Real-rooted atlas

A001263 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(2 + 10\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-6 - 10\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21918441
3100138608306
45621016656176162426
533047380632025604803220076
6200715335856940309208880124864171481
71249963853224924363357201304802844831361281500666
8793774278337641442243402336167126449728086016768025613386206
95120632201245403421008032860800195350407390656175795224883218432512121267476
1033463102145681220278392200306338400213897600983808002991744058636806950404352010241112674026
1122106000810559494442240786680277945800022301136001209035520448842240113034240186700801886720101376204810318939956
12147383030876638012961788103996824685554080223832664001398418560061393305601894717440403983360572211205001216233472409696572168916
13990418619255691336168141697186592215493394688217932100672154229989056782357621762864034201675102044161379096576169861120129925125324808192910896992856
14670154013914051776109501116467659580185461724812020698561884001637003400032942697339584399735033984124688788224282532490244529581056490706944331735041204224163848650566601401

Other recurrences

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

\(\left(2 + n\right)\,P_{n}(x) = \left(4 + 8\,x + 4\,x^{2} - 4\,n - 8\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(4 + 2\,x + 8\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
Technical details
Node id
n-14506fb9ff389955 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-cdca5f77258a98bd)
Fingerprint
f6-1afe30f8794f3fd0

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:       4,        2;
  2:      19,       18,        4;
  3:     100,      138,       60,        8;
  4:     562,     1016,      656,      176,       16;
  5:    3304,     7380,     6320,     2560,      480,      32;
  6:   20071,    53358,    56940,    30920,     8880,    1248,      64;
  7:  124996,   385322,   492436,   335720,   130480,   28448,    3136,    128;
  8:  793774,  2783376,  4144224,  3402336,  1671264,  497280,   86016,   7680,   256;
  9: 5120632, 20124540, 34210080, 32860800, 19535040, 7390656, 1757952, 248832, 18432, 512;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (10*n+2+6*n*x)*P(n-1,x) - (6+10*x+4*x^2)*d/dx 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[((10*n + 2 + 6*n*x)*P[n - 1, x] - (6 + 10*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((10*n + 2 + 6*n*x)*v[i - 1] - (6 + 10*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