Real-rooted atlas

A104684 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-21 - 10\,x + 13\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-12 - 14\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2527
33324461
4245258848595
5192126561296256166145
615525266101736052807203265527
71279052617202140208832018960192064712909
810679252540370249858013045203847206249649281287863667
99004545244166402806944017740800667296015052801935361228825687615745
107649952523290461030645792022747200010416672030183552544051257139229952512983726695
11653808673220826812032729092202790773760150727584053444966412453504018493440162432071680102411112210781
12561499576520835159906343466347403308452884020595136320864612672024767930884782412805985408044774401689602048126142119187
1348416454529195784325472355333613712381599299840269059879440130584655872444463810561059541401617334028801860812801202995239321640961437751935361
144188951748851833469750034363282297019243036508440643390412659920186834901968073722522931220949411225642479154432599016704055937024031629312905216819216443380994775
153634723102113171199301936883677174459030847636096971008414725721490082558674941222411489550456384379486903910492003309798416138592870419886490624163407462481629184206438416384188609259215725

Other recurrences

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

\(\left(1 - n\right)\,P_{n}(x) = \left(-18 - 24\,x - 8\,x^{2} + 9\,n + 12\,n x + 4\,n x^{2}\right)\,P_{n-2}(x) + \left(15 + 6\,x - 10\,n - 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 2\,x\)
Technical details
Node id
n-344301f9723c6726 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-7b640b1203ef46c2)
Fingerprint
f6-b86e4a1bda66e0b6

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        5,         2;
   3:       33,        24,         4;
   4:      245,       258,        84,         8;
   5:     1921,      2656,      1296,       256,        16;
   6:    15525,     26610,     17360,      5280,       720,       32;
   7:   127905,    261720,    214020,     88320,     18960,     1920,      64;
   8:  1067925,   2540370,   2498580,   1304520,    384720,    62496,    4928,    128;
   9:  9004545,  24416640,  28069440,  17740800,   6672960,  1505280,  193536,  12288,   256;
  10: 76499525, 232904610, 306457920, 227472000, 104166720, 30183552, 5440512, 571392, 29952, 512;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (13*n-21+(6*n-10)*x)*P(n-1,x) - (12+14*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((13*n - 21 + (6*n - 10)*x)*P[n - 1, x] - (12 + 14*x + 4*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((13*n - 21 + (6*n - 10)*x)*v[i - 1] - (12 + 14*x + 4*x^2)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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