Real-rooted atlas

A266213 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-18 + 9\,n\right)\,P_{n}(x) = \left(18 - 45\,n + 18\,n^{2}\right)\,P_{n-1}(x) + \left(375 + 423\,x + 108\,x^{2} - 123\,n - 126\,n x - 27\,n x^{2}\right)\,P_{n-1}'(x) + \left(164 + 291\,x + 162\,x^{2} + 27\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 9 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
29615
3496018127
4281498288541121
516013720318611881629857
69129262382973616578453648686703
75204917826025129818640876626165241458762623
829676111797861988928184145410102323280505832043746707905
916920017658400150239701667358011441250496692013311542012041312259001601
109647049489774301095901201417724101167922806350124622773960519193868234439366518968143
115500324930953790077795397011493986401105064370722074500324598914990798481964363422832281180984564756351
12313604441193755011454036763208976715110986600412075299474344070817648155828561441376290472551538755827235429440150827809
1317880352011203172524036871461522680260609808412861933073451240640464085496982137399714871194854421672110216262768050248005801062882353159916929
1410194593769742137455822478798389525028625788666911154414006793598632864912695833042645412302341060493271123125050227065780586009363990428077903231886463106335130095
Technical details
Node id
n-594bbffcdc3fa5ea · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-5d09cadc540fdad8

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:       1;
  2:       9,       6;
  3:      49,      60,       18;
  4:     281,     498,      288,       54;
  5:    1601,    3720,     3186,     1188,      162;
  6:    9129,   26238,    29736,    16578,     4536,     486;
  7:   52049,  178260,   251298,   186408,    76626,   16524,    1458;
  8:  296761, 1179786,  1988928,  1841454,  1010232,  328050,   58320,   4374;
  9: 1692001, 7658400, 15023970, 16673580, 11441250, 4966920, 1331154, 201204, 13122;
%F Formula
The row polynomials satisfy (9*n-18)*P(n,x) = (18*n^2-45*n+18)*P(n-1,x) - (123*n-375+(126*n-423)*x+(27*n-108)*x^2)*d/dx P(n-1,x) + (164+291*x+162*x^2+27*x^3)*d^2/dx^2 P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 9 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 9 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^2 - 45*n + 18)*P[n - 1, x] - (123*n - 375 + (126*n - 423)*x + (27*n - 108)*x^2)*D[P[n - 1, x], x] + (164 + 291*x + 162*x^2 + 27*x^3)*D[P[n - 1, x], {x, 2}])/(9*n - 18)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 9 + 6*x; for(n = 3, N, my(i = n + 1); v[i] = ((18*n^2 - 45*n + 18)*v[i - 1] - (123*n - 375 + (126*n - 423)*x + (27*n - 108)*x^2)*deriv(v[i - 1], x) + (164 + 291*x + 162*x^2 + 27*x^3)*deriv(deriv(v[i - 1], x), x))/(9*n - 18)); 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