Real-rooted atlas

A123519 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(3 + 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(21 + 12\,x + 19\,n + 10\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(12 + 16\,x + 4\,x^{2} + 6\,n + 8\,n x + 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 14 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
014822
1768824188
2372648360641444
3171839923360121616010446
47642221922505613760368038472714
5331121153441634401206404896010368896492760
61407125716329747369046404937601585922777620483273896
75890982734032545040060972164187680180902448025671680460821423994
8243722212720888290289603801862431500000171279366114304138240017971210240138520286
99986596578984081488281042233878402168588801423027206394214419433472382464044032022528886925652
104059484425881912873997868012524076801394405120107254425658169395222250496058844160102502401058816491525633150988
11163908702113975748035875440006759519360849178080074944465924765436928220017254473218816017132544026763264250675210649635535456518
12658017970495584560017029469760353524537604947151648049285516032359670958081947061248078202967042302136320482910208683704325857280229376222870328210
1326285592802131569680079397871840180062215040277775419520308379257088254102483712157892782080743578767362641565696069695774721324597248171458560135331844915201390807477040
1410454825552908211787203645091080008966240646401511785699200185089251916816999944389121192906192896644869969920268600217600855098449922043636940835499376644231987203096576010485768641409579728
Technical details
Node id
n-f06dded2c12b823a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-88ca96ddd6c3cdf3

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      14,        8;
  1:      76,       88,        24;
  2:     372,      648,       360,        64;
  3:    1718,     3992,      3360,      1216,       160;
  4:    7642,    22192,     25056,     13760,      3680,       384;
  5:   33112,   115344,    163440,    120640,     48960,     10368,      896;
  6:  140712,   571632,    974736,    904640,    493760,    158592,    27776,     2048;
  7:  589098,  2734032,   5450400,   6097216,   4187680,   1809024,   480256,    71680,    4608;
  8: 2437222, 12720888,  29028960,  38018624,  31500000,  17127936,  6114304,  1382400,  179712,  10240;
  9: 9986596, 57898408, 148828104, 223387840, 216858880, 142302720, 63942144, 19433472, 3824640, 440320, 22528;
%F Formula
The row polynomials satisfy (2*n^2+5*n+3)*P(n,x) = (4*n^2+19*n+21+(2*n^2+10*n+12)*x)*P(n-1,x) + (6*n+12+(8*n+16)*x+(2*n+4)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 8*x.
%t Mathematica
P[0, x_] := 14 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 19*n + 21 + (2*n^2 + 10*n + 12)*x)*P[n - 1, x] + (6*n + 12 + (8*n + 16)*x + (2*n + 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 + 5*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 14 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 19*n + 21 + (2*n^2 + 10*n + 12)*x)*v[i - 1] + (6*n + 12 + (8*n + 16)*x + (2*n + 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 + 5*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023