Real-rooted atlas

A257619 → D

nonnegative coefficients

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

Recurrence

\(\left(-16 + 9\,n\right)\,P_{n}(x) = \left(154 - 700\,x + 2800\,x^{2} - 198\,n + 1152\,n x - 6183\,n x^{2} - 324\,n^{2} x + 3888\,n^{2} x^{2} - 729\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(126\,x - 1134\,x^{2} + 1008\,x^{3} - 162\,n x + 2025\,n x^{2} - 1863\,n x^{3} - 729\,n^{2} x^{2} + 729\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-190 + 512\,x + 117\,n - 576\,n x + 162\,n^{2} x\right)\,P_{n-1}(x) + \left(-144\,x + 144\,x^{2} + 81\,n x - 81\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
244852
35641128241716
4643645120193086470928
571404127568819135322856161603546400
678694830940464110984856618809283934740384208528320
786600126927823924982167272664288969617319559805196007289614110416320
895268452148255691841933240095245048596168963222066825404447670755266687916420481080454735360
91047971916308896282440685174408765231299636705552391245458819401370348817530410811369885408383775328460892378879873280
1011527733572633302549236822881796607980817009830866889923677707312902360255147463003348853390858751955225332101969472103749602148102408724671988032000
1112680516350012857411362031273386868459672808446824768776505629192105977464036035030527172956843214781943345358884819569831589246080578583511291404126805163500022528902131083562508800
121394857005348259481442249024022882939567877996439338771809371770882058333493403900772039234190433711664384288166689076546108087867754361874354176686488187036339892129740721124512001534342705882849152101366729025750988800
131534342750938852178049564885096699413686498827207617481111933073104536013313003811365222434803802614789142895807664443638392066671177560821300806098184355895683933250184941448520602331378954996090692028697927260686802818412113011265610649612299163121791119974400
14168777703578116104693654534976728021073494359373998224875007699900063410983368070777154397989139966033116590203692534045020857672990968667253082904044155453604923378726694414527398877916380451938818751924975015852745840772694793177046601576628161927209860368021941101465155082293761602675563716473633587200
Technical details
Node id
n-eacf2d59efee0250 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-49151ecf04600104

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:          2;
  2:         44,            8;
  3:        564,         1128,            24;
  4:       6436,        45120,         19308,             64;
  5:      71404,      1275688,       1913532,         285616,            160;
  6:     786948,     30940464,     110984856,       61880928,        3934740,            384;
  7:    8660012,    692782392,    4982167272,     6642889696,     1731955980,       51960072,           896;
  8:   95268452,  14825569184,  193324009524,   504859616896,   322206682540,    44476707552,     666879164,       2048;
  9: 1047971916, 308896282440, 6851744087652, 31299636705552, 39124545881940, 13703488175304, 1081136988540, 8383775328, 4608;
%F Formula
The row polynomials satisfy (9*n-16)*P(n,x) = -(198*n-154+(324*n^2-1152*n+700)*x+(729*n^3-3888*n^2+6183*n-2800)*x^2)*P(n-2,x) - ((162*n-126)*x+(729*n^2-2025*n+1134)*x^2-(729*n^2-1863*n+1008)*x^3)*d/dx P(n-2,x) + (117*n-190+(162*n^2-576*n+512)*x)*P(n-1,x) + ((81*n-144)*x-(81*n-144)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(198*n - 154 + (324*n^2 - 1152*n + 700)*x + (729*n^3 - 3888*n^2 + 6183*n - 2800)*x^2)*P[n - 2, x] - ((162*n - 126)*x + (729*n^2 - 2025*n + 1134)*x^2 - (729*n^2 - 1863*n + 1008)*x^3)*D[P[n - 2, x], x] + (117*n - 190 + (162*n^2 - 576*n + 512)*x)*P[n - 1, x] + ((81*n - 144)*x - (81*n - 144)*x^2)*D[P[n - 1, x], x])/(9*n - 16)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(198*n - 154 + (324*n^2 - 1152*n + 700)*x + (729*n^3 - 3888*n^2 + 6183*n - 2800)*x^2)*v[i - 2] - ((162*n - 126)*x + (729*n^2 - 2025*n + 1134)*x^2 - (729*n^2 - 1863*n + 1008)*x^3)*deriv(v[i - 2], x) + (117*n - 190 + (162*n^2 - 576*n + 512)*x)*v[i - 1] + ((81*n - 144)*x - (81*n - 144)*x^2)*deriv(v[i - 1], x))/(9*n - 16)); 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.