Real-rooted atlas

A056939 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(144 + 144\,x + 72\,x^{2} + 84\,n + 156\,n x + 150\,n x^{2} + 24\,n x^{3} + 12\,n^{2} + 54\,n^{2} x + 105\,n^{2} x^{2} + 50\,n^{2} x^{3} + 6\,n^{3} x + 30\,n^{3} x^{2} + 35\,n^{3} x^{3} + 3\,n^{4} x^{2} + 10\,n^{4} x^{3} + n^{5} x^{3}\right)\,P_{n}(x) = \left(144 + 144\,x + 72\,x^{2} + 204\,n + 468\,n x + 414\,n x^{2} + 144\,n x^{3} + 12\,n x^{4} + 132\,n^{2} + 474\,n^{2} x + 603\,n^{2} x^{2} + 370\,n^{2} x^{3} + 65\,n^{2} x^{4} + 114\,n^{3} x + 306\,n^{3} x^{2} + 325\,n^{3} x^{3} + 98\,n^{3} x^{4} + 45\,n^{4} x^{2} + 110\,n^{4} x^{3} + 55\,n^{4} x^{4} + 11\,n^{5} x^{3} + 10\,n^{5} x^{4}\right)\,P_{n-1}(x) + \left(144 + 288\,x + 252\,x^{2} + 144\,x^{3} + 36\,x^{4} - 72\,n - 252\,n x - 324\,n x^{2} - 180\,n x^{3} - 60\,n x^{4} - 24\,n x^{5} - 108\,n^{2} x - 348\,n^{2} x^{2} - 402\,n^{2} x^{3} - 249\,n^{2} x^{4} - 87\,n^{2} x^{5} - 60\,n^{3} x^{2} - 153\,n^{3} x^{3} - 162\,n^{3} x^{4} - 69\,n^{3} x^{5} - 12\,n^{4} x^{3} - 27\,n^{4} x^{4} - 15\,n^{4} x^{5}\right)\,P_{n-1}'(x) + \left(48 + 192\,x + 348\,x^{2} + 372\,x^{3} + 240\,x^{4} + 84\,x^{5} + 12\,x^{6} + 48\,n x + 228\,n x^{2} + 440\,n x^{3} + 425\,n x^{4} + 202\,n x^{5} + 37\,n x^{6} + 36\,n^{2} x^{2} + 146\,n^{2} x^{3} + 212\,n^{2} x^{4} + 130\,n^{2} x^{5} + 28\,n^{2} x^{6} + 12\,n^{3} x^{3} + 30\,n^{3} x^{4} + 24\,n^{3} x^{5} + 6\,n^{3} x^{6}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2612321
32266394131
492368348965909
542221102760130020066798
620741244420835140803925372753737
7107547527815378313622056455102066378443341
8582024656161123500123468868663019084823716102493784233
93262402936160817434010742760753585028753205665385044815661033209232
101882960188296005942340090969600772590003779496010447290151632099900230011298225341
11111405601225461604323699607560273607551060004527230401630120803387648037273501864503267122730718719
12673299928079599043151857192619978304071266984805072782176226560180062273587299977328853072033105645121325423592085
13414499438538849269423031043932503474829006550981723054074218770289419410289983361960215090132427245933018369494563316608414240133157514
14259334158636306782204168735141939405904798208589985210815554655217116346851860355144706779360395706361056819312500693307615375254889239238036152296860845265
Technical details
Node id
n-89b57aba8253668a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-32ea2ae53c3bb750

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:      2,       2;
  2:      6,      12,       3;
  3:     22,      66,      39,        4;
  4:     92,     368,     348,       96,       5;
  5:    422,    2110,    2760,     1300,     200,       6;
  6:   2074,   12444,   20835,    14080,    3925,     372,      7;
  7:  10754,   75278,  153783,   136220,   56455,   10206,    637,     8;
  8:  58202,  465616, 1123500,  1234688,  686630,  190848,  23716,  1024,    9;
  9: 326240, 2936160, 8174340, 10742760, 7535850, 2875320, 566538, 50448, 1566, 10;
%F Formula
The row polynomials satisfy (12*n^2+84*n+144+(6*n^3+54*n^2+156*n+144)*x+(3*n^4+30*n^3+105*n^2+150*n+72)*x^2+(n^5+10*n^4+35*n^3+50*n^2+24*n)*x^3)*P(n,x) = (132*n^2+204*n+144+(114*n^3+474*n^2+468*n+144)*x+(45*n^4+306*n^3+603*n^2+414*n+72)*x^2+(11*n^5+110*n^4+325*n^3+370*n^2+144*n)*x^3+(10*n^5+55*n^4+98*n^3+65*n^2+12*n)*x^4)*P(n-1,x) - (72*n-144+(108*n^2+252*n-288)*x+(60*n^3+348*n^2+324*n-252)*x^2+(12*n^4+153*n^3+402*n^2+180*n-144)*x^3+(27*n^4+162*n^3+249*n^2+60*n-36)*x^4+(15*n^4+69*n^3+87*n^2+24*n)*x^5)*d/dx P(n-1,x) + (48+(48*n+192)*x+(36*n^2+228*n+348)*x^2+(12*n^3+146*n^2+440*n+372)*x^3+(30*n^3+212*n^2+425*n+240)*x^4+(24*n^3+130*n^2+202*n+84)*x^5+(6*n^3+28*n^2+37*n+12)*x^6)*d^2/dx^2 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[((132*n^2 + 204*n + 144 + (114*n^3 + 474*n^2 + 468*n + 144)*x + (45*n^4 + 306*n^3 + 603*n^2 + 414*n + 72)*x^2 + (11*n^5 + 110*n^4 + 325*n^3 + 370*n^2 + 144*n)*x^3 + (10*n^5 + 55*n^4 + 98*n^3 + 65*n^2 + 12*n)*x^4)*P[n - 1, x] - (72*n - 144 + (108*n^2 + 252*n - 288)*x + (60*n^3 + 348*n^2 + 324*n - 252)*x^2 + (12*n^4 + 153*n^3 + 402*n^2 + 180*n - 144)*x^3 + (27*n^4 + 162*n^3 + 249*n^2 + 60*n - 36)*x^4 + (15*n^4 + 69*n^3 + 87*n^2 + 24*n)*x^5)*D[P[n - 1, x], x] + (48 + (48*n + 192)*x + (36*n^2 + 228*n + 348)*x^2 + (12*n^3 + 146*n^2 + 440*n + 372)*x^3 + (30*n^3 + 212*n^2 + 425*n + 240)*x^4 + (24*n^3 + 130*n^2 + 202*n + 84)*x^5 + (6*n^3 + 28*n^2 + 37*n + 12)*x^6)*D[P[n - 1, x], {x, 2}])/(12*n^2 + 84*n + 144 + (6*n^3 + 54*n^2 + 156*n + 144)*x + (3*n^4 + 30*n^3 + 105*n^2 + 150*n + 72)*x^2 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*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] = ((132*n^2 + 204*n + 144 + (114*n^3 + 474*n^2 + 468*n + 144)*x + (45*n^4 + 306*n^3 + 603*n^2 + 414*n + 72)*x^2 + (11*n^5 + 110*n^4 + 325*n^3 + 370*n^2 + 144*n)*x^3 + (10*n^5 + 55*n^4 + 98*n^3 + 65*n^2 + 12*n)*x^4)*v[i - 1] - (72*n - 144 + (108*n^2 + 252*n - 288)*x + (60*n^3 + 348*n^2 + 324*n - 252)*x^2 + (12*n^4 + 153*n^3 + 402*n^2 + 180*n - 144)*x^3 + (27*n^4 + 162*n^3 + 249*n^2 + 60*n - 36)*x^4 + (15*n^4 + 69*n^3 + 87*n^2 + 24*n)*x^5)*deriv(v[i - 1], x) + (48 + (48*n + 192)*x + (36*n^2 + 228*n + 348)*x^2 + (12*n^3 + 146*n^2 + 440*n + 372)*x^3 + (30*n^3 + 212*n^2 + 425*n + 240)*x^4 + (24*n^3 + 130*n^2 + 202*n + 84)*x^5 + (6*n^3 + 28*n^2 + 37*n + 12)*x^6)*deriv(deriv(v[i - 1], x), x))/(12*n^2 + 84*n + 144 + (6*n^3 + 54*n^2 + 156*n + 144)*x + (3*n^4 + 30*n^3 + 105*n^2 + 150*n + 72)*x^2 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023