Real-rooted atlas

A121574 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n\right)\,P_{n}(x) = \left(40 + 16\,x - 32\,n - 8\,n x + 12\,n^{2}\right)\,P_{n-1}(x) + \left(200 + 176\,x + 32\,x^{2} - 74\,n - 56\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x) + \left(111 + 158\,x + 68\,x^{2} + 8\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
22822454
3155182688413
48591342760184163161
547609284705226004643224192
6263776168458792291767936112064185149
71461653985464570522853681046242249626241281417003
8809956252286633820962546056117601634096060544601625610844766
9448827515722210241244602126468011856800433356810375041568001356851282998377
10248712439677611016725874016897468011043352048756352147182082998144394240302081024635212469
11137821040589766216113366372012914338409682167205013631041828833924697254483217929681926656020484861479168
12763718929356454397675431794729565938160809320120048103850562058810496639060352142733056223636482332672145408409637206416521
134232057765213950984664942550890469057678704650854001604367885043221466198976781960716821106964484168325125852672055316483153928192284752311487
142345144561212766067764631967636782048792358475250695383812837918261004821048589734488065235840278990164486658112512117802393614981324812943360679936163842179298263014
Technical details
Node id
n-bb656bb28f34987e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-7e54cdbe5f0f29a0

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:       5,        2;
  2:      28,       22,        4;
  3:     155,      182,       68,        8;
  4:     859,     1342,      760,      184,       16;
  5:    4760,     9284,     7052,     2600,      464,      32;
  6:   26377,    61684,    58792,    29176,     7936,    1120,      64;
  7:  146165,   398546,   457052,   285368,   104624,   22496,    2624,    128;
  8:  809956,  2522866,  3382096,  2546056,  1176016,  340960,   60544,   6016,   256;
  9: 4488275, 15722210, 24124460, 21264680, 11856800, 4333568, 1037504, 156800, 13568, 512;
%F Formula
The row polynomials satisfy 4*n*P(n,x) = (12*n^2-32*n+40-(8*n-16)*x)*P(n-1,x) - (74*n-200+(56*n-176)*x+(8*n-32)*x^2)*d/dx P(n-1,x) + (111+158*x+68*x^2+8*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[((12*n^2 - 32*n + 40 - (8*n - 16)*x)*P[n - 1, x] - (74*n - 200 + (56*n - 176)*x + (8*n - 32)*x^2)*D[P[n - 1, x], x] + (111 + 158*x + 68*x^2 + 8*x^3)*D[P[n - 1, x], {x, 2}])/(4*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] = ((12*n^2 - 32*n + 40 - (8*n - 16)*x)*v[i - 1] - (74*n - 200 + (56*n - 176)*x + (8*n - 32)*x^2)*deriv(v[i - 1], x) + (111 + 158*x + 68*x^2 + 8*x^3)*deriv(deriv(v[i - 1], x), x))/(4*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

Candidate equalities ?

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