A038314 → basis A013609

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(P_{n}(x) = \left(22 + 24\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 484 + 1056\,x + 576\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	484	1056	576								2116
1	10648	34848	38016	13824							97336
2	234256	1022208	1672704	1216512	331776						4477456
3	5153632	28110720	61332480	66908160	36495360	7962624					205962976
4	113379904	742123008	2023971840	2943959040	2408693760	1051066368	191102976				9474296896
5	2494357888	19047823872	62338332672	113342423040	123646279680	80932110336	29429858304	4586471424			435817657216
6	54875873536	478916714496	1828591091712	3989653291008	5440436305920	4748017139712	2589827530752	807218970624	110075314176		20047612231936
7	1207269217792	11853188683776	51723005165568	131658558603264	215441277714432	235026848415744	170928617029632	79914678091776	21794912206848	2641807540224	922190162669056

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-2a3c7d06bf210d3a · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-60cc3cc9b4461a16

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): yes

OEIS matches ?

A009990: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:             484,             1056,               576;
  1:           10648,            34848,             38016,              13824;
  2:          234256,          1022208,           1672704,            1216512,             331776;
  3:         5153632,         28110720,          61332480,           66908160,           36495360,            7962624;
  4:       113379904,        742123008,        2023971840,         2943959040,         2408693760,         1051066368,          191102976;
  5:      2494357888,      19047823872,       62338332672,       113342423040,       123646279680,        80932110336,        29429858304,         4586471424;
  6:     54875873536,     478916714496,     1828591091712,      3989653291008,      5440436305920,      4748017139712,      2589827530752,       807218970624,       110075314176;
  7:   1207269217792,   11853188683776,    51723005165568,    131658558603264,    215441277714432,    235026848415744,    170928617029632,     79914678091776,     21794912206848,     2641807540224;
  8:  26559922791424,  289744612270080,  1422382642053120,   4137840413245440,   7899513516195840,  10341181330292736,   9401073936629760,   5860409726730240,   2397440342753280,   581197658849280,    63403380965376;
  9: 584318301411328, 7011819616935936, 38246288819650560, 125169672500674560, 273097467274199040, 417094313655140352, 455011978532880384, 354554788467179520, 193393520982097920, 70324916720762880, 15343618193620992, 1521681143169024;

%F Formula

The row polynomials satisfy P(n,x) = (22+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1056*x + 576*x^2.

%t Mathematica

P[0, x_] := 484 + 1056*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 24*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 484 + 1056*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 24*x)*v[i - 1]); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-61350e73fec19600
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-61350e73fec19600
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-61350e73fec19600

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A013609 e-61350e73fec19600

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-50ef4b08e4862a18 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-08ecf9f1e7a97c44 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.

References

[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