Real-rooted atlas

A144445 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(7\,x + 21\,x^{2} + 21\,x^{3} + 7\,x^{4}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(7\,x + 21\,x^{2} + 21\,x^{3} + 7\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
21252551
312096214141245
4116811137819394969742151
51134571872696145267345202938081843581
6110766529419221660728935650582328163251093877599062559
71861329451920214138361921421447090226034848216917696464833627566316817517
8168906416863859389848815669508770425031243817631491568582569619891228792424728071981466299514767
9155125137103638341492279784931761705170190465601690173142811372802854840118750425293806469252713855143761171419039115179186621
1014410011051560018729625186201132309547671872777822681886490525627128826984605201096124132222730976164967502310468089289631085936178579262173675177426079583
11135280088492344382980341116665387593192170919945339006011311339025011179406309905522004058629477327550325611170646880868044086703692718226065371280197897782476750549052686954382354045974058989
12128224070801352005837924982605243754854149522874834839920534589038622209748812167912828524343446610936974750457219410780965206960233886120425175485680089608281939211558883527320369220100651731053394870016775288543634177262557996591
1312257925664175282874695866295789500732439557615770906957914296951810194459836174007011029083829807164744840122024813734020241292698135426626043902313974031138066354247001372170195808289658352847054436898303994308298784473375671119122459668420577955567885419608961463107760541
141180634053134579265417453928021282389362305131829470856984567830859326993669755054374567845415856336376399991392249321970088173798366017335811082621339775680314331695017574902628669418841505067910856604841416500877137358545520477792394380171786018481561299505776659331273603534278653952276161051479727818291181518143616767
Technical details
Node id
n-24ae9d775ee03472 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9024966cc99eeda3

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: 1,        2;
  2: 1,       25,          25;
  3: 1,      209,         621,          414;
  4: 1,     1681,       11378,        19394,          9697;
  5: 1,    13457,      187269,       614526,        734520,        293808;
  6: 1,   107665,     2941922,     16607289,      35650582,      32816325,       10938775;
  7: 1,   861329,    45192021,    413836192,    1421447090,    2260348482,     1691769646,      483362756;
  8: 1,  6890641,   686385938,   9848815669,   50877042503,  124381763149,   156858256961,    98912287924,   24728071981;
  9: 1, 55125137, 10363834149, 227978493176, 1705170190465, 6016901731428, 11372802854840, 11875042529380, 6469252713855, 1437611714190;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (7*x+21*x^2+21*x^3+7*x^4)*P(n-2,x) + (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (7*x+21*x^2+21*x^3+7*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((7*x + 21*x^2 + 21*x^3 + 7*x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (7*x + 21*x^2 + 21*x^3 + 7*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*x + 21*x^2 + 21*x^3 + 7*x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (7*x + 21*x^2 + 21*x^3 + 7*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^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