A013618 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(12 + 11\,x\right)\,P_{n-1}(x) + \left(11\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 144 + 385\,x + 121\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	144	385	121								650
1	1728	10439	8349	1331							21847
2	20736	259105	398695	151734	14641						844911
3	248832	6187511	16405785	11213675	2488970	161051					36705824
4	2985984	145049905	625859311	685079010	262732745	38169087	1771561				1761647603
5	35831808	3368993639	22874765529	37713007871	22248902830	5447389024	558041715	19487171			92266419587
6	429981696	77881003585	814800958015	1948707775014	1660781645061	609712995738	103448533034	7872817084	214358881		5223850068108
7	5159780352	1795992881111	28559923611945	96654660413795	114439557648570	59119368810117	14775828529770	1838614583850	108036876024	2357947691	317299501083225

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-a747dc70c2aab5a2 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-271bbcbc6f9a34cb

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

OEIS matches ?

No OEIS matches.

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,             385,               121;
  1:         1728,           10439,              8349,               1331;
  2:        20736,          259105,            398695,             151734,              14641;
  3:       248832,         6187511,          16405785,           11213675,            2488970,             161051;
  4:      2985984,       145049905,         625859311,          685079010,          262732745,           38169087,            1771561;
  5:     35831808,      3368993639,       22874765529,        37713007871,        22248902830,         5447389024,          558041715,          19487171;
  6:    429981696,     77881003585,      814800958015,      1948707775014,      1660781645061,       609712995738,       103448533034,        7872817084,        214358881;
  7:   5159780352,   1795992881111,    28559923611945,     96654660413795,    114439557648570,     59119368810117,     14775828529770,     1838614583850,     108036876024,      2357947691;
  8:  61917364224,  41364593849425,   990793324498351,   4663618878352170,   7471816492871665,   5219832844412109,   1802827682233347,   326170811790120,   31028448024750,   1450137829965,    25937424601;
  9: 743008370688, 952066749543239, 34141983565287609, 220761576095329511, 469721531262687110, 431918781997199618, 198038720502734265, 48860306753887497, 6690723732166320, 502278227398365, 19115881930937, 285311670611;

%F Formula

The row polynomials satisfy P(n,x) = (12+11*x)*P(n-1,x) + 11*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 144 + 385*x + 121*x^2.

%t Mathematica

P[0, x_] := 144 + 385*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 11*x)*P[n - 1, x] + 11*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 385*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 11*x)*v[i - 1] + 11*x*deriv(v[i - 1], x)); 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-e469bcf1fd1b08af
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-e469bcf1fd1b08af
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-e469bcf1fd1b08af

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 A008277 e-e469bcf1fd1b08af

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- 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 ?

No candidate equalities or related sequences.

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741