Real-rooted atlas

A290595 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 - 24\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(-352 - 3668\,x + 1680\,x^{2} + 1548\,n + 6024\,n x - 6264\,n x^{2} - 1683\,n^{2} - 5184\,n^{2} x + 7857\,n^{2} x^{2} + 648\,n^{3} + 2160\,n^{3} x - 4050\,n^{3} x^{2} - 81\,n^{4} - 324\,n^{4} x + 729\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-303\,x - 937\,x^{2} + 1840\,x^{3} - 360\,n x - 2550\,n x^{2} - 6312\,n x^{3} + 621\,n^{2} x + 3780\,n^{2} x^{2} + 6939\,n^{2} x^{3} - 162\,n^{3} x - 1836\,n^{3} x^{2} - 2862\,n^{3} x^{3} + 324\,n^{4} x^{2} + 324\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(261\,x^{2} - 597\,x^{3} + 336\,x^{4} + 54\,n x^{2} + 1026\,n x^{3} - 1080\,n x^{4} - 81\,n^{2} x^{2} - 972\,n^{2} x^{3} + 1053\,n^{2} x^{4} + 324\,n^{3} x^{3} - 324\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-124 - 256\,x + 345\,n + 576\,n x - 261\,n^{2} - 432\,n^{2} x + 54\,n^{3} + 108\,n^{3} x\right)\,P_{n-1}(x) + \left(-39\,x + 64\,x^{2} - 90\,n x - 288\,n x^{2} + 54\,n^{2} x + 324\,n^{2} x^{2} - 108\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2189
347636116
42888813231282367
528012776371521419240064800
63640220096102209484060812087511522208465
7582404429344294190924100281613892100906012313689710740
8110656010210476890489970818802873601135391000190841400628194781924220920935

All 15 rows

Technical details
Node id
n-3ca93ea710386bee · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1c61ac6f25279709

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:         1,           8;
   3:         4,          76,            36;
   4:        28,         888,          1323,           128;
   5:       280,       12776,         37152,         14192,           400;
   6:      3640,      220096,       1022094,        840608,        120875,          1152;
   7:     58240,     4429344,      29419092,      41002816,      13892100,        906012,         3136;
   8:   1106560,   102104768,     904899708,    1880287360,    1135391000,     190841400,      6281947,        8192;
   9:  24344320,  2654422720,   29931906384,   85636988224,   79734580000,   25045725240,   2329328680,    41377216,     20736;
  10: 608608000, 76855645440, 1065255473664, 3975903159936, 5224911959700, 2609278142400, 475151297250, 26242459776, 262856259, 51200;
%F Formula
The row polynomials satisfy (9*n^2-24*n+16)*P(n,x) = -(81*n^4-648*n^3+1683*n^2-1548*n+352+(324*n^4-2160*n^3+5184*n^2-6024*n+3668)*x-(729*n^4-4050*n^3+7857*n^2-6264*n+1680)*x^2)*P(n-2,x) - ((162*n^3-621*n^2+360*n+303)*x-(324*n^4-1836*n^3+3780*n^2-2550*n-937)*x^2-(324*n^4-2862*n^3+6939*n^2-6312*n+1840)*x^3)*d/dx P(n-2,x) - ((81*n^2-54*n-261)*x^2-(324*n^3-972*n^2+1026*n-597)*x^3+(324*n^3-1053*n^2+1080*n-336)*x^4)*d^2/dx^2 P(n-2,x) + (54*n^3-261*n^2+345*n-124+(108*n^3-432*n^2+576*n-256)*x)*P(n-1,x) + ((54*n^2-90*n-39)*x-(108*n^3-324*n^2+288*n-64)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(81*n^4 - 648*n^3 + 1683*n^2 - 1548*n + 352 + (324*n^4 - 2160*n^3 + 5184*n^2 - 6024*n + 3668)*x - (729*n^4 - 4050*n^3 + 7857*n^2 - 6264*n + 1680)*x^2)*P[n - 2, x] - ((162*n^3 - 621*n^2 + 360*n + 303)*x - (324*n^4 - 1836*n^3 + 3780*n^2 - 2550*n - 937)*x^2 - (324*n^4 - 2862*n^3 + 6939*n^2 - 6312*n + 1840)*x^3)*D[P[n - 2, x], x] - ((81*n^2 - 54*n - 261)*x^2 - (324*n^3 - 972*n^2 + 1026*n - 597)*x^3 + (324*n^3 - 1053*n^2 + 1080*n - 336)*x^4)*D[P[n - 2, x], {x, 2}] + (54*n^3 - 261*n^2 + 345*n - 124 + (108*n^3 - 432*n^2 + 576*n - 256)*x)*P[n - 1, x] + ((54*n^2 - 90*n - 39)*x - (108*n^3 - 324*n^2 + 288*n - 64)*x^2)*D[P[n - 1, x], x])/(9*n^2 - 24*n + 16)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 8*x; for(n = 3, N, my(i = n); v[i] = (-(81*n^4 - 648*n^3 + 1683*n^2 - 1548*n + 352 + (324*n^4 - 2160*n^3 + 5184*n^2 - 6024*n + 3668)*x - (729*n^4 - 4050*n^3 + 7857*n^2 - 6264*n + 1680)*x^2)*v[i - 2] - ((162*n^3 - 621*n^2 + 360*n + 303)*x - (324*n^4 - 1836*n^3 + 3780*n^2 - 2550*n - 937)*x^2 - (324*n^4 - 2862*n^3 + 6939*n^2 - 6312*n + 1840)*x^3)*deriv(v[i - 2], x) - ((81*n^2 - 54*n - 261)*x^2 - (324*n^3 - 972*n^2 + 1026*n - 597)*x^3 + (324*n^3 - 1053*n^2 + 1080*n - 336)*x^4)*deriv(deriv(v[i - 2], x), x) + (54*n^3 - 261*n^2 + 345*n - 124 + (108*n^3 - 432*n^2 + 576*n - 256)*x)*v[i - 1] + ((54*n^2 - 90*n - 39)*x - (108*n^3 - 324*n^2 + 288*n - 64)*x^2)*deriv(v[i - 1], x))/(9*n^2 - 24*n + 16)); 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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.