A159854 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 + 6\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 9 + 36\,x + 36\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	9	36	36															81
1	36	198	360	216														810
2	144	1008	2628	3024	1296													8100
3	576	4896	16560	27864	23328	7776												81000
4	2304	23040	95616	210816	260496	171072	46656											810000
5	9216	105984	520704	1416960	2306880	2247264	1213056	279936										8100000
6	36864	479232	2718720	8792064	17729280	22830336	18335808	8398080	1679616									81000000
7	147456	2138112	13750272	51480576	123669504	197697024	210325248	143607168	57106944	10077696								810000000
8	589824	9437184	67829760	288423936	803561472	1532805120	2027483136	1836380160	1090070784	382952448	60466176							8100000000
9	2359296	41287680	327942144	1560674304	4944789504	10952589312	17306763264	19510419456	15378564096	8072234496	2539579392	362797056						81000000000
10	9437184	179306496	1559494656	8210350080	29143203840	73479094272	134942588928	181882257408	178576773120	124560322560	58591724544	16688664576	2176782336					810000000000
11	37748736	773849088	7313817600	42198368256	165834915840	468775600128	980644921344	1537184563200	1805600636928	1569701928960	981728833536	418305005568	108839116800	13060694016				8100000000000
12	150994944	3321888768	33898364928	212676378624	916529872896	2870111895552	6735233286144	12032607780864	16445509926912	17112411537408	13345126907904	7563593023488	2945186500608	705277476864	78364164096			81000000000000
13	603979776	14193524736	155524792320	1054095704064	4942177763328	16979626819584	44161604517888	88541830840320	137977686392832	167122705711104	156054976856064	110325133541376	57162304143360	20492228911104	4545121517568	470184984576		810000000000000
14	2415919104	60397977600	707260317696	5149531570176	26093285277696	97571573858304	278524178989056	619136950468608	1083161730613248	1496356941201408	1626956141690880	1377630395301888	890600017821696	424942740504576	141133859536896	29151469043712	2821109907456	8100000000000000

Technical details

Node id: n-12545cb3cb924bbb · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-99f5d32b38d6538a

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

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:       9,       36,        36;
  1:      36,      198,       360,        216;
  2:     144,     1008,      2628,       3024,       1296;
  3:     576,     4896,     16560,      27864,      23328,        7776;
  4:    2304,    23040,     95616,     210816,     260496,      171072,       46656;
  5:    9216,   105984,    520704,    1416960,    2306880,     2247264,     1213056,      279936;
  6:   36864,   479232,   2718720,    8792064,   17729280,    22830336,    18335808,     8398080,     1679616;
  7:  147456,  2138112,  13750272,   51480576,  123669504,   197697024,   210325248,   143607168,    57106944,   10077696;
  8:  589824,  9437184,  67829760,  288423936,  803561472,  1532805120,  2027483136,  1836380160,  1090070784,  382952448,   60466176;
  9: 2359296, 41287680, 327942144, 1560674304, 4944789504, 10952589312, 17306763264, 19510419456, 15378564096, 8072234496, 2539579392, 362797056;

%F Formula

The row polynomials satisfy P(n,x) = (4+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 36*x + 36*x^2.

%t Mathematica

P[0, x_] := 9 + 36*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 6*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] = 9 + 36*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (4 + 6*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)

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_A013609 of Riordan array (1-x,x/(1-x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    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
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

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.