Real-rooted atlas

A142462 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-20 + 14\,x + 7\,n\right)\,P_{n}(x) = \left(104 - 723\,x + 5312\,x^{2} - 6195\,x^{3} + 1274\,x^{4} - 56\,n + 511\,n x - 6342\,n x^{2} + 5194\,n x^{3} - 686\,n x^{4} - 98\,n^{2} x + 2597\,n^{2} x^{2} - 1029\,n^{2} x^{3} - 343\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(91\,x - 2009\,x^{2} + 4515\,x^{3} - 3283\,x^{4} + 686\,x^{5} - 49\,n x + 1666\,n x^{2} - 2646\,n x^{3} + 1029\,n x^{4} - 343\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-173 + 786\,x - 651\,x^{2} + 98\,x^{3} + 63\,n - 511\,n x + 245\,n x^{2} + 98\,n^{2} x\right)\,P_{n-1}(x) + \left(-140\,x + 238\,x^{2} - 98\,x^{3} + 49\,n x - 49\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
21718136
31444291461720
41167974677881170119872
5935818936136000812743093621695520
67489333176581350164710445620199584374898129410560
7599180546328934294429866707102152692263283061193159918611455822720
8479348386682858612202212946350446857742739889748465098371104645719264793490182496620800
938347914134482226733212725002561572380898508220128756632499601638680615179041595270117206843834792215263284407040
103067833692058265668508028986882013632355088839121472256122845621173056006266203364288794909034641547973521055209420533067833781373214712499200
11245426701631245192903491935099677853622349607674503517858308435288080811216349450405498560217885391038010832700014162587803227102956815855751818172454267026129110747574937600
121963413619947199115402338454618455889442482378766873177546451489766760705627913059579647249028741581605797916632247996862122967460337855932715566691743429012417818238057161312481963413621012476652832144691200
13157073089670710912217328657104877078328713192276603448082278918621970664477237762870656865137629022217758190525408419982057644298742358543828316104027595060361867381031204147664626815387497202121790003572624491955221570730896821228205868104760832000
14125658471744510689286173545994238797978331543722789893556431011691708100713352519478094902142945062163442977051507849632145455942647547648719498076016453437624556680534103869475594197200990839759990783101116727828658850585834010037331228453603503957410471256584717458122638022115992274534400
Technical details
Node id
n-87a630b6538ab782 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-49ff540d672629c6

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:       17,          18,            1;
  3:      144,         429,          146,             1;
  4:     1167,        9746,         7788,          1170,             1;
  5:     9358,      189361,       360008,        127430,          9362,            1;
  6:    74893,     3317658,     13501647,      10445620,       1995843,        74898,            1;
  7:   599180,    54632893,    429442986,     670710215,     269226328,     30611931,       599186,          1;
  8:  4793483,   866828586,  12202212946,   35044685774,   27398897484,   6509837110,    464571926,    4793490,        1;
  9: 38347914, 13448222673, 321272500256, 1572380898508, 2201287566324, 996016386806, 151790415952, 7011720684, 38347922, 1;
%F Formula
The row polynomials satisfy (7*n-20+14*x)*P(n,x) = -(56*n-104+(98*n^2-511*n+723)*x+(343*n^3-2597*n^2+6342*n-5312)*x^2+(1029*n^2-5194*n+6195)*x^3+(686*n-1274)*x^4)*P(n-2,x) - ((49*n-91)*x+(343*n^2-1666*n+2009)*x^2-(343*n^2-2646*n+4515)*x^3-(1029*n-3283)*x^4-686*x^5)*d/dx P(n-2,x) + (63*n-173+(98*n^2-511*n+786)*x+(245*n-651)*x^2+98*x^3)*P(n-1,x) + ((49*n-140)*x-(49*n-238)*x^2-98*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[(-(56*n - 104 + (98*n^2 - 511*n + 723)*x + (343*n^3 - 2597*n^2 + 6342*n - 5312)*x^2 + (1029*n^2 - 5194*n + 6195)*x^3 + (686*n - 1274)*x^4)*P[n - 2, x] - ((49*n - 91)*x + (343*n^2 - 1666*n + 2009)*x^2 - (343*n^2 - 2646*n + 4515)*x^3 - (1029*n - 3283)*x^4 - 686*x^5)*D[P[n - 2, x], x] + (63*n - 173 + (98*n^2 - 511*n + 786)*x + (245*n - 651)*x^2 + 98*x^3)*P[n - 1, x] + ((49*n - 140)*x - (49*n - 238)*x^2 - 98*x^3)*D[P[n - 1, x], x])/(7*n - 20 + 14*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] = (-(56*n - 104 + (98*n^2 - 511*n + 723)*x + (343*n^3 - 2597*n^2 + 6342*n - 5312)*x^2 + (1029*n^2 - 5194*n + 6195)*x^3 + (686*n - 1274)*x^4)*v[i - 2] - ((49*n - 91)*x + (343*n^2 - 1666*n + 2009)*x^2 - (343*n^2 - 2646*n + 4515)*x^3 - (1029*n - 3283)*x^4 - 686*x^5)*deriv(v[i - 2], x) + (63*n - 173 + (98*n^2 - 511*n + 786)*x + (245*n - 651)*x^2 + 98*x^3)*v[i - 1] + ((49*n - 140)*x - (49*n - 238)*x^2 - 98*x^3)*deriv(v[i - 1], x))/(7*n - 20 + 14*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

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.