Real-rooted atlas

A257617 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 70\,n + 49\,n^{2}\right)\,P_{n}(x) = \left(-100 - 12720\,x - 1200\,x^{2} - 210\,n + 5404\,n x - 14840\,n x^{2} + 490\,n^{2} + 5880\,n^{2} x + 59633\,n^{2} x^{2} - 2744\,n^{3} x - 66542\,n^{3} x^{2} + 21609\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(2485\,x - 11850\,x^{2} - 1175\,x^{3} + 588\,n x - 1435\,n x^{2} - 20860\,n x^{3} - 1372\,n^{2} x + 6468\,n^{2} x^{2} + 70217\,n^{2} x^{3} + 2744\,n^{3} x^{2} - 61054\,n^{3} x^{3} + 9604\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(2205\,x^{2} - 2030\,x^{3} - 175\,x^{4} + 1029\,n x^{2} + 3381\,n x^{3} - 4410\,n x^{4} - 2401\,n^{2} x^{2} - 10976\,n^{2} x^{3} + 13377\,n^{2} x^{4} + 9604\,n^{3} x^{3} - 9604\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(100 - 500\,x - 35\,n + 2100\,n x - 147\,n^{2} - 2940\,n^{2} x + 1372\,n^{3} x\right)\,P_{n-1}(x) + \left(-140\,x - 200\,x^{2} - 637\,n x - 140\,n x^{2} + 686\,n^{2} x + 1568\,n^{2} x^{2} - 1372\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
2414436184
38155234921285180
4161470411174458816400185680
5321342562548836453126483910011528054640
664121304049843656208438144138454600109173603136408870000
7128109286248984840767452405632116443838003593936304133875644819223734022400

All 15 rows

Technical details
Node id
n-c6b14968e3b3956f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b45811e67d078769

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:   2,         8;
  2:   4,       144,           36;
  3:   8,      1552,         3492,           128;
  4:  16,     14704,       111744,         58816,            400;
  5:  32,    134256,      2548836,       4531264,         839100,           1152;
  6:  64,   1213040,     49843656,     208438144,      138454600,       10917360,           3136;
  7: 128,  10928624,    898484076,    7452405632,    11644383800,     3593936304,      133875644,          8192;
  8: 256,  98383728,  15457678992,  230505615424,   698663028000,   518637634704,    84158474512,    1574139648,       20736;
  9: 512, 885512944, 258612396660, 6510763155904, 34285996731700, 49371835293648, 19939212164956, 1839021487360, 17931637116, 51200;
%F Formula
The row polynomials satisfy (49*n^2-70*n+25)*P(n,x) = (490*n^2-210*n-100-(2744*n^3-5880*n^2-5404*n+12720)*x+(21609*n^4-66542*n^3+59633*n^2-14840*n-1200)*x^2)*P(n-2,x) - ((1372*n^2-588*n-2485)*x-(2744*n^3+6468*n^2-1435*n-11850)*x^2-(9604*n^4-61054*n^3+70217*n^2-20860*n-1175)*x^3)*d/dx P(n-2,x) - ((2401*n^2-1029*n-2205)*x^2-(9604*n^3-10976*n^2+3381*n-2030)*x^3+(9604*n^3-13377*n^2+4410*n+175)*x^4)*d^2/dx^2 P(n-2,x) - (147*n^2+35*n-100-(1372*n^3-2940*n^2+2100*n-500)*x)*P(n-1,x) + ((686*n^2-637*n-140)*x-(1372*n^3-1568*n^2+140*n+200)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((490*n^2 - 210*n - 100 - (2744*n^3 - 5880*n^2 - 5404*n + 12720)*x + (21609*n^4 - 66542*n^3 + 59633*n^2 - 14840*n - 1200)*x^2)*P[n - 2, x] - ((1372*n^2 - 588*n - 2485)*x - (2744*n^3 + 6468*n^2 - 1435*n - 11850)*x^2 - (9604*n^4 - 61054*n^3 + 70217*n^2 - 20860*n - 1175)*x^3)*D[P[n - 2, x], x] - ((2401*n^2 - 1029*n - 2205)*x^2 - (9604*n^3 - 10976*n^2 + 3381*n - 2030)*x^3 + (9604*n^3 - 13377*n^2 + 4410*n + 175)*x^4)*D[P[n - 2, x], {x, 2}] - (147*n^2 + 35*n - 100 - (1372*n^3 - 2940*n^2 + 2100*n - 500)*x)*P[n - 1, x] + ((686*n^2 - 637*n - 140)*x - (1372*n^3 - 1568*n^2 + 140*n + 200)*x^2)*D[P[n - 1, x], x])/(49*n^2 - 70*n + 25)]];
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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((490*n^2 - 210*n - 100 - (2744*n^3 - 5880*n^2 - 5404*n + 12720)*x + (21609*n^4 - 66542*n^3 + 59633*n^2 - 14840*n - 1200)*x^2)*v[i - 2] - ((1372*n^2 - 588*n - 2485)*x - (2744*n^3 + 6468*n^2 - 1435*n - 11850)*x^2 - (9604*n^4 - 61054*n^3 + 70217*n^2 - 20860*n - 1175)*x^3)*deriv(v[i - 2], x) - ((2401*n^2 - 1029*n - 2205)*x^2 - (9604*n^3 - 10976*n^2 + 3381*n - 2030)*x^3 + (9604*n^3 - 13377*n^2 + 4410*n + 175)*x^4)*deriv(deriv(v[i - 2], x), x) - (147*n^2 + 35*n - 100 - (1372*n^3 - 2940*n^2 + 2100*n - 500)*x)*v[i - 1] + ((686*n^2 - 637*n - 140)*x - (1372*n^3 - 1568*n^2 + 140*n + 200)*x^2)*deriv(v[i - 1], x))/(49*n^2 - 70*n + 25)); 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. [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.