Real-rooted atlas

A142458 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-8 + 6\,x + 3\,n\right)\,P_{n}(x) = \left(20 - 119\,x + 404\,x^{2} - 423\,x^{3} + 90\,x^{4} - 12\,n + 87\,n x - 450\,n x^{2} + 378\,n x^{3} - 54\,n x^{4} - 18\,n^{2} x + 189\,n^{2} x^{2} - 81\,n^{2} x^{3} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(15\,x - 153\,x^{2} + 327\,x^{3} - 243\,x^{4} + 54\,x^{5} - 9\,n x + 126\,n x^{2} - 198\,n x^{3} + 81\,n x^{4} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-37 + 134\,x - 111\,x^{2} + 18\,x^{3} + 15\,n - 87\,n x + 45\,n x^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-24\,x + 42\,x^{2} - 18\,x^{3} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
2910120
340117421200
41671258104417012640
567811641219288190682143120
62725970743760952983406079527301837760
71091675853355028348548375357872043929910922118849600
843683569177872057130200883894159840020400219823129822436901481712000
91747544163343387403089640728868685682679524265371636642953923222127364174762113776963200
1069904129964745010050626205741672459601692548361301366079170684081940349044961776401557655656990501435853017600
112796192213458130111128797793012484436926054410623651160575077706250629420198259485968513103704633372812010938544652796202115109571276800
1211184799151070444101199303758176198234604225301039015509792752071200240934921698815024566805866141673134884318196050454727312408507670962584111848101569514609664000
1344739230106476322777126727315474323012394330809862266047228698590661205523134569582315965759965764535811966785676110668188696373811632328447613547918789614005375266252244739242123187380536320000
141789569577485107447381319973342114194426909102451340465605351001325811921468763945665663474254550275666192893456759815687121123455844110816152005054206690631815783769620839774836438016586837649233431917895697011014061442121728000
Technical details
Node id
n-edcac0fcb6fe78a6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2b6b8baf2ca9e23d

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:      2,        1;
  2:      9,       10,         1;
  3:     40,      117,        42,          1;
  4:    167,     1258,      1044,        170,          1;
  5:    678,    11641,     21928,       8190,        682,          1;
  6:   2725,    97074,    376095,     298340,      60795,       2730,         1;
  7:  10916,   758533,   5502834,    8548375,    3578720,     439299,     10922,        1;
  8:  43683,  5691778,  72057130,  200883894,  159840020,   40021982,   3129822,    43690,      1;
  9: 174754, 41633433, 874030896, 4072886868, 5682679524, 2653716366, 429539232, 22127364, 174762, 1;
%F Formula
The row polynomials satisfy (3*n-8+6*x)*P(n,x) = -(12*n-20+(18*n^2-87*n+119)*x+(27*n^3-189*n^2+450*n-404)*x^2+(81*n^2-378*n+423)*x^3+(54*n-90)*x^4)*P(n-2,x) - ((9*n-15)*x+(27*n^2-126*n+153)*x^2-(27*n^2-198*n+327)*x^3-(81*n-243)*x^4-54*x^5)*d/dx P(n-2,x) + (15*n-37+(18*n^2-87*n+134)*x+(45*n-111)*x^2+18*x^3)*P(n-1,x) + ((9*n-24)*x-(9*n-42)*x^2-18*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 20 + (18*n^2 - 87*n + 119)*x + (27*n^3 - 189*n^2 + 450*n - 404)*x^2 + (81*n^2 - 378*n + 423)*x^3 + (54*n - 90)*x^4)*P[n - 2, x] - ((9*n - 15)*x + (27*n^2 - 126*n + 153)*x^2 - (27*n^2 - 198*n + 327)*x^3 - (81*n - 243)*x^4 - 54*x^5)*D[P[n - 2, x], x] + (15*n - 37 + (18*n^2 - 87*n + 134)*x + (45*n - 111)*x^2 + 18*x^3)*P[n - 1, x] + ((9*n - 24)*x - (9*n - 42)*x^2 - 18*x^3)*D[P[n - 1, x], x])/(3*n - 8 + 6*x)]];
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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n - 20 + (18*n^2 - 87*n + 119)*x + (27*n^3 - 189*n^2 + 450*n - 404)*x^2 + (81*n^2 - 378*n + 423)*x^3 + (54*n - 90)*x^4)*v[i - 2] - ((9*n - 15)*x + (27*n^2 - 126*n + 153)*x^2 - (27*n^2 - 198*n + 327)*x^3 - (81*n - 243)*x^4 - 54*x^5)*deriv(v[i - 2], x) + (15*n - 37 + (18*n^2 - 87*n + 134)*x + (45*n - 111)*x^2 + 18*x^3)*v[i - 1] + ((9*n - 24)*x - (9*n - 42)*x^2 - 18*x^3)*deriv(v[i - 1], x))/(3*n - 8 + 6*x)); 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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.