Real-rooted atlas

A257627 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 - 56\,n + 49\,n^{2}\right)\,P_{n}(x) = \left(-144 - 10208\,x - 2288\,x^{2} - 84\,n + 5572\,n x - 1624\,n x^{2} + 588\,n^{2} + 7056\,n^{2} x + 33761\,n^{2} x^{2} - 4116\,n^{3} x - 54194\,n^{3} x^{2} + 21609\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(2464\,x - 10008\,x^{2} - 2896\,x^{3} + 294\,n x + 357\,n x^{2} - 4424\,n x^{3} - 2058\,n^{2} x + 6664\,n^{2} x^{2} + 45227\,n^{2} x^{3} + 4116\,n^{3} x^{2} - 55566\,n^{3} x^{3} + 9604\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(1960\,x^{2} - 1400\,x^{3} - 560\,x^{4} + 343\,n x^{2} + 833\,n x^{3} - 1176\,n x^{4} - 2401\,n^{2} x^{2} - 6860\,n^{2} x^{3} + 9261\,n^{2} x^{4} + 9604\,n^{3} x^{3} - 9604\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(96 - 256\,x - 140\,n + 1344\,n x - 49\,n^{2} - 2352\,n^{2} x + 1372\,n^{3} x\right)\,P_{n-1}(x) + \left(-168\,x - 192\,x^{2} - 441\,n x + 224\,n x^{2} + 686\,n^{2} x + 980\,n^{2} x^{2} - 1372\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131215
2924081330
3273012677743210248
481327122304181308482025396084
52433371645683554101040962107275874818241080
672934085761201376074849966083337155753067718435721972972000
721873421698023337725671826083560028532555625933509026841915800513996858915771200

All 15 rows

Technical details
Node id
n-910c53358e788124 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-245d5db50a17a5b1

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,         12;
  2:     9,        240,           81;
  3:    27,       3012,         6777,            432;
  4:    81,      32712,       230418,         130848,           2025;
  5:   243,     337164,      5683554,       10104096,        2107275,            8748;
  6:   729,    3408576,    120137607,      484996608,      333715575,        30677184,          35721;
  7:  2187,   34216980,   2333772567,    18260835600,    28532555625,      9335090268,      419158005,        139968;
  8:  6561,  342624696,  43138602864,   595919356704,  1769018448750,   1340818552584,   234865726704,    5481995136,      531441;
  9: 19683, 3427795356, 773443338120, 17753152790016, 90222283715550, 129920088550392, 54369030419424, 5500041515520, 69412855959, 1968300;
%F Formula
The row polynomials satisfy (49*n^2-56*n+16)*P(n,x) = (588*n^2-84*n-144-(4116*n^3-7056*n^2-5572*n+10208)*x+(21609*n^4-54194*n^3+33761*n^2-1624*n-2288)*x^2)*P(n-2,x) - ((2058*n^2-294*n-2464)*x-(4116*n^3+6664*n^2+357*n-10008)*x^2-(9604*n^4-55566*n^3+45227*n^2-4424*n-2896)*x^3)*d/dx P(n-2,x) - ((2401*n^2-343*n-1960)*x^2-(9604*n^3-6860*n^2+833*n-1400)*x^3+(9604*n^3-9261*n^2+1176*n+560)*x^4)*d^2/dx^2 P(n-2,x) - (49*n^2+140*n-96-(1372*n^3-2352*n^2+1344*n-256)*x)*P(n-1,x) + ((686*n^2-441*n-168)*x-(1372*n^3-980*n^2-224*n+192)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[((588*n^2 - 84*n - 144 - (4116*n^3 - 7056*n^2 - 5572*n + 10208)*x + (21609*n^4 - 54194*n^3 + 33761*n^2 - 1624*n - 2288)*x^2)*P[n - 2, x] - ((2058*n^2 - 294*n - 2464)*x - (4116*n^3 + 6664*n^2 + 357*n - 10008)*x^2 - (9604*n^4 - 55566*n^3 + 45227*n^2 - 4424*n - 2896)*x^3)*D[P[n - 2, x], x] - ((2401*n^2 - 343*n - 1960)*x^2 - (9604*n^3 - 6860*n^2 + 833*n - 1400)*x^3 + (9604*n^3 - 9261*n^2 + 1176*n + 560)*x^4)*D[P[n - 2, x], {x, 2}] - (49*n^2 + 140*n - 96 - (1372*n^3 - 2352*n^2 + 1344*n - 256)*x)*P[n - 1, x] + ((686*n^2 - 441*n - 168)*x - (1372*n^3 - 980*n^2 - 224*n + 192)*x^2)*D[P[n - 1, x], x])/(49*n^2 - 56*n + 16)]];
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 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = ((588*n^2 - 84*n - 144 - (4116*n^3 - 7056*n^2 - 5572*n + 10208)*x + (21609*n^4 - 54194*n^3 + 33761*n^2 - 1624*n - 2288)*x^2)*v[i - 2] - ((2058*n^2 - 294*n - 2464)*x - (4116*n^3 + 6664*n^2 + 357*n - 10008)*x^2 - (9604*n^4 - 55566*n^3 + 45227*n^2 - 4424*n - 2896)*x^3)*deriv(v[i - 2], x) - ((2401*n^2 - 343*n - 1960)*x^2 - (9604*n^3 - 6860*n^2 + 833*n - 1400)*x^3 + (9604*n^3 - 9261*n^2 + 1176*n + 560)*x^4)*deriv(deriv(v[i - 2], x), x) - (49*n^2 + 140*n - 96 - (1372*n^3 - 2352*n^2 + 1344*n - 256)*x)*v[i - 1] + ((686*n^2 - 441*n - 168)*x - (1372*n^3 - 980*n^2 - 224*n + 192)*x^2)*deriv(v[i - 1], x))/(49*n^2 - 56*n + 16)); 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.