Real-rooted atlas

A113413 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(5 - 24\,x - 62\,x^{2} - 20\,x^{3} + 16\,x^{4} - 18\,n - 18\,n x - 32\,n x^{2} - 12\,n x^{3} + 16\,n x^{4} + 8\,n^{2} - 8\,n^{2} x + 16\,n^{2} x^{3} + 8\,n^{3} x\right)\,P_{n}(x) = \left(-11 - 220\,x - 790\,x^{2} - 556\,x^{3} + 80\,x^{4} + 128\,x^{5} - 72\,n - 294\,n x - 148\,n x^{2} + 116\,n x^{3} + 80\,n x^{4} + 56\,n^{2} + 224\,n^{2} x + 112\,n^{2} x^{2} - 48\,n^{2} x^{3} - 8\,n^{3} - 24\,n^{3} x\right)\,P_{n-1}(x) + \left(-1 - 26\,x - 160\,x^{2} - 308\,x^{3} - 44\,x^{4} + 160\,x^{5} + 64\,x^{6} - 8\,n - 64\,n x - 136\,n x^{2} + 144\,n x^{4} + 64\,n x^{5} + 4\,n^{2} + 32\,n^{2} x + 72\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
2315616103
310938033664889
43451840307217922567305
51031746019008206088960102458091
629672720894416155392124160430084096451247
78317925084085289156481104640697344200704163843444073
822865299232160883245962247539200715161637273609175046553625928369
961927932548591820820625664431303685551104043294720192020484128768262144193067435
1016574328236402067585685223936217592320356806656376750080249167872961413121835008010485761424746071
11439341835526069374672330548544999288320200107827226894336002402975744137841868847054848080740352419430410435395577
121155305242639522253438721219522816426678144010122485760166097469441886873190414594015232738721792022607298563523215361677721675949093753
133017639693726447127222404320698752171901504004723178188891831984128127304007680125043277824851902464003857081958410695475200152672665667108864549757389899
147836839195746536220458516814804189440660504389122065308282884652407889927641318031369153468825607910473072644812701696001970484019204994577203265766686722684354563960669854815
Technical details
Node id
n-5bcc3e5fe0a51dd4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-79ac97648dc9a6af

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:     7,      4;
  2:    31,     56,      16;
  3:   109,    380,     336,       64;
  4:   345,   1840,    3072,     1792,      256;
  5:  1031,   7460,   19008,    20608,     8960,     1024;
  6:  2967,  27208,   94416,   155392,   124160,    43008,     4096;
  7:  8317,  92508,  408528,   915648,  1104640,   697344,   200704,    16384;
  8: 22865, 299232, 1608832,  4596224,  7539200,  7151616,  3727360,   917504,   65536;
  9: 61927, 932548, 5918208, 20625664, 43130368, 55511040, 43294720, 19202048, 4128768, 262144;
%F Formula
The row polynomials satisfy (8*n^2-18*n+5+(8*n^3-8*n^2-18*n-24)*x-(32*n+62)*x^2+(16*n^2-12*n-20)*x^3+(16*n+16)*x^4)*P(n,x) = -(8*n^3-56*n^2+72*n+11+(24*n^3-224*n^2+294*n+220)*x-(112*n^2-148*n-790)*x^2+(48*n^2-116*n+556)*x^3-(80*n+80)*x^4-128*x^5)*P(n-1,x) + (4*n^2-8*n-1+(32*n^2-64*n-26)*x+(72*n^2-136*n-160)*x^2+(32*n^2-308)*x^3+(144*n-44)*x^4+(64*n+160)*x^5+64*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[(-(8*n^3 - 56*n^2 + 72*n + 11 + (24*n^3 - 224*n^2 + 294*n + 220)*x - (112*n^2 - 148*n - 790)*x^2 + (48*n^2 - 116*n + 556)*x^3 - (80*n + 80)*x^4 - 128*x^5)*P[n - 1, x] + (4*n^2 - 8*n - 1 + (32*n^2 - 64*n - 26)*x + (72*n^2 - 136*n - 160)*x^2 + (32*n^2 - 308)*x^3 + (144*n - 44)*x^4 + (64*n + 160)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(8*n^2 - 18*n + 5 + (8*n^3 - 8*n^2 - 18*n - 24)*x - (32*n + 62)*x^2 + (16*n^2 - 12*n - 20)*x^3 + (16*n + 16)*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] = (-(8*n^3 - 56*n^2 + 72*n + 11 + (24*n^3 - 224*n^2 + 294*n + 220)*x - (112*n^2 - 148*n - 790)*x^2 + (48*n^2 - 116*n + 556)*x^3 - (80*n + 80)*x^4 - 128*x^5)*v[i - 1] + (4*n^2 - 8*n - 1 + (32*n^2 - 64*n - 26)*x + (72*n^2 - 136*n - 160)*x^2 + (32*n^2 - 308)*x^3 + (144*n - 44)*x^4 + (64*n + 160)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(8*n^2 - 18*n + 5 + (8*n^3 - 8*n^2 - 18*n - 24)*x - (32*n + 62)*x^2 + (16*n^2 - 12*n - 20)*x^3 + (16*n + 16)*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.