Real-rooted atlas

A144442 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-21 + 20\,n\right)\,P_{n}(x) = \left(25\,x^{2} - 500\,n x^{2}\right)\,P_{n-4}(x) + \left(-40\,x + 35\,x^{2} + 800\,n x + 1850\,n x^{2} - 1000\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(25\,x^{2} - 25\,x^{3} - 500\,n x^{2} + 500\,n x^{3}\right)\,P_{n-3}'(x) + \left(-4 - 138\,x - 189\,x^{2} + 80\,n - 1280\,n x - 660\,n x^{2} + 800\,n^{2} x + 1325\,n^{2} x^{2} - 500\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-20\,x + 125\,x^{2} - 105\,x^{3} + 400\,n x + 25\,n x^{2} - 425\,n x^{3} - 500\,n^{2} x^{2} + 500\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-17 + 168\,x - 60\,n - 370\,n x + 200\,n^{2} x\right)\,P_{n-1}(x) + \left(-105\,x + 105\,x^{2} + 100\,n x - 100\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
\(P_{2}(x) = 1 + 34\,x + 3\,x^{2}\)
\(P_{3}(x) = 1 + 236\,x + 354\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
2134338
312363544595
41145880432916512423
518800125235166980220006323022
6152862166572053969802776200158586710050356
71317244204505381338267401672834254090107611103548363889386
8119035462401042292861284624705963497542919269365602432017614184915022711705
91114213682746317333558411990362420805018802904966022569772209831373235128885139615610696273049241
10168528310309261813901026883111420726542003102515148181127162101715880434352053766222840927785441703426415501135789954431314
111411169972345002108940181390698142601991325681594006655788174716227765086203835593186121090550404081290708208511500070298002261434846122020281773709537
1212467019954382699166352331159142091908051178292787746002595116871893440248231898853689103346015582919125369212092701794280778978552297700140323027662511480211972413124222994381891167
131148021198564230722174967952479981818844441255042907434528259277223826169611722589020493761524872295888056429888556813915835739254417582510085814869056501443199500018222116922888699871696213779064148263611527139885840
1418881271927846675233586294887152423411883956297209743608838688031079325772752234012124787071092645697771213315011480112270056160440519976258754277517988762879203900206538614359394451136261457270235651868202259345540610862168903494615591282616155112743272
Technical details
Node id
n-3cf14a762fddaaaa · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-0d6ccd89a44d9fdd

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: 1,        2;
  2: 1,       34,          3;
  3: 1,      236,        354,           4;
  4: 1,     1458,       8043,        2916,            5;
  5: 1,     8800,     125235,      166980,        22000,            6;
  6: 1,    52862,    1665720,     5396980,      2776200,       158586,           7;
  7: 1,   317244,   20450538,   133826740,    167283425,     40901076,     1110354,          8;
  8: 1,  1903546,  240104229,  2861284624,   7059634975,   4291926936,   560243201,    7614184,        9;
  9: 1, 11421368, 2746317333, 55841199036, 242080501880, 290496602256, 97722098313, 7323512888, 51396156, 10;
%F Formula
The row polynomials satisfy (20*n-21)*P(n,x) = -(500*n-25)*x^2*P(n-4,x) + ((800*n-40)*x-(1000*n^2-1850*n-35)*x^2)*P(n-3,x) - ((500*n-25)*x^2-(500*n-25)*x^3)*d/dx P(n-3,x) + (80*n-4+(800*n^2-1280*n-138)*x-(500*n^3-1325*n^2+660*n+189)*x^2)*P(n-2,x) + ((400*n-20)*x-(500*n^2-25*n-125)*x^2+(500*n^2-425*n-105)*x^3)*d/dx P(n-2,x) - (60*n+17-(200*n^2-370*n+168)*x)*P(n-1,x) + ((100*n-105)*x-(100*n-105)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 34*x + 3*x^2, P(3,x) = 1 + 236*x + 354*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 34*x + 3*x^2;
P[3, x_] := 1 + 236*x + 354*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(500*n - 25)*x^2*P[n - 4, x] + ((800*n - 40)*x - (1000*n^2 - 1850*n - 35)*x^2)*P[n - 3, x] - ((500*n - 25)*x^2 - (500*n - 25)*x^3)*D[P[n - 3, x], x] + (80*n - 4 + (800*n^2 - 1280*n - 138)*x - (500*n^3 - 1325*n^2 + 660*n + 189)*x^2)*P[n - 2, x] + ((400*n - 20)*x - (500*n^2 - 25*n - 125)*x^2 + (500*n^2 - 425*n - 105)*x^3)*D[P[n - 2, x], x] - (60*n + 17 - (200*n^2 - 370*n + 168)*x)*P[n - 1, x] + ((100*n - 105)*x - (100*n - 105)*x^2)*D[P[n - 1, x], x])/(20*n - 21)]];
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] = 1 + 2*x; v[3] = 1 + 34*x + 3*x^2; v[4] = 1 + 236*x + 354*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(500*n - 25)*x^2*v[i - 4] + ((800*n - 40)*x - (1000*n^2 - 1850*n - 35)*x^2)*v[i - 3] - ((500*n - 25)*x^2 - (500*n - 25)*x^3)*deriv(v[i - 3], x) + (80*n - 4 + (800*n^2 - 1280*n - 138)*x - (500*n^3 - 1325*n^2 + 660*n + 189)*x^2)*v[i - 2] + ((400*n - 20)*x - (500*n^2 - 25*n - 125)*x^2 + (500*n^2 - 425*n - 105)*x^3)*deriv(v[i - 2], x) - (60*n + 17 - (200*n^2 - 370*n + 168)*x)*v[i - 1] + ((100*n - 105)*x - (100*n - 105)*x^2)*deriv(v[i - 1], x))/(20*n - 21)); 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. [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.