Real-rooted atlas

A144443 → D

nonnegative coefficients

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

Recurrence

\(\left(-5 + 6\,n\right)\,P_{n}(x) = \left(-36\,x^{2} - 216\,n x^{2}\right)\,P_{n-4}(x) + \left(-12\,x - 12\,x^{2} - 72\,n x + 1152\,n x^{2} - 432\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-36\,x^{2} + 36\,x^{3} - 216\,n x^{2} + 216\,n x^{3}\right)\,P_{n-3}'(x) + \left(-7 - 26\,x + 425\,x^{2} - 42\,n + 264\,n x - 1170\,n x^{2} - 72\,n^{2} x + 972\,n^{2} x^{2} - 216\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x - 144\,x^{2} + 150\,x^{3} - 36\,n x + 396\,n x^{2} - 360\,n x^{3} - 216\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-34 + 110\,x + 48\,n - 192\,n x + 72\,n^{2} x\right)\,P_{n-1}(x) + \left(-30\,x + 30\,x^{2} + 36\,n x - 36\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 20 + 2\,x\)
\(P_{3}(x) = 159 + 318\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
220222
31593183480
4113885083414413064
57997155756233634319885429380
656016243833489546644876668280080616605768
7392155352672982629779453506372608816824523529307739795840
8274513448816653666584553181760350031211097425530146449960819215938837334008384
91921599365869523521539212994727174074845048967593556303078425989442305433323215372794492105744968080
101345120128754648949833537566154962558190552909655731106301400383728582936447825432102824350185957992121060810810131304136410080
1194158415111531312767107021532524369583306105457828029162079821978703499449578637444145785684551199018724086731652051890907451959415841510118970384232933376
12659108913015107194007796143001242836426225440757666635708135371343423184740259360532784540936189519880792458636508815153332714164290037285092786755359700389807250198043012666384801970664832
13461376239891972880270545802856292257212470274144824234788814057610060111095011551660249773113617188019369580686325533860921760961777520184816682585452827483159520974190708234010850841488001905536514878681353485456470421218880
1432296336800825715595378814945626589403354247922087076386065692669222975655868831161324095401424317792395760216675994627534876985121272018990903898610136413561805638960903832521769096516423167302063082781229890904154293572272889644198523784104144612642481834699665024
Technical details
Node id
n-f4d3223bc24595a5 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ae7e1082727f92c8

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:        0;
  1:        1;
  2:       20,          2;
  3:      159,        318,            3;
  4:     1138,       8508,         3414,            4;
  5:     7997,     155756,       233634,        31988,            5;
  6:    56016,    2438334,      8954664,      4876668,       280080,            6;
  7:   392155,   35267298,    262977945,    350637260,     88168245,      2352930,           7;
  8:  2745134,  488166536,   6658455318,  17603500312,  11097425530,   1464499608,    19215938,         8;
  9: 19215993, 6586952352, 153921299472, 717407484504, 896759355630, 307842598944, 23054333232, 153727944, 9;
%F Formula
The row polynomials satisfy (6*n-5)*P(n,x) = -(216*n+36)*x^2*P(n-4,x) - ((72*n+12)*x+(432*n^2-1152*n+12)*x^2)*P(n-3,x) - ((216*n+36)*x^2-(216*n+36)*x^3)*d/dx P(n-3,x) - (42*n+7+(72*n^2-264*n+26)*x+(216*n^3-972*n^2+1170*n-425)*x^2)*P(n-2,x) - ((36*n+6)*x+(216*n^2-396*n+144)*x^2-(216*n^2-360*n+150)*x^3)*d/dx P(n-2,x) + (48*n-34+(72*n^2-192*n+110)*x)*P(n-1,x) + ((36*n-30)*x-(36*n-30)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 20 + 2*x, P(3,x) = 159 + 318*x + 3*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 20 + 2*x;
P[3, x_] := 159 + 318*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(216*n + 36)*x^2*P[n - 4, x] - ((72*n + 12)*x + (432*n^2 - 1152*n + 12)*x^2)*P[n - 3, x] - ((216*n + 36)*x^2 - (216*n + 36)*x^3)*D[P[n - 3, x], x] - (42*n + 7 + (72*n^2 - 264*n + 26)*x + (216*n^3 - 972*n^2 + 1170*n - 425)*x^2)*P[n - 2, x] - ((36*n + 6)*x + (216*n^2 - 396*n + 144)*x^2 - (216*n^2 - 360*n + 150)*x^3)*D[P[n - 2, x], x] + (48*n - 34 + (72*n^2 - 192*n + 110)*x)*P[n - 1, x] + ((36*n - 30)*x - (36*n - 30)*x^2)*D[P[n - 1, x], x])/(6*n - 5)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 20 + 2*x; v[4] = 159 + 318*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(216*n + 36)*x^2*v[i - 4] - ((72*n + 12)*x + (432*n^2 - 1152*n + 12)*x^2)*v[i - 3] - ((216*n + 36)*x^2 - (216*n + 36)*x^3)*deriv(v[i - 3], x) - (42*n + 7 + (72*n^2 - 264*n + 26)*x + (216*n^3 - 972*n^2 + 1170*n - 425)*x^2)*v[i - 2] - ((36*n + 6)*x + (216*n^2 - 396*n + 144)*x^2 - (216*n^2 - 360*n + 150)*x^3)*deriv(v[i - 2], x) + (48*n - 34 + (72*n^2 - 192*n + 110)*x)*v[i - 1] + ((36*n - 30)*x - (36*n - 30)*x^2)*deriv(v[i - 1], x))/(6*n - 5)); 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.