Real-rooted atlas

A144443 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(961 - 1860\,n + 900\,n^{2}\right)\,P_{n}(x) = \left(90396\,x^{2} - 311040\,n x^{2} + 291600\,n^{2} x^{2}\right)\,P_{n-4}(x) + \left(44640\,x^{3} - 138240\,n x^{3} + 129600\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(133176\,x - 273420\,x^{2} + 23040\,n x + 1316520\,n x^{2} - 21600\,n^{2} x - 1756080\,n^{2} x^{2} + 583200\,n^{3} x^{2}\right)\,P_{n-3}(x) + \left(40920\,x^{2} - 273048\,x^{3} - 23040\,n x^{2} + 1046304\,n x^{3} + 21600\,n^{2} x^{2} - 1205280\,n^{2} x^{3} + 259200\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(44640\,x^{3} - 44640\,x^{4} - 138240\,n x^{3} + 138240\,n x^{4} + 129600\,n^{2} x^{3} - 129600\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(155 - 221092\,x + 31713\,x^{2} - 4800\,n + 46296\,n x - 649512\,n x^{2} + 4500\,n^{2} + 84240\,n^{2} x + 1479384\,n^{2} x^{2} - 21600\,n^{3} x - 1153440\,n^{3} x^{2} + 291600\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(38688\,x - 193316\,x^{2} + 49972\,x^{3} + 11520\,n x - 67176\,n x^{2} - 811704\,n x^{3} - 10800\,n^{2} x + 66960\,n^{2} x^{2} + 1569024\,n^{2} x^{3} + 21600\,n^{3} x^{2} - 937440\,n^{3} x^{3} + 129600\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(32364\,x^{2} - 43896\,x^{3} + 11532\,x^{4} + 34560\,n x^{2} + 126144\,n x^{3} - 160704\,n x^{4} - 32400\,n^{2} x^{2} - 246240\,n^{2} x^{3} + 278640\,n^{2} x^{4} + 129600\,n^{3} x^{3} - 129600\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(806 - 19220\,x + 2940\,n + 60264\,n x - 3600\,n^{2} - 62640\,n^{2} x + 21600\,n^{3} x\right)\,P_{n-1}(x) + \left(372\,x - 3844\,x^{2} - 16920\,n x - 15624\,n x^{2} + 10800\,n^{2} x + 41040\,n^{2} x^{2} - 21600\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
\(P_{2}(x) = 1 + 80\,x + 9\,x^{2}\)
\(P_{3}(x) = 1 + 636\,x + 1431\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
2180990
316361431162084
41455238286182082561072
51319887009021246048199925362178900
6122406410972503477582083047917520165764991450576
7115686201587028411402549040219148287563481136419215595644408330400
8110980536219674941235511761696110021876950799014638161196008013217568857681239778601200
917686397229641285584820913597184448379677815064566673605362514047891376210782475264155649543310014517482747600
1015380480483939592027411788670194931215988690955685040126396537008031337834273142671546807797248354563282466913451201200121967916308681696
11137663366045189090745195374481734633040520663159111425020996697471824664285788382255391261332897687325248018958137815797655765656383550011393168227114470444403030543040
1212636435652067982373035082762673295127606415900473541647317597467367264693012821181110177404176441732753195816764672515175342749373087476670809454754009139852374716589491168347201695555994721358408320
13118455049595688779612174561015233558705133174446340515146743008754147924327998840831613558706480427873702017709330913211916672093328297379973918711185362060586972035001152037877075696314142047379479297607797258454141196471977646149737388864
14112918534720321157201792046672330008476817889322241304422741291057918251654247222555843615328779111631928637898708315246623659460458957824128791922829019734276274595131173766232264800631340606784872032433270076291361003900162172967809509862436330082012956822542958330949094299453952
Technical details
Node id
n-4aa6c410d6c94a43 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b8d0c092bbd5a0ce

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,        4;
  2: 1,       80,           9;
  3: 1,      636,        1431,           16;
  4: 1,     4552,       38286,        18208,            25;
  5: 1,    31988,      700902,      1246048,        199925,            36;
  6: 1,   224064,    10972503,     47758208,      30479175,       2016576,            49;
  7: 1,  1568620,   158702841,   1402549040,    2191482875,     634811364,      19215595,           64;
  8: 1, 10980536,  2196749412,  35511761696,  110021876950,   79901463816,   11960080132,    175688576,         81;
  9: 1, 76863972, 29641285584, 820913597184, 4483796778150, 6456667360536, 2514047891376, 210782475264, 1556495433, 100;
%F Formula
The row polynomials satisfy (900*n^2-1860*n+961)*P(n,x) = (291600*n^2-311040*n+90396)*x^2*P(n-4,x) + (129600*n^2-138240*n+44640)*x^3*d/dx P(n-4,x) - ((21600*n^2-23040*n-133176)*x-(583200*n^3-1756080*n^2+1316520*n-273420)*x^2)*P(n-3,x) + ((21600*n^2-23040*n+40920)*x^2+(259200*n^3-1205280*n^2+1046304*n-273048)*x^3)*d/dx P(n-3,x) + ((129600*n^2-138240*n+44640)*x^3-(129600*n^2-138240*n+44640)*x^4)*d^2/dx^2 P(n-3,x) + (4500*n^2-4800*n+155-(21600*n^3-84240*n^2-46296*n+221092)*x+(291600*n^4-1153440*n^3+1479384*n^2-649512*n+31713)*x^2)*P(n-2,x) - ((10800*n^2-11520*n-38688)*x-(21600*n^3+66960*n^2-67176*n-193316)*x^2-(129600*n^4-937440*n^3+1569024*n^2-811704*n+49972)*x^3)*d/dx P(n-2,x) - ((32400*n^2-34560*n-32364)*x^2-(129600*n^3-246240*n^2+126144*n-43896)*x^3+(129600*n^3-278640*n^2+160704*n-11532)*x^4)*d^2/dx^2 P(n-2,x) - (3600*n^2-2940*n-806-(21600*n^3-62640*n^2+60264*n-19220)*x)*P(n-1,x) + ((10800*n^2-16920*n+372)*x-(21600*n^3-41040*n^2+15624*n+3844)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 80*x + 9*x^2, P(3,x) = 1 + 636*x + 1431*x^2 + 16*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 1 + 80*x + 9*x^2;
P[3, x_] := 1 + 636*x + 1431*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((291600*n^2 - 311040*n + 90396)*x^2*P[n - 4, x] + (129600*n^2 - 138240*n + 44640)*x^3*D[P[n - 4, x], x] - ((21600*n^2 - 23040*n - 133176)*x - (583200*n^3 - 1756080*n^2 + 1316520*n - 273420)*x^2)*P[n - 3, x] + ((21600*n^2 - 23040*n + 40920)*x^2 + (259200*n^3 - 1205280*n^2 + 1046304*n - 273048)*x^3)*D[P[n - 3, x], x] + ((129600*n^2 - 138240*n + 44640)*x^3 - (129600*n^2 - 138240*n + 44640)*x^4)*D[P[n - 3, x], {x, 2}] + (4500*n^2 - 4800*n + 155 - (21600*n^3 - 84240*n^2 - 46296*n + 221092)*x + (291600*n^4 - 1153440*n^3 + 1479384*n^2 - 649512*n + 31713)*x^2)*P[n - 2, x] - ((10800*n^2 - 11520*n - 38688)*x - (21600*n^3 + 66960*n^2 - 67176*n - 193316)*x^2 - (129600*n^4 - 937440*n^3 + 1569024*n^2 - 811704*n + 49972)*x^3)*D[P[n - 2, x], x] - ((32400*n^2 - 34560*n - 32364)*x^2 - (129600*n^3 - 246240*n^2 + 126144*n - 43896)*x^3 + (129600*n^3 - 278640*n^2 + 160704*n - 11532)*x^4)*D[P[n - 2, x], {x, 2}] - (3600*n^2 - 2940*n - 806 - (21600*n^3 - 62640*n^2 + 60264*n - 19220)*x)*P[n - 1, x] + ((10800*n^2 - 16920*n + 372)*x - (21600*n^3 - 41040*n^2 + 15624*n + 3844)*x^2)*D[P[n - 1, x], x])/(900*n^2 - 1860*n + 961)]];
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 + 4*x; v[3] = 1 + 80*x + 9*x^2; v[4] = 1 + 636*x + 1431*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((291600*n^2 - 311040*n + 90396)*x^2*v[i - 4] + (129600*n^2 - 138240*n + 44640)*x^3*deriv(v[i - 4], x) - ((21600*n^2 - 23040*n - 133176)*x - (583200*n^3 - 1756080*n^2 + 1316520*n - 273420)*x^2)*v[i - 3] + ((21600*n^2 - 23040*n + 40920)*x^2 + (259200*n^3 - 1205280*n^2 + 1046304*n - 273048)*x^3)*deriv(v[i - 3], x) + ((129600*n^2 - 138240*n + 44640)*x^3 - (129600*n^2 - 138240*n + 44640)*x^4)*deriv(deriv(v[i - 3], x), x) + (4500*n^2 - 4800*n + 155 - (21600*n^3 - 84240*n^2 - 46296*n + 221092)*x + (291600*n^4 - 1153440*n^3 + 1479384*n^2 - 649512*n + 31713)*x^2)*v[i - 2] - ((10800*n^2 - 11520*n - 38688)*x - (21600*n^3 + 66960*n^2 - 67176*n - 193316)*x^2 - (129600*n^4 - 937440*n^3 + 1569024*n^2 - 811704*n + 49972)*x^3)*deriv(v[i - 2], x) - ((32400*n^2 - 34560*n - 32364)*x^2 - (129600*n^3 - 246240*n^2 + 126144*n - 43896)*x^3 + (129600*n^3 - 278640*n^2 + 160704*n - 11532)*x^4)*deriv(deriv(v[i - 2], x), x) - (3600*n^2 - 2940*n - 806 - (21600*n^3 - 62640*n^2 + 60264*n - 19220)*x)*v[i - 1] + ((10800*n^2 - 16920*n + 372)*x - (21600*n^3 - 41040*n^2 + 15624*n + 3844)*x^2)*deriv(v[i - 1], x))/(900*n^2 - 1860*n + 961)); 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.