Real-rooted atlas

A166961 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(6 - 10\,x + 15\,x^{2} - 12\,n + 24\,n x - 46\,n x^{2} - 8\,n^{2} x + 36\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(9\,x - 23\,x^{2} + 6\,x^{3} - 18\,n x + 60\,n x^{2} - 16\,n x^{3} - 28\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2\,x^{2} - 6\,x^{3} - 4\,n x^{2} + 16\,n x^{3} - 8\,n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(-19 + 18\,x + 14\,n - 24\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-27\,x + 6\,x^{2} + 18\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(-6\,x^{2} + 4\,n x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
29211
359843146
4361185046242677
5217532804268112028563823
6130615312101095150270000793561917362
77837982554923783553522371400221121029112770781135
8470289125870234119561059814607394203416896101591645210273283140399343
92821751190216218035931566307832688960043898483273543185898561051738103534489164514396100
1016930525286283722341048882505646436991413616837179462805658140278704344788934102465507074411945971010027961701947
1110158317143006894346030098733650055217623786240480316395559230020123153427112952142940135624374833405520243911746365398430011702482946605275
1260949904964553006450828550215070901501047530834355618024962552901190320161427325912669443295750608775324220975243959024454953865653922652297550131512261255963369310087400
1336569943199685754663248424143944168844283493049420306311900187012879377912994519183146180267176646403215173539208647524150569319188831017866898275284061798139602012821737490143044276135024439193463386085
14219419659411453046049203210679372022086460622228632543505116754241351384241216974839952127172576335255583021104090878979380628002102529862888804027215204531040204799344040062060075590034814645222203071335688916013989881114504395350307208696415
Technical details
Node id
n-951ab4fe319085d0 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c850b20c6ccbc440

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:       0;
  1:       1;
  2:       9,          2;
  3:      59,         84,           3;
  4:     361,       1850,         462,           4;
  5:    2175,      32804,       26811,        2028,           5;
  6:   13061,     531210,     1095150,      270000,        7935,          6;
  7:   78379,    8255492,    37835535,    22371400,     2211210,      29112,         7;
  8:  470289,  125870234,  1195610598,  1460739420,   341689610,   15916452,    102732,      8;
  9: 2821751, 1902162180, 35931566307, 83268896004, 38984832735, 4318589856, 105173810, 353448, 9;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = -(12*n-6+(8*n^2-24*n+10)*x+(8*n^3-36*n^2+46*n-15)*x^2)*P(n-2,x) - ((18*n-9)*x+(28*n^2-60*n+23)*x^2-(8*n^2-16*n+6)*x^3)*d/dx P(n-2,x) - ((4*n-2)*x^2+(8*n^2-16*n+6)*x^3)*d^2/dx^2 P(n-2,x) + (14*n-19+(8*n^2-24*n+18)*x)*P(n-1,x) + ((18*n-27)*x-(4*n-6)*x^2)*d/dx P(n-1,x) + (4*n-6)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 6 + (8*n^2 - 24*n + 10)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*P[n - 2, x] - ((18*n - 9)*x + (28*n^2 - 60*n + 23)*x^2 - (8*n^2 - 16*n + 6)*x^3)*D[P[n - 2, x], x] - ((4*n - 2)*x^2 + (8*n^2 - 16*n + 6)*x^3)*D[P[n - 2, x], {x, 2}] + (14*n - 19 + (8*n^2 - 24*n + 18)*x)*P[n - 1, x] + ((18*n - 27)*x - (4*n - 6)*x^2)*D[P[n - 1, x], x] + (4*n - 6)*x^2*D[P[n - 1, x], {x, 2}])/(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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n - 6 + (8*n^2 - 24*n + 10)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*v[i - 2] - ((18*n - 9)*x + (28*n^2 - 60*n + 23)*x^2 - (8*n^2 - 16*n + 6)*x^3)*deriv(v[i - 2], x) - ((4*n - 2)*x^2 + (8*n^2 - 16*n + 6)*x^3)*deriv(deriv(v[i - 2], x), x) + (14*n - 19 + (8*n^2 - 24*n + 18)*x)*v[i - 1] + ((18*n - 27)*x - (4*n - 6)*x^2)*deriv(v[i - 1], x) + (4*n - 6)*x^2*deriv(deriv(v[i - 1], x), 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.