Real-rooted atlas

A081577 → basis A013611 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(4\,n\right)\,P_{n}(x) = \left(18\,x^{2} + 6\,x^{3} + 16\,n - 8\,n x - 30\,n x^{2} - 9\,n x^{3} + 16\,n^{2} x + 12\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(76\,x^{2} + 66\,x^{3} + 12\,x^{4} - 16\,n x - 56\,n x^{2} - 36\,n x^{3} - 6\,n x^{4}\right)\,P_{n-1}'(x) + \left(16\,x^{2} + 52\,x^{3} + 24\,x^{4} + 3\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21624646
3641609616336
4256896928352442476
51024460869124352120012018216
64096225284454440448181763936328134056
7163841064962621443153922022407040012544896986496
8655364915201449984218726418524169123842588163916824487259536
926214422282247667712139591681483161696030723827712915456120384668853422176
10104857699614723919052883755008107642880872693764552089615212544314342436544018272393128416
1141943044404019219503513647919923272509030471093452846311014420171980857977856105425921098240499202892992256
12167772161929379849500098562640314368460718080053225717764187750400227180544084809318421369856034688512327321613638421289237696
1367108864838860800454662553614109638656279330160643726376960034533801984225894727681047596236834182430727664476161123409929687808372608156665348736
1426843545636238786562144128204873517760512162973876224247059185664264457682944203572641792113632739328459532533761330805145626871644163589959682850355210179841152884469376
Technical details
Node id
n-d805e802700c1fb4 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-eb711c461076fbdf

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:     16,      24,       6;
  3:     64,     160,      96,       16;
  4:    256,     896,     928,      352,       44;
  5:   1024,    4608,    6912,     4352,     1200,     120;
  6:   4096,   22528,   44544,    40448,    18176,    3936,     328;
  7:  16384,  106496,  262144,   315392,   202240,   70400,   12544,    896;
  8:  65536,  491520, 1449984,  2187264,  1852416,  912384,  258816,  39168,   2448;
  9: 262144, 2228224, 7667712, 13959168, 14831616, 9603072, 3827712, 915456, 120384, 6688;
%F Formula
The row polynomials satisfy 4*n*P(n,x) = (16*n+(16*n^2-8*n)*x+(12*n^2-30*n+18)*x^2+(3*n^2-9*n+6)*x^3)*P(n-1,x) - (16*n*x+(56*n-76)*x^2+(36*n-66)*x^3+(6*n-12)*x^4)*d/dx P(n-1,x) + (16*x^2+52*x^3+24*x^4+3*x^5)*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[((16*n + (16*n^2 - 8*n)*x + (12*n^2 - 30*n + 18)*x^2 + (3*n^2 - 9*n + 6)*x^3)*P[n - 1, x] - (16*n*x + (56*n - 76)*x^2 + (36*n - 66)*x^3 + (6*n - 12)*x^4)*D[P[n - 1, x], x] + (16*x^2 + 52*x^3 + 24*x^4 + 3*x^5)*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] = ((16*n + (16*n^2 - 8*n)*x + (12*n^2 - 30*n + 18)*x^2 + (3*n^2 - 9*n + 6)*x^3)*v[i - 1] - (16*n*x + (56*n - 76)*x^2 + (36*n - 66)*x^3 + (6*n - 12)*x^4)*deriv(v[i - 1], x) + (16*x^2 + 52*x^3 + 24*x^4 + 3*x^5)*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

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