Real-rooted atlas

A099040 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-10\,n - 104\,n x - 224\,n x^{2} + 128\,n x^{3} - 24\,n^{2} + 128\,n^{2} x^{2} + 16\,n^{3}\right)\,P_{n}(x) = \left(-120 - 816\,x - 1472\,x^{2} - 256\,x^{3} + 1024\,x^{4} - 268\,n - 752\,n x - 64\,n x^{2} + 768\,n x^{3} + 240\,n^{2} + 384\,n^{2} x - 256\,n^{2} x^{2} - 32\,n^{3}\right)\,P_{n-1}(x) + \left(-15 - 140\,x - 352\,x^{2} + 128\,x^{3} + 1280\,x^{4} + 1024\,x^{5} - 36\,n - 192\,n x + 1024\,n x^{3} + 1024\,n x^{4} + 24\,n^{2} + 128\,n^{2} x + 128\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110818
24616864278
3176118420485123920
462060801964822016409652460
520722678412761626777622118432768678200
666961074886787842109952326041621299202621448555400
7211204052483190784131891203024486436896768199229442097152105968000
86542414607361378560071225344217583616394067968396623872182452224167772161294042000
91998405089920560117763475333121329078272320946176047946137604104126464164416716813421772815624500000
10603616172724482171781121573629952723647692821896396800436294123525540465868841238396928146297323521073741824186917500000
1118065925738035281161420867275735043616781107213170756812832900749721655727829811261498562969640480066764812884901888085899345922218984800000
1253659521873233922944050176274797772801691588526087207009648642167867703296460465163468867768208588806609417797632389875656294411252814315526871947673626171991800000
13158355206026777601041982873610815816499275030229811236627550044161289700927078432997577326592609036428902407919731789004869176890753024369624885493769758165696512549755813888306975102000000
1446474880191515392036125001728412798181376318664890777617541214371840708055741235202121393084825604717290088038407692675658547208956295302348807081971574374403457964069355528411263952486443980465111043583253986000000
Technical details
Node id
n-a1cc921413cb5bbf · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-894b4c242e35e90e

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:     10,       8;
  2:     46,     168,       64;
  3:    176,    1184,     2048,       512;
  4:    620,    6080,    19648,     22016,       4096;
  5:   2072,   26784,   127616,    267776,     221184,      32768;
  6:   6696,  107488,   678784,   2109952,    3260416,    2129920,     262144;
  7:  21120,  405248,  3190784,  13189120,   30244864,   36896768,   19922944,    2097152;
  8:  65424, 1460736, 13785600,  71225344,  217583616,  394067968,  396623872,  182452224,   16777216;
  9: 199840, 5089920, 56011776, 347533312, 1329078272, 3209461760, 4794613760, 4104126464, 1644167168, 134217728;
%F Formula
The row polynomials satisfy (16*n^3-24*n^2-10*n-104*n*x+(128*n^2-224*n)*x^2+128*n*x^3)*P(n,x) = -(32*n^3-240*n^2+268*n+120-(384*n^2-752*n-816)*x+(256*n^2+64*n+1472)*x^2-(768*n-256)*x^3-1024*x^4)*P(n-1,x) + (24*n^2-36*n-15+(128*n^2-192*n-140)*x+(128*n^2-352)*x^2+(1024*n+128)*x^3+(1024*n+1280)*x^4+1024*x^5)*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[(-(32*n^3 - 240*n^2 + 268*n + 120 - (384*n^2 - 752*n - 816)*x + (256*n^2 + 64*n + 1472)*x^2 - (768*n - 256)*x^3 - 1024*x^4)*P[n - 1, x] + (24*n^2 - 36*n - 15 + (128*n^2 - 192*n - 140)*x + (128*n^2 - 352)*x^2 + (1024*n + 128)*x^3 + (1024*n + 1280)*x^4 + 1024*x^5)*D[P[n - 1, x], x])/(16*n^3 - 24*n^2 - 10*n - 104*n*x + (128*n^2 - 224*n)*x^2 + 128*n*x^3)]];
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] = (-(32*n^3 - 240*n^2 + 268*n + 120 - (384*n^2 - 752*n - 816)*x + (256*n^2 + 64*n + 1472)*x^2 - (768*n - 256)*x^3 - 1024*x^4)*v[i - 1] + (24*n^2 - 36*n - 15 + (128*n^2 - 192*n - 140)*x + (128*n^2 - 352)*x^2 + (1024*n + 128)*x^3 + (1024*n + 1280)*x^4 + 1024*x^5)*deriv(v[i - 1], x))/(16*n^3 - 24*n^2 - 10*n - 104*n*x + (128*n^2 - 224*n)*x^2 + 128*n*x^3)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.