Real-rooted atlas

A281000 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 2\,n x + 8\,n^{2} + 10\,n^{2} x + 18\,n^{3} x + n^{3} x^{2} - 20\,n^{4} x - 3\,n^{5} x^{2} + 2\,n^{6} x^{2}\right)\,P_{n}(x) = \left(-8 + 36\,x + 36\,x^{2} + 2\,x^{3} - 156\,n x - 110\,n x^{2} - 7\,n x^{3} + 32\,n^{2} + 100\,n^{2} x + 24\,n^{2} x^{2} + 3\,n^{2} x^{3} + 112\,n^{3} x + 160\,n^{3} x^{2} + 12\,n^{3} x^{3} - 80\,n^{4} x - 102\,n^{4} x^{2} - 12\,n^{4} x^{3} - 16\,n^{5} x^{2} - 4\,n^{5} x^{3} + 8\,n^{6} x^{2} + 6\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-16\,x - 36\,x^{2} + 6\,x^{3} + x^{4} + 64\,n x + 88\,n x^{2} - 14\,n x^{3} - 3\,n x^{4} + 24\,n^{2} x^{2} - 6\,n^{2} x^{3} - 160\,n^{3} x^{2} + 16\,n^{3} x^{3} + 6\,n^{3} x^{4} + 12\,n^{4} x^{3} - 3\,n^{4} x^{4} + 16\,n^{5} x^{3} - 4\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2347
31040959
43528018916520
512616802268576254675
64629240207901056013753642463
717164804816216213728035750280849387813
8643524024011351341441440625625982805145643552363
92431011668807351344130690568508500222768023324087048132589795
10923785542680448957081064194569699690038093328664734049612813851100299197869
113527162586584026189163079814592096996900053330659213959414017364480969570210001212747481009
121352078118982864147241294256091921608764362750642507465623544878284393213444088353517710003061314425227871914
1352003005408312008031343320373946144007303635625068840085600336355404008786426880122650605088550000306130043200169231588559069
14200583002433740400427121440202386456300805696835787506711908346004216455243001468588492802870024157031081050001790860505054400593191962125182906265
1577558760108582264002223221855401468032209280420530059950060556328632804755224524050212945331456055487133702084126042000727089363034201440080278387957622519493519876059
Technical details
Node id
n-8b1a9aa04967006b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-aeca54f94370d2fd

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     3,       4;
   3:    10,      40,        9;
   4:    35,     280,      189,        16;
   5:   126,    1680,     2268,       576,       25;
   6:   462,    9240,    20790,     10560,     1375,       36;
   7:  1716,   48048,   162162,    137280,    35750,     2808,      49;
   8:  6435,  240240,  1135134,   1441440,   625625,    98280,    5145,     64;
   9: 24310, 1166880,  7351344,  13069056,  8508500,  2227680,  233240,   8704,    81;
  10: 92378, 5542680, 44895708, 106419456, 96996900, 38093328, 6647340, 496128, 13851, 100;
%F Formula
The row polynomials satisfy (8*n^2+4*n-(20*n^4-18*n^3-10*n^2+2*n)*x+(2*n^6-3*n^5+n^3)*x^2)*P(n,x) = (32*n^2-8-(80*n^4-112*n^3-100*n^2+156*n-36)*x+(8*n^6-16*n^5-102*n^4+160*n^3+24*n^2-110*n+36)*x^2+(6*n^6-4*n^5-12*n^4+12*n^3+3*n^2-7*n+2)*x^3)*P(n-1,x) + ((64*n-16)*x-(160*n^3-24*n^2-88*n+36)*x^2+(16*n^5+12*n^4+16*n^3-6*n^2-14*n+6)*x^3-(4*n^5+3*n^4-6*n^3+3*n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((32*n^2 - 8 - (80*n^4 - 112*n^3 - 100*n^2 + 156*n - 36)*x + (8*n^6 - 16*n^5 - 102*n^4 + 160*n^3 + 24*n^2 - 110*n + 36)*x^2 + (6*n^6 - 4*n^5 - 12*n^4 + 12*n^3 + 3*n^2 - 7*n + 2)*x^3)*P[n - 1, x] + ((64*n - 16)*x - (160*n^3 - 24*n^2 - 88*n + 36)*x^2 + (16*n^5 + 12*n^4 + 16*n^3 - 6*n^2 - 14*n + 6)*x^3 - (4*n^5 + 3*n^4 - 6*n^3 + 3*n - 1)*x^4)*D[P[n - 1, x], x])/(8*n^2 + 4*n - (20*n^4 - 18*n^3 - 10*n^2 + 2*n)*x + (2*n^6 - 3*n^5 + n^3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((32*n^2 - 8 - (80*n^4 - 112*n^3 - 100*n^2 + 156*n - 36)*x + (8*n^6 - 16*n^5 - 102*n^4 + 160*n^3 + 24*n^2 - 110*n + 36)*x^2 + (6*n^6 - 4*n^5 - 12*n^4 + 12*n^3 + 3*n^2 - 7*n + 2)*x^3)*v[i - 1] + ((64*n - 16)*x - (160*n^3 - 24*n^2 - 88*n + 36)*x^2 + (16*n^5 + 12*n^4 + 16*n^3 - 6*n^2 - 14*n + 6)*x^3 - (4*n^5 + 3*n^4 - 6*n^3 + 3*n - 1)*x^4)*deriv(v[i - 1], x))/(8*n^2 + 4*n - (20*n^4 - 18*n^3 - 10*n^2 + 2*n)*x + (2*n^6 - 3*n^5 + n^3)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.