Real-rooted atlas

A111125 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-900 - 1080\,x + 8424\,x^{2} + 9720\,x^{3} - 2916\,x^{4} + 600\,n + 720\,n x - 17766\,n x^{2} - 19602\,n x^{3} + 10692\,n x^{4} + 12150\,n^{2} x^{2} + 13122\,n^{2} x^{3} - 15309\,n^{2} x^{4} - 2700\,n^{3} x^{2} - 2916\,n^{3} x^{3} + 10692\,n^{3} x^{4} - 3645\,n^{4} x^{4} + 486\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-600 - 1170\,x + 1296\,x^{2} + 3726\,x^{3} + 1944\,x^{4} + 900\,n - 45\,n x - 12204\,n x^{2} - 22599\,n x^{3} - 10692\,n x^{4} + 1575\,n^{2} x + 14850\,n^{2} x^{2} + 25758\,n^{2} x^{3} + 8262\,n^{2} x^{4} - 2916\,n^{2} x^{5} - 4050\,n^{3} x^{2} - 6399\,n^{3} x^{3} + 4617\,n^{3} x^{4} + 5832\,n^{3} x^{5} - 810\,n^{4} x^{3} - 4860\,n^{4} x^{4} - 3645\,n^{4} x^{5} + 729\,n^{5} x^{4} + 729\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(125 + 900\,x + 1440\,x^{2} - 1728\,x^{3} - 4617\,x^{4} - 1944\,x^{5} + 3150\,n x^{2} + 16200\,n x^{3} + 23166\,n x^{4} + 8748\,n x^{5} - 1575\,n^{2} x^{2} - 8100\,n^{2} x^{3} - 9963\,n^{2} x^{4} + 1458\,n^{2} x^{5} + 2916\,n^{2} x^{6} - 1620\,n^{3} x^{4} - 5832\,n^{3} x^{5} - 2916\,n^{3} x^{6} + 405\,n^{4} x^{4} + 1458\,n^{4} x^{5} + 729\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
241216
3118481176
4293728104321643
576138048605616202513957
619946802308542336324008748111448
7521150489598524451229362516621235721851624
81364467163664441203120199462517583487858621399686296447

All 15 rows

Technical details
Node id
n-79f9fc35447e5f9f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-144f627d80fdb61a

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,     12;
   3:   11,     84,      81;
   4:   29,    372,     810,      432;
   5:   76,   1380,    4860,     5616,     2025;
   6:  199,   4680,   23085,    42336,    32400,     8748;
   7:  521,  15048,   95985,   244512,   293625,   166212,    35721;
   8: 1364,  46716,  366444,  1203120,  1994625,  1758348,   785862,   139968;
   9: 3571, 141468, 1318680,  5319216, 11329875, 13725612,  9501786,  3499200,   531441;
  10: 9349, 420540, 4544505, 21787488, 56928825, 88363548, 83765745, 47589120, 14880348, 1968300;
%F Formula
The row polynomials satisfy (600*n-900+(720*n-1080)*x-(2700*n^3-12150*n^2+17766*n-8424)*x^2-(2916*n^3-13122*n^2+19602*n-9720)*x^3+(486*n^5-3645*n^4+10692*n^3-15309*n^2+10692*n-2916)*x^4)*P(n,x) = (900*n-600+(1575*n^2-45*n-1170)*x-(4050*n^3-14850*n^2+12204*n-1296)*x^2-(810*n^4+6399*n^3-25758*n^2+22599*n-3726)*x^3+(729*n^5-4860*n^4+4617*n^3+8262*n^2-10692*n+1944)*x^4+(729*n^5-3645*n^4+5832*n^3-2916*n^2)*x^5)*P(n-1,x) + (125+900*x-(1575*n^2-3150*n-1440)*x^2-(8100*n^2-16200*n+1728)*x^3+(405*n^4-1620*n^3-9963*n^2+23166*n-4617)*x^4+(1458*n^4-5832*n^3+1458*n^2+8748*n-1944)*x^5+(729*n^4-2916*n^3+2916*n^2)*x^6)*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[((900*n - 600 + (1575*n^2 - 45*n - 1170)*x - (4050*n^3 - 14850*n^2 + 12204*n - 1296)*x^2 - (810*n^4 + 6399*n^3 - 25758*n^2 + 22599*n - 3726)*x^3 + (729*n^5 - 4860*n^4 + 4617*n^3 + 8262*n^2 - 10692*n + 1944)*x^4 + (729*n^5 - 3645*n^4 + 5832*n^3 - 2916*n^2)*x^5)*P[n - 1, x] + (125 + 900*x - (1575*n^2 - 3150*n - 1440)*x^2 - (8100*n^2 - 16200*n + 1728)*x^3 + (405*n^4 - 1620*n^3 - 9963*n^2 + 23166*n - 4617)*x^4 + (1458*n^4 - 5832*n^3 + 1458*n^2 + 8748*n - 1944)*x^5 + (729*n^4 - 2916*n^3 + 2916*n^2)*x^6)*D[P[n - 1, x], x])/(600*n - 900 + (720*n - 1080)*x - (2700*n^3 - 12150*n^2 + 17766*n - 8424)*x^2 - (2916*n^3 - 13122*n^2 + 19602*n - 9720)*x^3 + (486*n^5 - 3645*n^4 + 10692*n^3 - 15309*n^2 + 10692*n - 2916)*x^4)]];
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] = ((900*n - 600 + (1575*n^2 - 45*n - 1170)*x - (4050*n^3 - 14850*n^2 + 12204*n - 1296)*x^2 - (810*n^4 + 6399*n^3 - 25758*n^2 + 22599*n - 3726)*x^3 + (729*n^5 - 4860*n^4 + 4617*n^3 + 8262*n^2 - 10692*n + 1944)*x^4 + (729*n^5 - 3645*n^4 + 5832*n^3 - 2916*n^2)*x^5)*v[i - 1] + (125 + 900*x - (1575*n^2 - 3150*n - 1440)*x^2 - (8100*n^2 - 16200*n + 1728)*x^3 + (405*n^4 - 1620*n^3 - 9963*n^2 + 23166*n - 4617)*x^4 + (1458*n^4 - 5832*n^3 + 1458*n^2 + 8748*n - 1944)*x^5 + (729*n^4 - 2916*n^3 + 2916*n^2)*x^6)*deriv(v[i - 1], x))/(600*n - 900 + (720*n - 1080)*x - (2700*n^3 - 12150*n^2 + 17766*n - 8424)*x^2 - (2916*n^3 - 13122*n^2 + 19602*n - 9720)*x^3 + (486*n^5 - 3645*n^4 + 10692*n^3 - 15309*n^2 + 10692*n - 2916)*x^4)); 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