Real-rooted atlas

A109692 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(4 + 48\,x - 324\,x^{2} - 16\,n - 176\,n x + 1080\,n x^{2} + 20\,n^{2} + 192\,n^{2} x - 1008\,n^{2} x^{2} - 8\,n^{3} - 64\,n^{3} x + 288\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-48\,x^{2} - 144\,x^{3} + 176\,n x^{2} + 480\,n x^{3} - 192\,n^{2} x^{2} - 448\,n^{2} x^{3} + 64\,n^{3} x^{2} + 128\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(2 + 24\,x - 12\,n - 64\,n x + 8\,n^{2} + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(-24\,x^{2} + 64\,n x^{2} - 32\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
2880108196
348800244819205216
43849088447847628842000172544
538401185288159042195968256560010886406788480
646080176025615657984582625281062736009434188832598720308941056
7645120294359043221867521539422208385474560052996700163795036672110702592015948168192

All 15 rows

Technical details
Node id
n-e8422031461468ed · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-cfe1d0f41456d01a

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:         8,          80,          108;
  3:        48,         800,         2448,          1920;
  4:       384,        9088,        44784,         76288,         42000;
  5:      3840,      118528,       815904,       2195968,       2565600,        1088640;
  6:     46080,     1760256,     15657984,      58262528,     106273600,       94341888,       32598720;
  7:    645120,    29435904,    322186752,    1539422208,    3854745600,     5299670016,     3795036672,     1107025920;
  8:  10321920,   548388864,   7141911552,   41814048768,  133836345600,   251319730176,   277123779072,   166415892480,   42032390400;
  9: 185794560, 11274780672, 170506156032, 1184341753856, 4630425561600, 11076382623744, 16618746647552, 15302044090368, 7917666900480, 1764322560000;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-20*n^2+16*n-4+(64*n^3-192*n^2+176*n-48)*x-(288*n^3-1008*n^2+1080*n-324)*x^2)*P(n-2,x) + ((64*n^3-192*n^2+176*n-48)*x^2+(128*n^3-448*n^2+480*n-144)*x^3)*d/dx P(n-2,x) + (8*n^2-12*n+2+(32*n^2-64*n+24)*x)*P(n-1,x) - (32*n^2-64*n+24)*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[(-(8*n^3 - 20*n^2 + 16*n - 4 + (64*n^3 - 192*n^2 + 176*n - 48)*x - (288*n^3 - 1008*n^2 + 1080*n - 324)*x^2)*P[n - 2, x] + ((64*n^3 - 192*n^2 + 176*n - 48)*x^2 + (128*n^3 - 448*n^2 + 480*n - 144)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 12*n + 2 + (32*n^2 - 64*n + 24)*x)*P[n - 1, x] - (32*n^2 - 64*n + 24)*x^2*D[P[n - 1, x], x])/(2*n - 3)]];
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] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (64*n^3 - 192*n^2 + 176*n - 48)*x - (288*n^3 - 1008*n^2 + 1080*n - 324)*x^2)*v[i - 2] + ((64*n^3 - 192*n^2 + 176*n - 48)*x^2 + (128*n^3 - 448*n^2 + 480*n - 144)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 12*n + 2 + (32*n^2 - 64*n + 24)*x)*v[i - 1] - (32*n^2 - 64*n + 24)*x^2*deriv(v[i - 1], x))/(2*n - 3)); 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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.