A013618 → 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

\(P_{n}(x) = \left(12 + 22\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 144 + 528\,x + 484\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	144	528	484								1156
1	1728	9504	17424	10648							39304
2	20736	152064	418176	511104	234256						1336336
3	248832	2280960	8363520	15333120	14055360	5153632					45435424
4	2985984	32845824	150543360	367994880	505992960	371061504	113379904				1544804416
5	35831808	459841536	2529128448	7727892480	14167802880	15584583168	9523911936	2494357888			52523350144
6	429981696	6306398208	40466055168	148375535616	340027269120	498706661376	457147772928	239458357248	54875873536		1785793904896
7	5159780352	85136375808	624333422592	2670759641088	7344589012992	13465079857152	16457319825408	12930751291392	5926594341888	1207269217792	60716992766464

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-43a1e62341e55229 · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-57ca34e86b523c55

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 ?

A009978: 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:          144,            528,             484;
  1:         1728,           9504,           17424,           10648;
  2:        20736,         152064,          418176,          511104,           234256;
  3:       248832,        2280960,         8363520,        15333120,         14055360,          5153632;
  4:      2985984,       32845824,       150543360,       367994880,        505992960,        371061504,         113379904;
  5:     35831808,      459841536,      2529128448,      7727892480,      14167802880,      15584583168,        9523911936,        2494357888;
  6:    429981696,     6306398208,     40466055168,    148375535616,     340027269120,     498706661376,      457147772928,      239458357248,       54875873536;
  7:   5159780352,    85136375808,    624333422592,   2670759641088,    7344589012992,   13465079857152,    16457319825408,    12930751291392,     5926594341888,    1207269217792;
  8:  61917364224,  1135151677440,   9365001338880,  45784450990080,  146891780259840,  323161916571648,   493719594762240,   517230051655680,   355595660513280,  144872306135040,   26559922791424;
  9: 743008370688, 14984002142208, 137353352970240, 755443441336320, 2769959284899840, 7109562164576256, 13034197301723136, 17068591704637440, 15646209062584320, 9561572204912640, 3505909808467968, 584318301411328;

%F Formula

The row polynomials satisfy P(n,x) = (12+22*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 528*x + 484*x^2.

%t Mathematica

P[0, x_] := 144 + 528*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 22*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] = 144 + 528*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 22*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-4aabe19e5745ac70
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-4aabe19e5745ac70
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-4aabe19e5745ac70

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-4aabe19e5745ac70

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle of coefficients in expansion of (1+11x)^n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-5d158048399db1f0 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-70681fa17f5604ff 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