Real-rooted atlas

A123519 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(3 + 2\,n\right)\,P_{n}(x) = \left(18 + 15\,x + 2\,x^{2} + 12\,n + 12\,n x + 2\,n x^{2}\right)\,P_{n-1}(x) + \left(-6\,x - 8\,x^{2} - 2\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 36 + 42\,x + 11\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
036421189
121639622841881
212963240291611161538721
37776246243024017964515457186329
44665617884827864022550499864229262131854569
527993612597122379456244296014709605190489933679538459361
616796168678016192689282399673618318960877262425723444221362968183739041
7100776965878656015004569621979641620352124812346862449053600123031441767294110771828931049
86046617639303014411337408001910283264208104422415309000007698771362612606405724399673116664134038205571449
936279705626000455688364487680159378762241997735270417289780480105393415684523603760133945293626054283629959788154284181226783441
10217678233617051461632605266421761286921779201824818803201817378104321303141271046776808883225361255520665980812011646860761217845325757961804062262961
11130606940161110158991364310029025281011659590656160129550592018044165076481488858651648910575260928412700546880136880267040322878960005128908660491726106214890037959395846169
12783641640967183381708803027904229376777691769241613593884755968171029888916481596833606016011237148361728598819019788824043157009287158871886401532652278402230130520019740539108019805178789896198729
13470184984576462348568166421027717365760586621071728641122399764029441559818127831041626206764216321294918179033607938887598259237468079736192134998853886723646259854560714580846560959206356007885677840299303201779939566141121
14282110990745629621654028288144581882757120435365174329344905060382879744137707386461798415860574249062401410330624215040978332191448832531126833985792224883441078912734727849811201815663730896032805819720004086953042403136447665611170147537720605765212481
Technical details
Node id
n-a95bf72987741784 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-02ef7371f378b284

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:        36,         42,         11;
  1:       216,        396,        228,          41;
  2:      1296,       3240,       2916,        1116,         153;
  3:      7776,      24624,      30240,       17964,        5154,         571;
  4:     46656,     178848,     278640,      225504,       99864,       22926,        2131;
  5:    279936,    1259712,    2379456,     2442960,     1470960,      519048,       99336,       7953;
  6:   1679616,    8678016,   19268928,    23996736,    18318960,     8772624,     2572344,     422136,      29681;
  7:  10077696,   58786560,  150045696,   219796416,   203521248,   123468624,    49053600,   12303144,    1767294,    110771;
  8:  60466176,  393030144, 1133740800,  1910283264,  2081044224,  1530900000,   769877136,  261260640,   57243996,   7311666,   413403;
  9: 362797056, 2600045568, 8364487680, 15937876224, 19977352704, 17289780480, 10539341568, 4523603760, 1339452936, 260542836, 29959788, 1542841;
%F Formula
The row polynomials satisfy (2*n+3)*P(n,x) = (12*n+18+(12*n+15)*x+(2*n+2)*x^2)*P(n-1,x) - (6*x+8*x^2+2*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 36 + 42*x + 11*x^2.
%t Mathematica
P[0, x_] := 36 + 42*x + 11*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 18 + (12*n + 15)*x + (2*n + 2)*x^2)*P[n - 1, x] - (6*x + 8*x^2 + 2*x^3)*D[P[n - 1, x], x])/(2*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 36 + 42*x + 11*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 18 + (12*n + 15)*x + (2*n + 2)*x^2)*v[i - 1] - (6*x + 8*x^2 + 2*x^3)*deriv(v[i - 1], x))/(2*n + 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

Same fingerprint ?

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023