Real-rooted atlas

A059473 → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(16 + 8\,x - 11\,n - 4\,n x + 3\,n^{2}\right)\,P_{n-1}(x) + \left(67 + 68\,x + 16\,x^{2} - 22\,n - 20\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(33 + 52\,x + 26\,x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18412
2706816154
3608880416641968
45284101607216224025625156
54592011012810438448832112641024321552
6399064114680013595208504962959365427240964110184
73468032116151041653017612952576603238416691202539521638452537728
830138640115270016191446144180282368105246976389918728949760116326465536671554576
926191731211262774402138290688235187968016511098887670671362357657604621926452428802621448584032192
102276170336108700634242322119219229215678976239690091521339591577651640606721355595776231866368233308161048576109723932064
111978086656010387052262424658464512034927464140832792896716821425020518499379011584327213875207493648384113665638410276044841943041402527506688
12171903954496984412709120257096665830440487145543684281207073792320196904140817366009405446880769638401976483184644013529497654683238404487905281677721617927569400896
13149391683532892652640929282640037406412845755569497088538091871764484533779772620828153958580224130448638443524509341843456114847016550420945095884825899827200194615705667108864229156107877632
14129827584096008669187215744026762749967104050622220619264065537787606016061427577305907242981034135552022804961062092892192387891200282565337415686462839455744106956849152012106439065683886080002684354562929148988540544

Other recurrences

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

\(P_{n}(x) = \left(6 + 4\,x\right)\,P_{n-2}(x) + \left(8 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 4\,x\)
Technical details
Node id
n-da6732524307ae2d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-58fb064ba3810865)
Fingerprint
f6-f3b4375d2792bb93

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:         8,          4;
  2:        70,         68,         16;
  3:       608,        880,        416,         64;
  4:      5284,      10160,       7216,       2240,        256;
  5:     45920,     110128,     104384,      48832,      11264,      1024;
  6:    399064,    1146800,    1359520,     850496,     295936,     54272,      4096;
  7:   3468032,   11615104,   16530176,   12952576,    6032384,   1669120,    253952,    16384;
  8:  30138640,  115270016,  191446144,  180282368,  105246976,  38991872,   8949760,  1163264,   65536;
  9: 261917312, 1126277440, 2138290688, 2351879680, 1651109888, 767067136, 235765760, 46219264, 5242880, 262144;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n^2-11*n+16-(4*n-8)*x)*P(n-1,x) - (22*n-67+(20*n-68)*x+(4*n-16)*x^2)*d/dx P(n-1,x) + (33+52*x+26*x^2+4*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 - 11*n + 16 - (4*n - 8)*x)*P[n - 1, x] - (22*n - 67 + (20*n - 68)*x + (4*n - 16)*x^2)*D[P[n - 1, x], x] + (33 + 52*x + 26*x^2 + 4*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 - 11*n + 16 - (4*n - 8)*x)*v[i - 1] - (22*n - 67 + (20*n - 68)*x + (4*n - 16)*x^2)*deriv(v[i - 1], x) + (33 + 52*x + 26*x^2 + 4*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