Real-rooted atlas

A371081 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 12\,x + 9\,x^{2} - 12\,n - 16\,n x - 13\,n^{2} - 4\,n^{2} x - 6\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(-23\,x + 21\,x^{2} + 4\,x^{3} - 42\,n x + 16\,n x^{2} - 23\,n^{2} x + 4\,n^{2} x^{2} - 4\,n^{3} x\right)\,P_{n-2}'(x) + \left(-15\,x^{2} + 25\,x^{3} - 20\,n x^{2} + 8\,n x^{3} - 5\,n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(-2\,x^{3} + 4\,x^{4} - 2\,n x^{3}\right)\,P_{n-2}'''(x) + \left(10 + 4\,x + 8\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(9\,x - 4\,x^{2} + 4\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 16 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
116420
24002089617
3144001179210441627252
47056008122881110963456251632465
5451584006861772812937248615680900036127338092
63657830400704240640017197968961116574722483000200164912534194233
736578304000086676249600026349616742421927529472661132800807912039788641518638484668

All 15 rows

Technical details
Node id
n-b5c009c5db7508b7 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-580bcf0aafe7c36c

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:               16,                 4;
  2:              400,               208,                9;
  3:            14400,             11792,             1044,               16;
  4:           705600,            812288,           111096,             3456,             25;
  5:         45158400,          68617728,         12937248,           615680,           9000,            36;
  6:       3657830400,        7042406400,       1719796896,        111657472,        2483000,         20016,          49;
  7:     365783040000,      866762496000,     263496167424,      21927529472,      661132800,       8079120,       39788,       64;
  8:   44259747840000,   126276931584000,   46481067910656,    4766233407488,   183016644800,    3020285952,    22495312,    72704,     81;
  9: 6373403688960000, 21517840429056000, 9394412406552576, 1155035526303744, 54299500768000, 1136406608640, 11399425856, 55627776, 124416, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4+6*n^3+13*n^2+12*n+4+(4*n^2+16*n+12)*x-9*x^2)*P(n-2,x) - ((4*n^3+23*n^2+42*n+23)*x-(4*n^2+16*n+21)*x^2-4*x^3)*d/dx P(n-2,x) - ((5*n^2+20*n+15)*x^2-(8*n+25)*x^3)*d^2/dx^2 P(n-2,x) - ((2*n+2)*x^3-4*x^4)*d^3/dx^3 P(n-2,x) + (2*n^2+8*n+10+4*x)*P(n-1,x) + ((4*n+9)*x-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) = 16 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 16 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (4*n^2 + 16*n + 12)*x - 9*x^2)*P[n - 2, x] - ((4*n^3 + 23*n^2 + 42*n + 23)*x - (4*n^2 + 16*n + 21)*x^2 - 4*x^3)*D[P[n - 2, x], x] - ((5*n^2 + 20*n + 15)*x^2 - (8*n + 25)*x^3)*D[P[n - 2, x], {x, 2}] - ((2*n + 2)*x^3 - 4*x^4)*D[P[n - 2, x], {x, 3}] + (2*n^2 + 8*n + 10 + 4*x)*P[n - 1, x] + ((4*n + 9)*x - 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] = 16 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (4*n^2 + 16*n + 12)*x - 9*x^2)*v[i - 2] - ((4*n^3 + 23*n^2 + 42*n + 23)*x - (4*n^2 + 16*n + 21)*x^2 - 4*x^3)*deriv(v[i - 2], x) - ((5*n^2 + 20*n + 15)*x^2 - (8*n + 25)*x^3)*deriv(deriv(v[i - 2], x), x) - ((2*n + 2)*x^3 - 4*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2*n^2 + 8*n + 10 + 4*x)*v[i - 1] + ((4*n + 9)*x - 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.