Real-rooted atlas

A257618 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-11 + 8\,x + 4\,n\right)\,P_{n}(x) = \left(140 - 888\,x + 3884\,x^{2} - 4272\,x^{3} + 896\,x^{4} - 80\,n + 640\,n x - 4464\,n x^{2} + 3712\,n x^{3} - 512\,n x^{4} - 128\,n^{2} x + 1856\,n^{2} x^{2} - 768\,n^{2} x^{3} - 256\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(112\,x - 1472\,x^{2} + 3216\,x^{3} - 2368\,x^{4} + 512\,x^{5} - 64\,n x + 1216\,n x^{2} - 1920\,n x^{3} + 768\,n x^{4} - 256\,n^{2} x^{2} + 256\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-124 + 492\,x - 408\,x^{2} + 64\,x^{3} + 48\,n - 320\,n x + 160\,n x^{2} + 64\,n^{2} x\right)\,P_{n-1}(x) + \left(-88\,x + 152\,x^{2} - 64\,x^{3} + 32\,n x - 32\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
24448496
3480141649682400
44944389123177649921680640
549856917792173593663328049984323386880
64996481951104077207232605900801200364849996864170311680
7499916838917030429219089924549705600187036928022207987249999361289963233280
84999808074649845769888569036827923005030422050632448053971025920405747814449999872256664215552000
949999564813976950425631003918827521475536688537620617671763968949779288371214976956579847363471564849999974451249683323289600
10499999027225778505113609220255788748869968974607155216105831156101121292723957489664379483734927360405967796305921331425164288499999948810244119199167283200
1149999978496471013077831682642057303879680306627600031027201098999570490982401434334692268154887262284881344102414430230096547840108457476845568024025653667840499999989762048374847124222771200
1249999995289685562353660723273732079834259456126772277262565376067735949788059033601358685196139613388811096931754066722816377913046563835084853073553751537664028710457038602240433061767766016499999997952409637138699692225331200
1349999998976001548722365052518420188387765484912645017486806860452659238638384585040355328011393254126619729510401419365158134547218432775832997255318700032186186673966880342016190797844275014451207555181183195545607801111823990784499999999590481923979146395595571200000
14499999997788162797100256895303685451846948848322150419211592986339710074882076324897587560990310486905103019127636164608157873497697027590242304131044267963919315238912503254174144832041943048815080854589912973312675146616112863461376198041733126847201281404800128417955844999999999180816384458397664772609802240000
Technical details
Node id
n-bac49424b4de44e2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-84d0fce31cd67972

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:         4,            2;
  2:        44,           48,             4;
  3:       480,         1416,           496,              8;
  4:      4944,        38912,         31776,           4992,             16;
  5:     49856,       917792,       1735936,         633280,          49984,            32;
  6:    499648,     19511040,      77207232,       60590080,       12003648,        499968,            64;
  7:   4999168,    389170304,    2921908992,     4549705600,     1870369280,     222079872,       4999936,         128;
  8:  49998080,   7464984576,   98885690368,   279230050304,   220506324480,   53971025920,    4057478144,    49999872,       256;
  9: 499995648, 139769504256, 3100391882752, 14755366885376, 20617671763968, 9497792883712, 1497695657984, 73634715648, 499999744, 512;
%F Formula
The row polynomials satisfy (4*n-11+8*x)*P(n,x) = -(80*n-140+(128*n^2-640*n+888)*x+(256*n^3-1856*n^2+4464*n-3884)*x^2+(768*n^2-3712*n+4272)*x^3+(512*n-896)*x^4)*P(n-2,x) - ((64*n-112)*x+(256*n^2-1216*n+1472)*x^2-(256*n^2-1920*n+3216)*x^3-(768*n-2368)*x^4-512*x^5)*d/dx P(n-2,x) + (48*n-124+(64*n^2-320*n+492)*x+(160*n-408)*x^2+64*x^3)*P(n-1,x) + ((32*n-88)*x-(32*n-152)*x^2-64*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(80*n - 140 + (128*n^2 - 640*n + 888)*x + (256*n^3 - 1856*n^2 + 4464*n - 3884)*x^2 + (768*n^2 - 3712*n + 4272)*x^3 + (512*n - 896)*x^4)*P[n - 2, x] - ((64*n - 112)*x + (256*n^2 - 1216*n + 1472)*x^2 - (256*n^2 - 1920*n + 3216)*x^3 - (768*n - 2368)*x^4 - 512*x^5)*D[P[n - 2, x], x] + (48*n - 124 + (64*n^2 - 320*n + 492)*x + (160*n - 408)*x^2 + 64*x^3)*P[n - 1, x] + ((32*n - 88)*x - (32*n - 152)*x^2 - 64*x^3)*D[P[n - 1, x], x])/(4*n - 11 + 8*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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(80*n - 140 + (128*n^2 - 640*n + 888)*x + (256*n^3 - 1856*n^2 + 4464*n - 3884)*x^2 + (768*n^2 - 3712*n + 4272)*x^3 + (512*n - 896)*x^4)*v[i - 2] - ((64*n - 112)*x + (256*n^2 - 1216*n + 1472)*x^2 - (256*n^2 - 1920*n + 3216)*x^3 - (768*n - 2368)*x^4 - 512*x^5)*deriv(v[i - 2], x) + (48*n - 124 + (64*n^2 - 320*n + 492)*x + (160*n - 408)*x^2 + 64*x^3)*v[i - 1] + ((32*n - 88)*x - (32*n - 152)*x^2 - 64*x^3)*deriv(v[i - 1], x))/(4*n - 11 + 8*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.