Real-rooted atlas

A081577 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-12\,n - 12\,n x - 160\,n x^{2} + 16\,n x^{3} + 64\,n x^{4} + 24\,n^{2} + 128\,n^{2} x^{2} + 64\,n^{2} x^{3} + 48\,n^{3} x\right)\,P_{n}(x) = \left(-78 - 636\,x - 1304\,x^{2} + 144\,x^{3} + 768\,x^{4} + 256\,x^{5} - 18\,n + 276\,n x + 1416\,n x^{2} + 528\,n x^{3} - 64\,n x^{4} + 240\,n^{2} + 864\,n^{2} x - 160\,n^{2} x^{2} - 320\,n^{2} x^{3} - 72\,n^{3} - 240\,n^{3} x\right)\,P_{n-1}(x) + \left(-9 - 84\,x - 144\,x^{2} + 656\,x^{3} + 2192\,x^{4} + 1408\,x^{5} + 256\,x^{6} + 96\,n x + 816\,n x^{2} + 2048\,n x^{3} + 1344\,n x^{4} + 256\,n x^{5} + 36\,n^{2} + 288\,n^{2} x + 624\,n^{2} x^{2} + 192\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
2305616102
311241635264944
43962208361619202568396
51320990425088253449728102472408
6426440288139520218624157184471044096611080
71344015334467712014638081626112901120221184163845072512
841616556800299596883220481278873610887168489062410158086553641564304
912707219512961239859242240000837611529858662467633152254935044587520262144337041056
103837126652288487810561972961284818677767331266566939279363972792321288437762044723210485762709654368
111148160221834241844761608648570882517270528472871731257865666564558684160223451545663543705690177536419430421628227840
123409600726502406757841923606071296122071203842739830784041445752832421800181762837001011212140412928307232768039426457616777216171582907072
13100604162343697922411683840144394199045579805491214601279897626432608665633264395878428866091417616902940262464122519552146129551361711276032671088641354080609664
1429521536746495488842048921655919370240243074318336727749689344153864123187223192341053442487990353920187679794790497166924185633081314508868551704576738197504026843545610637288025216
Technical details
Node id
n-2e2e6e7b58e35791 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-c2264ee630c24250

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:      6,       4;
  2:     30,      56,       16;
  3:    112,     416,      352,       64;
  4:    396,    2208,     3616,     1920,      256;
  5:   1320,    9904,    25088,    25344,     9728,     1024;
  6:   4264,   40288,   139520,   218624,   157184,    47104,     4096;
  7:  13440,  153344,   677120,  1463808,  1626112,   901120,   221184,    16384;
  8:  41616,  556800,  2995968,  8322048, 12788736, 10887168,  4890624,  1015808,   65536;
  9: 127072, 1951296, 12398592, 42240000, 83761152, 98586624, 67633152, 25493504, 4587520, 262144;
%F Formula
The row polynomials satisfy (24*n^2-12*n+(48*n^3-12*n)*x+(128*n^2-160*n)*x^2+(64*n^2+16*n)*x^3+64*n*x^4)*P(n,x) = -(72*n^3-240*n^2+18*n+78+(240*n^3-864*n^2-276*n+636)*x+(160*n^2-1416*n+1304)*x^2+(320*n^2-528*n-144)*x^3+(64*n-768)*x^4-256*x^5)*P(n-1,x) + (36*n^2-9+(288*n^2+96*n-84)*x+(624*n^2+816*n-144)*x^2+(192*n^2+2048*n+656)*x^3+(1344*n+2192)*x^4+(256*n+1408)*x^5+256*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(72*n^3 - 240*n^2 + 18*n + 78 + (240*n^3 - 864*n^2 - 276*n + 636)*x + (160*n^2 - 1416*n + 1304)*x^2 + (320*n^2 - 528*n - 144)*x^3 + (64*n - 768)*x^4 - 256*x^5)*P[n - 1, x] + (36*n^2 - 9 + (288*n^2 + 96*n - 84)*x + (624*n^2 + 816*n - 144)*x^2 + (192*n^2 + 2048*n + 656)*x^3 + (1344*n + 2192)*x^4 + (256*n + 1408)*x^5 + 256*x^6)*D[P[n - 1, x], x])/(24*n^2 - 12*n + (48*n^3 - 12*n)*x + (128*n^2 - 160*n)*x^2 + (64*n^2 + 16*n)*x^3 + 64*n*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(72*n^3 - 240*n^2 + 18*n + 78 + (240*n^3 - 864*n^2 - 276*n + 636)*x + (160*n^2 - 1416*n + 1304)*x^2 + (320*n^2 - 528*n - 144)*x^3 + (64*n - 768)*x^4 - 256*x^5)*v[i - 1] + (36*n^2 - 9 + (288*n^2 + 96*n - 84)*x + (624*n^2 + 816*n - 144)*x^2 + (192*n^2 + 2048*n + 656)*x^3 + (1344*n + 2192)*x^4 + (256*n + 1408)*x^5 + 256*x^6)*deriv(v[i - 1], x))/(24*n^2 - 12*n + (48*n^3 - 12*n)*x + (128*n^2 - 160*n)*x^2 + (64*n^2 + 16*n)*x^3 + 64*n*x^4)); 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.