Real-rooted atlas

A133214 → basis A007318 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(18 + 8\,x + 13\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-12 - 14\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 33 + 24\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
03324461
1245258848595
2192126561296256166145
315525266101736052807203265527
41279052617202140208832018960192064712909
510679252540370249858013045203847206249649281287863667
69004545244166402806944017740800667296015052801935361228825687615745
77649952523290461030645792022747200010416672030183552544051257139229952512983726695
8653808673220826812032729092202790773760150727584053444966412453504018493440162432071680102411112210781
9561499576520835159906343466347403308452884020595136320864612672024767930884782412805985408044774401689602048126142119187
1048416454529195784325472355333613712381599299840269059879440130584655872444463810561059541401617334028801860812801202995239321640961437751935361
114188951748851833469750034363282297019243036508440643390412659920186834901968073722522931220949411225642479154432599016704055937024031629312905216819216443380994775
123634723102113171199301936883677174459030847636096971008414725721490082558674941222411489550456384379486903910492003309798416138592870419886490624163407462481629184206438416384188609259215725
1331616937184725159454149341490369042811652340518983111712520494851544176080337965473253408170225343928640641181530246401811168042112038093300044805858842429446379656192046573363202072985604669440327682168833084841395
1427562110280294514818876574284803676496045287680557787835680768057812089314470404330500257341440241835806974361610235408547020803303738768844808114282102784015003399290880204562287820819868319744012991856640519045120104857606553624994269200292865

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(2 + n\right)\,P_{n}(x) = \left(-9 - 12\,x - 4\,x^{2} - 9\,n - 12\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(15 + 6\,x + 10\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 33 + 24\,x + 4\,x^{2}\)
\(P_{1}(x) = 245 + 258\,x + 84\,x^{2} + 8\,x^{3}\)
Technical details
Node id
n-72d7b1d737e4797d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-f21e850e4557f0ef)
Fingerprint
f6-b86e4a1bda66e0b6

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:         33,          24,           4;
  1:        245,         258,          84,           8;
  2:       1921,        2656,        1296,         256,          16;
  3:      15525,       26610,       17360,        5280,         720,         32;
  4:     127905,      261720,      214020,       88320,       18960,       1920,         64;
  5:    1067925,     2540370,     2498580,     1304520,      384720,      62496,       4928,       128;
  6:    9004545,    24416640,    28069440,    17740800,     6672960,    1505280,     193536,     12288,      256;
  7:   76499525,   232904610,   306457920,   227472000,   104166720,   30183552,    5440512,    571392,    29952,     512;
  8:  653808673,  2208268120,  3272909220,  2790773760,  1507275840,  534449664,  124535040,  18493440,  1624320,   71680,   1024;
  9: 5614995765, 20835159906, 34346634740, 33084528840, 20595136320, 8646126720, 2476793088, 478241280, 59854080, 4477440, 168960, 2048;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (13*n+18+(6*n+8)*x)*P(n-1,x) - (12+14*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 33 + 24*x + 4*x^2.
%t Mathematica
P[0, x_] := 33 + 24*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((13*n + 18 + (6*n + 8)*x)*P[n - 1, x] - (12 + 14*x + 4*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 33 + 24*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((13*n + 18 + (6*n + 8)*x)*v[i - 1] - (12 + 14*x + 4*x^2)*deriv(v[i - 1], x))/(n + 2)); 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. [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