Real-rooted atlas

A113413 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,n\right)\,P_{n}(x) = \left(8 + 2\,x^{2} + 2\,x^{3} + 8\,n + 6\,n x - 6\,n x^{2} - 3\,n x^{3} + 6\,n^{2} x + 4\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-8\,x + 10\,x^{2} + 18\,x^{3} + 4\,x^{4} - 8\,n x - 20\,n x^{2} - 12\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}'(x) + \left(8\,x^{2} + 18\,x^{3} + 8\,x^{4} + x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
21624747
3641449217317
4256768768304412137
510243840524832649329914407
6409618432320002739212240272823997127
71638486016181248198400122048423847740577654797
865536393216974848130252810265604899841388802147213934414417
926214417694725046272797081676861444699648182707243699258564336329760487
10104857678643202536243246268416528711683957964819694080644761613330081576248119200635007
1141943043460300812451840025775308834095104030277632018442035277186560218023043967760419740196011352612477
1216777216150994944599785472138936320020897464322150760448155442380879594291228706688071255296115779841107984473219118849897
13671088646543114242843738112728970035212293242880144040919041207127244873330769923233319936102355072022649587233234624290339611424361476161767
14268435456281857228813304332288373964144646990961049691951202304877860618246184330854432337629184125050531843525172992703497472940927527561032275807414451220087
Technical details
Node id
n-c2a7afb6d35683f0 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-b4b323c4cf375c4a

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,       3;
  2:     16,      24,       7;
  3:     64,     144,      92,      17;
  4:    256,     768,     768,     304,      41;
  5:   1024,    3840,    5248,    3264,     932,      99;
  6:   4096,   18432,   32000,   27392,   12240,    2728,     239;
  7:  16384,   86016,  181248,  198400,  122048,   42384,    7740,    577;
  8:  65536,  393216,  974848, 1302528, 1026560,  489984,  138880,  21472,  1393;
  9: 262144, 1769472, 5046272, 7970816, 7686144, 4699648, 1827072, 436992, 58564, 3363;
%F Formula
The row polynomials satisfy (2*n+2)*P(n,x) = (8*n+8+(6*n^2+6*n)*x+(4*n^2-6*n+2)*x^2+(n^2-3*n+2)*x^3)*P(n-1,x) - ((8*n+8)*x+(20*n-10)*x^2+(12*n-18)*x^3+(2*n-4)*x^4)*d/dx P(n-1,x) + (8*x^2+18*x^3+8*x^4+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[((8*n + 8 + (6*n^2 + 6*n)*x + (4*n^2 - 6*n + 2)*x^2 + (n^2 - 3*n + 2)*x^3)*P[n - 1, x] - ((8*n + 8)*x + (20*n - 10)*x^2 + (12*n - 18)*x^3 + (2*n - 4)*x^4)*D[P[n - 1, x], x] + (8*x^2 + 18*x^3 + 8*x^4 + x^5)*D[P[n - 1, x], {x, 2}])/(2*n + 2)]];
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] = ((8*n + 8 + (6*n^2 + 6*n)*x + (4*n^2 - 6*n + 2)*x^2 + (n^2 - 3*n + 2)*x^3)*v[i - 1] - ((8*n + 8)*x + (20*n - 10)*x^2 + (12*n - 18)*x^3 + (2*n - 4)*x^4)*deriv(v[i - 1], x) + (8*x^2 + 18*x^3 + 8*x^4 + x^5)*deriv(deriv(v[i - 1], x), x))/(2*n + 2)); 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