-128*n^3-192*n^2+256*n+256)*x^5)*P(n-2,x) + (75+1940*x-(420*n^2-2100*n-12068)*x^2-(3520*n^2-10400*n-27872)*x^3+(48*n^4-480*n^3-6976*n^2+17744*n+26144)*x^4+(224*n^4-1088*n^3-4000*n^2+10752*n+8832)*x^5+(128*n^4-512*n^3-256*n^2+1536*n+1152)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(25 + 580*x - (140*n^2 - 560*n - 3036)*x^2 - (1040*n^2 - 1760*n - 4608)*x^3 + (16*n^4 - 128*n^3 - 1536*n^2 + 1696*n + 1792)*x^4 + (64*n^4 - 128*n^3 - 192*n^2 + 256*n + 256)*x^5)*P[n - 2, x] + (75 + 1940*x - (420*n^2 - 2100*n - 12068)*x^2 - (3520*n^2 - 10400*n - 27872)*x^3 + (48*n^4 - 480*n^3 - 6976*n^2 + 17744*n + 26144)*x^4 + (224*n^4 - 1088*n^3 - 4000*n^2 + 10752*n + 8832)*x^5 + (128*n^4 - 512*n^3 - 256*n^2 + 1536*n + 1152)*x^6)*P[n - 1, x])/(25 + 580*x - (140*n^2 - 840*n - 2336)*x^2 - (1040*n^2 - 3840*n - 1808)*x^3 + (16*n^4 - 192*n^3 - 1056*n^2 + 4320*n - 1296)*x^4 + (64*n^4 - 384*n^3 + 576*n^2)*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(25 + 580*x - (140*n^2 - 560*n - 3036)*x^2 - (1040*n^2 - 1760*n - 4608)*x^3 + (16*n^4 - 128*n^3 - 1536*n^2 + 1696*n + 1792)*x^4 + (64*n^4 - 128*n^3 - 192*n^2 + 256*n + 256)*x^5)*v[i - 2] + (75 + 1940*x - (420*n^2 - 2100*n - 12068)*x^2 - (3520*n^2 - 10400*n - 27872)*x^3 + (48*n^4 - 480*n^3 - 6976*n^2 + 17744*n + 26144)*x^4 + (224*n^4 - 1088*n^3 - 4000*n^2 + 10752*n + 8832)*x^5 + (128*n^4 - 512*n^3 - 256*n^2 + 1536*n + 1152)*x^6)*v[i - 1])/(25 + 580*x - (140*n^2 - 840*n - 2336)*x^2 - (1040*n^2 - 3840*n - 1808)*x^3 + (16*n^4 - 192*n^3 - 1056*n^2 + 4320*n - 1296)*x^4 + (64*n^4 - 384*n^3 + 576*n^2)*x^5)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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