Real-rooted atlas

A123519 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(246 + 96\,x - 96\,x^{2} + 452\,n + 64\,n x - 64\,n x^{2} + 288\,n^{2} + 64\,n^{3}\right)\,P_{n}(x) = \left(1374 + 3072\,x - 224\,x^{2} - 512\,x^{3} + 1912\,n + 4112\,n x - 128\,n x^{2} - 256\,n x^{3} + 864\,n^{2} + 1792\,n^{2} x + 128\,n^{3} + 256\,n^{3} x\right)\,P_{n-1}(x) + \left(171 + 864\,x + 608\,x^{2} - 512\,x^{3} - 256\,x^{4} + 192\,n + 1024\,n x + 1024\,n x^{2} + 48\,n^{2} + 256\,n^{2} x + 256\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 67 + 184\,x + 64\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
06718464315
1345171222405124809
21641118562822424064409669881
374437074424697638297624166432768982571
432699385640178112040570884648960232652826214413494179
5140401197795211382912343705605785600052494336217579522097152182077265
6592529970896066956160252382208563507200750714880563347456199229440167772162423216049
724671634610090437069785616771624964678406144826739916891034746885821693952179516211213421772831896782211
810162291213283096195962054410352052224346891919367660722585611186942771210490793164858435043328159719096321073741824416089590091
94148922596632091299881859206036228659223636031488062728463974411447734108161423253176320116180543078457310969856014066017894485899345925387195067289
101681373374303485424494291885443364025779201508024934400467871544115210262663921664159435813027841724840411136012462116044800551581174988812283606466566871947673669306701018665
11677123811188917544882387208758401806971228160912819277824032436369948672831931275018241553443533619202099625464954882009742704640001302180397056005226116205772810651518894080549755813888886784598003739
1227122699318192560148011295328294409413523517440529213899161602120052402421766229304766627841360209616306176220954711660953626417475448995842266868706443264133119934359142448866419906969691809220919296439804651110411292928595389491
131081354032135156538592052519893721604778974831616029596601135104013200098429501444373735332642816109424063797002242078592890988134429864654050164736320024090603683842488055656716697613358310363234304451789327853158478725032548761635184372088832143217708610182561
14429363169611494958161728240551523417602373119548518401605452496117760789086946970828829116251513552896821572586700800001788210809269125123004309410362163203868190590663393283754703188037468162668043992196710401319271016819589124136362743693312067202150689669122814749767106561809670955368586849
Technical details
Node id
n-850be5acc9f192c8 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-addbf527d1f3ca48

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:       67,       184,         64;
  1:      345,      1712,       2240,         512;
  2:     1641,     11856,      28224,       24064,         4096;
  3:     7443,     70744,     246976,      382976,       241664,        32768;
  4:    32699,    385640,    1781120,     4057088,      4648960,      2326528,        262144;
  5:   140401,   1977952,   11382912,    34370560,     57856000,     52494336,      21757952,       2097152;
  6:   592529,   9708960,   66956160,   252382208,    563507200,    750714880,     563347456,     199229440,      16777216;
  7:  2467163,  46100904,  370697856,  1677162496,   4678406144,   8267399168,    9103474688,    5821693952,    1795162112,    134217728;
  8: 10162291, 213283096, 1959620544, 10352052224,  34689191936,  76607225856,  111869427712,  104907931648,   58435043328,  15971909632,   1073741824;
  9: 41489225, 966320912, 9988185920, 60362286592, 236360314880, 627284639744, 1144773410816, 1423253176320, 1161805430784, 573109698560, 140660178944, 8589934592;
%F Formula
The row polynomials satisfy (64*n^3+288*n^2+452*n+246+(64*n+96)*x-(64*n+96)*x^2)*P(n,x) = (128*n^3+864*n^2+1912*n+1374+(256*n^3+1792*n^2+4112*n+3072)*x-(128*n+224)*x^2-(256*n+512)*x^3)*P(n-1,x) + (48*n^2+192*n+171+(256*n^2+1024*n+864)*x+(256*n^2+1024*n+608)*x^2-512*x^3-256*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 67 + 184*x + 64*x^2.
%t Mathematica
P[0, x_] := 67 + 184*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((128*n^3 + 864*n^2 + 1912*n + 1374 + (256*n^3 + 1792*n^2 + 4112*n + 3072)*x - (128*n + 224)*x^2 - (256*n + 512)*x^3)*P[n - 1, x] + (48*n^2 + 192*n + 171 + (256*n^2 + 1024*n + 864)*x + (256*n^2 + 1024*n + 608)*x^2 - 512*x^3 - 256*x^4)*D[P[n - 1, x], x])/(64*n^3 + 288*n^2 + 452*n + 246 + (64*n + 96)*x - (64*n + 96)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 67 + 184*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((128*n^3 + 864*n^2 + 1912*n + 1374 + (256*n^3 + 1792*n^2 + 4112*n + 3072)*x - (128*n + 224)*x^2 - (256*n + 512)*x^3)*v[i - 1] + (48*n^2 + 192*n + 171 + (256*n^2 + 1024*n + 864)*x + (256*n^2 + 1024*n + 608)*x^2 - 512*x^3 - 256*x^4)*deriv(v[i - 1], x))/(64*n^3 + 288*n^2 + 452*n + 246 + (64*n + 96)*x - (64*n + 96)*x^2)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023