A200536 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(45 + 114\,x + 72\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	45	114	72													231
2	2025	10260	19476	16416	5184											53361
3	91125	692550	2191860	3697704	3506976	1772928	373248									12326391
4	4100625	41553000	184145400	466132320	737166096	745811712	471412224	170201088	26873856							2847396321
5	184528125	2337356250	13318830000	44960346000	99570027600	151159989024	159312044160	115098485760	54553927680	15318097920	1934917632					657748550151
6	8303765625	126217237500	879091987500	3709851840000	10565086446000	21390327564480	31570322723136	34224524103168	27046621301760	15195553136640	5761217249280	1323483660288	139314069504			151939915084881
7	373669453125	6626404968750	54545775637500	276247460475000	961646622930000	2434093927725600	4619848089003840	6679223959722624	7391756942406144	6231280454977536	3938904567521280	1810415356968960	571953912348672	111172627464192	10030613004288	35098120384607511

All 15 rows

Technical details

Node id: n-25f8fdf7d4ce854d · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-d4d934822159f131

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:               1;
  1:              45,               114,                 72;
  2:            2025,             10260,              19476,               16416,                5184;
  3:           91125,            692550,            2191860,             3697704,             3506976,              1772928,               373248;
  4:         4100625,          41553000,          184145400,           466132320,           737166096,            745811712,            471412224,             170201088,              26873856;
  5:       184528125,        2337356250,        13318830000,         44960346000,         99570027600,         151159989024,         159312044160,          115098485760,           54553927680,           15318097920,            1934917632;
  6:      8303765625,      126217237500,       879091987500,       3709851840000,      10565086446000,       21390327564480,       31570322723136,        34224524103168,        27046621301760,        15195553136640,         5761217249280,         1323483660288,          139314069504;
  7:    373669453125,     6626404968750,     54545775637500,     276247460475000,     961646622930000,     2434093927725600,     4619848089003840,      6679223959722624,      7391756942406144,      6231280454977536,      3938904567521280,      1810415356968960,       571953912348672,       111172627464192,       10030613004288;
  8:  16815125390625,   340786541250000,   3236874270750000,   19126455301800000,   78693604371900000,   239051758915872000,   554618428616851200,   1002482523130199040,   1426689656224932096,   1603972037008318464,   1419823177259139072,    979156004519411712,    515726405611683840,    200555419945402368,    54305738805215232,     9147919059910656,     722204136308736;
  9: 756680642578125, 17252318650781250, 185719696914375000, 1254230786416500000, 5954683048634700000, 21105504831340440000, 57875669318944512000, 125549941045122777600, 218416569027377921280, 307000104140390920704, 349466510443804674048, 321407849075514310656, 237058741530396721152, 138317036462672707584, 62439377324051791872, 21042500817559486464, 4985375152939204608, 740981443852763136, 51998697814228992;

%F Formula

The row polynomials satisfy P(n,x) = (45+114*x+72*x^2)*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[(45 + 114*x + 72*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (45 + 114*x + 72*x^2)*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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.