Real-rooted atlas

A121574 → x ↦ x+1 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(8 + 2\,x - 7\,n - n x + 3\,n^{2}\right)\,P_{n-1}(x) + \left(60 + 36\,x + 4\,x^{2} - 24\,n - 12\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(48 + 48\,x + 14\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
2189128
38060141155
4356356127191859
5158419809522192414760
67048105726398199333629126377
7313605488040048157603604478341146165
8139536279072238440113472328415910645391809956
9620864139694413678087638962678046104190368374414488275
102762528690632076241284888776201426655662910438513107105449124871243
11122918403380256041535392300748321424534446306681053282167523182481296541137821040
1254692416164077248222098704179236400959735163589554096091951853973255730245841563591763718929
1324335334479089836811693454081040729360621695984263062356810695521841341530809123749063224018556414232057765
1410827982083789793728607655465659128343523898696728184171000464245449416804035333102496488543453157641341217269123451445612

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(2 + x\right)\,P_{n-2}(x) + \left(4 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + x\)
Technical details
Node id
n-47772097820bf4fe · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-fdfc0c621074efe4)
Fingerprint
f6-55e87460da7bf930

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:      4,       1;
  2:     18,       9,       1;
  3:     80,      60,      14,      1;
  4:    356,     356,     127,     19,      1;
  5:   1584,    1980,     952,    219,     24,     1;
  6:   7048,   10572,    6398,   1993,    336,    29,    1;
  7:  31360,   54880,   40048,  15760,   3604,   478,   34,   1;
  8: 139536,  279072,  238440, 113472,  32841,  5910,  645,  39,  1;
  9: 620864, 1396944, 1367808, 763896, 267804, 61041, 9036, 837, 44, 1;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n^2-7*n+8-(n-2)*x)*P(n-1,x) - (24*n-60+(12*n-36)*x+(n-4)*x^2)*d/dx P(n-1,x) + (48+48*x+14*x^2+x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 7*n + 8 - (n - 2)*x)*P[n - 1, x] - (24*n - 60 + (12*n - 36)*x + (n - 4)*x^2)*D[P[n - 1, x], x] + (48 + 48*x + 14*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^2 - 7*n + 8 - (n - 2)*x)*v[i - 1] - (24*n - 60 + (12*n - 36)*x + (n - 4)*x^2)*deriv(v[i - 1], x) + (48 + 48*x + 14*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n)); 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. [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