Real-rooted atlas

A099040 → basis A007318 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(6 + 2\,x - 2\,n\right)\,P_{n-1}(x) + \left(16 + 12\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
22018442
396128568288
4464816532152161980
522404896424018163843213608
61081628256305121741655369286493528
75222415872020531214643262144156802176128642816
8252160873984131724811270405984162017924217649922564418064
91217536473958481556488139008518918421910086123521091841126451230365280
105878784253936644911872056009984416824962114640074037121765760274432250881024208700064
1128385280134725632289363968371144704315784704187092992787390722353190448926726737925529620481434392064
1213705625670900531216741693442385584128228417817615478919687610499202734762247126374413129728162304012083240969858552768
136617661443705806848954159513614953981952159133091841213986560068291258882867610496898641920207526912343203843848192262144819267757668992
14319528960019256893440536926822409178979328010746908160091146004992577362183682774469849610161796096282243788858508697687762944900300856524816384465697330560

Other recurrences

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

\(P_{n}(x) = \left(4 + 2\,x\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
Technical details
Node id
n-2b25cdd134778a9f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-64c660101cf38859)
Fingerprint
f6-7dc17607242719e1

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,       2;
  2:      20,      18,       4;
  3:      96,     128,      56,       8;
  4:     464,     816,     532,     152,      16;
  5:    2240,    4896,    4240,    1816,     384,      32;
  6:   10816,   28256,   30512,   17416,    5536,     928,     64;
  7:   52224,  158720,  205312,  146432,   62144,   15680,   2176,    128;
  8:  252160,  873984, 1317248, 1127040,  598416,  201792,  42176,   4992,   256;
  9: 1217536, 4739584, 8155648, 8139008, 5189184, 2191008, 612352, 109184, 11264, 512;
%F Formula
The row polynomials satisfy n*P(n,x) = -(2*n-6-2*x)*P(n-1,x) + (16+12*x+2*x^2)*d/dx 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[(-(2*n - 6 - 2*x)*P[n - 1, x] + (16 + 12*x + 2*x^2)*D[P[n - 1, x], x])/(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] = (-(2*n - 6 - 2*x)*v[i - 1] + (16 + 12*x + 2*x^2)*deriv(v[i - 1], 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

Same fingerprint ?

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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741