Real-rooted atlas

A166960 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-7\,x^{2} - 2\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(7\,x^{2} - 13\,x^{3} + 9\,n x^{2} - 14\,n x^{3} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(13\,x^{3} - 3\,x^{4} + 17\,n x^{3} - 4\,n x^{4}\right)\,P_{n-2}''(x) + \left(3\,x^{4} + 4\,n x^{4}\right)\,P_{n-2}'''(x) + \left(1 + 4\,n x\right)\,P_{n-1}(x) + \left(3\,x - 4\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
2124934
3110818916314
414482430928253832
511812259022780835253659084
61727225350362912021847511448491119868
712911623796991231928093938751355940335656425511540

All 15 rows

Technical details
Node id
n-c48848a8d81350db · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-935fde2140e0013c

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: 1,      4;
  2: 1,     24,         9;
  3: 1,    108,       189,         16;
  4: 1,    448,      2430,        928,          25;
  5: 1,   1812,     25902,      27808,        3525,         36;
  6: 1,   7272,    253503,     629120,      218475,      11448,        49;
  7: 1,  29116,   2379699,   12319280,     9393875,    1355940,     33565,       64;
  8: 1, 116496,  21875868,  222491936,   331091250,  102922560,   7181440,    91776,     81;
  9: 1, 466020, 198979740, 3832104000, 10363143150, 6089069160, 912246720, 34013184, 238869, 100;
%F Formula
The row polynomials satisfy P(n,x) = (9*n^2-2*n-7)*x^2*P(n-2,x) + ((9*n+7)*x^2+(4*n^2-14*n-13)*x^3)*d/dx P(n-2,x) + ((17*n+13)*x^3-(4*n+3)*x^4)*d^2/dx^2 P(n-2,x) + (4*n+3)*x^4*d^3/dx^3 P(n-2,x) + (1+4*n*x)*P(n-1,x) + (3*x-(4*n+4)*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[(9*n^2 - 2*n - 7)*x^2*P[n - 2, x] + ((9*n + 7)*x^2 + (4*n^2 - 14*n - 13)*x^3)*D[P[n - 2, x], x] + ((17*n + 13)*x^3 - (4*n + 3)*x^4)*D[P[n - 2, x], {x, 2}] + (4*n + 3)*x^4*D[P[n - 2, x], {x, 3}] + (1 + 4*n*x)*P[n - 1, x] + (3*x - (4*n + 4)*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (9*n^2 - 2*n - 7)*x^2*v[i - 2] + ((9*n + 7)*x^2 + (4*n^2 - 14*n - 13)*x^3)*deriv(v[i - 2], x) + ((17*n + 13)*x^3 - (4*n + 3)*x^4)*deriv(deriv(v[i - 2], x), x) + (4*n + 3)*x^4*deriv(deriv(deriv(v[i - 2], x), x), x) + (1 + 4*n*x)*v[i - 1] + (3*x - (4*n + 4)*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.