Real-rooted atlas

A059473 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n + 6\,n x + 4\,n x^{2} + 8\,n x^{3} + 2\,n x^{4} + 6\,n^{2} + 9\,n^{2} x + 10\,n^{2} x^{2} + 2\,n^{2} x^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(-12 + 24\,x + 112\,x^{2} + 108\,x^{3} + 36\,x^{4} + 4\,x^{5} + 90\,n + 138\,n x + 42\,n x^{2} - 12\,n x^{3} - 4\,n x^{4} + 12\,n^{2} - 18\,n^{2} x - 34\,n^{2} x^{2} - 8\,n^{2} x^{3} - 18\,n^{3} - 12\,n^{3} x\right)\,P_{n-1}(x) + \left(72 + 336\,x + 612\,x^{2} + 548\,x^{3} + 248\,x^{4} + 52\,x^{5} + 4\,x^{6} + 108\,n + 336\,n x + 390\,n x^{2} + 206\,n x^{3} + 48\,n x^{4} + 4\,n x^{5} + 36\,n^{2} + 72\,n^{2} x + 42\,n^{2} x^{2} + 6\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1628
23626466
3200232808520
410681728964216163992
5554411576906432885443230048
62819272328734243744810016131264222784
71411204301445384323567361341442835230721281632128
8697680246566436770883009600147105643680076224704025611841408
9341475213736352238047362325062414051424542313613282561972481587251285222912
10165751687480566414782713616802323212128889657895968184245123836288495616353281024609208832
11798969603998882568879375361152300672968832384552641664217510656588148481064345612165127782420484329762816
12382846912210494412851900684167576196224727886265648385881602276148288761043584178749952285921282929664169984409630619144192
131825150080109365600002965488627248114377856520371127043954870336021686871168866662489625162803205220894727480320069447683686408192215590780928
1486623856645619202470416621424307229681926540835705274828830563724582419168124390489303076992309863188487944680448147566387219142451216244736794624163841512177377280
Technical details
Node id
n-225abc33ea3874fb · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-45335b284fc98198

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:       6,        2;
  2:      36,       26,        4;
  3:     200,      232,       80,        8;
  4:    1068,     1728,      964,      216,       16;
  5:    5544,    11576,     9064,     3288,      544,      32;
  6:   28192,    72328,    73424,    37448,    10016,    1312,      64;
  7:  141120,   430144,   538432,   356736,   134144,   28352,    3072,    128;
  8:  697680,  2465664,  3677088,  3009600,  1471056,  436800,   76224,   7040,   256;
  9: 3414752, 13736352, 23804736, 23250624, 14051424, 5423136, 1328256, 197248, 15872, 512;
%F Formula
The row polynomials satisfy (6*n^2+6*n+(3*n^3+9*n^2+6*n)*x+(10*n^2+4*n)*x^2+(2*n^2+8*n)*x^3+2*n*x^4)*P(n,x) = -(18*n^3-12*n^2-90*n+12+(12*n^3+18*n^2-138*n-24)*x+(34*n^2-42*n-112)*x^2+(8*n^2+12*n-108)*x^3+(4*n-36)*x^4-4*x^5)*P(n-1,x) + (36*n^2+108*n+72+(72*n^2+336*n+336)*x+(42*n^2+390*n+612)*x^2+(6*n^2+206*n+548)*x^3+(48*n+248)*x^4+(4*n+52)*x^5+4*x^6)*d/dx 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[(-(18*n^3 - 12*n^2 - 90*n + 12 + (12*n^3 + 18*n^2 - 138*n - 24)*x + (34*n^2 - 42*n - 112)*x^2 + (8*n^2 + 12*n - 108)*x^3 + (4*n - 36)*x^4 - 4*x^5)*P[n - 1, x] + (36*n^2 + 108*n + 72 + (72*n^2 + 336*n + 336)*x + (42*n^2 + 390*n + 612)*x^2 + (6*n^2 + 206*n + 548)*x^3 + (48*n + 248)*x^4 + (4*n + 52)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(6*n^2 + 6*n + (3*n^3 + 9*n^2 + 6*n)*x + (10*n^2 + 4*n)*x^2 + (2*n^2 + 8*n)*x^3 + 2*n*x^4)]];
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] = (-(18*n^3 - 12*n^2 - 90*n + 12 + (12*n^3 + 18*n^2 - 138*n - 24)*x + (34*n^2 - 42*n - 112)*x^2 + (8*n^2 + 12*n - 108)*x^3 + (4*n - 36)*x^4 - 4*x^5)*v[i - 1] + (36*n^2 + 108*n + 72 + (72*n^2 + 336*n + 336)*x + (42*n^2 + 390*n + 612)*x^2 + (6*n^2 + 206*n + 548)*x^3 + (48*n + 248)*x^4 + (4*n + 52)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(6*n^2 + 6*n + (3*n^3 + 9*n^2 + 6*n)*x + (10*n^2 + 4*n)*x^2 + (2*n^2 + 8*n)*x^3 + 2*n*x^4)); 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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.