A272867 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 12\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	4	12	6																											22
2	16	96	192	144	36																									484
3	64	576	2016	3456	3024	1296	216																							10648
4	256	3072	15360	41472	65664	62208	34560	10368	1296																					234256
5	1024	15360	99840	368640	852480	1285632	1278720	829440	336960	77760	7776																			5153632
6	4096	73728	589824	2764800	8432640	17584128	25657344	26376192	18973440	9331200	2985984	559872	46656																	113379904
7	16384	344064	3268608	18579456	70447104	188116992	364234752	518897664	546352128	423263232	237758976	94058496	24820992	3919104	279936															2494357888
8	65536	1572864	17301504	115605504	524353536	1709309952	4137025536	7575109632	10597588992	11362664448	9308307456	5768921088	2654539776	877879296	197074944	26873856	1679616													54875873536
9	262144	7077888	88473600	679477248	3588489216	13823115264	40205942784	90200604672	158113824768	218072383488	237170737152	202951360512	135695056896	69979521024	27250089984	7739670528	1511654400	181398528	10077696											1207269217792
10	1048576	31457280	440401920	3822059520	23038525440	102431195136	348232089600	926212423680	1956098211840	3310859059200	4514234499072	4966288588800	4401220976640	3125966929920	1762924953600	777836888064	262423203840	65303470080	11287019520	1209323520	60466176									26559922791424
11	4194304	138412032	2145386496	20761804800	140661227520	709119442944	2760333852672	8498221940736	21028334469120	42273889320960	69523835977728	93901122699264	104285753966592	95116250972160	70970628833280	43022248574976	20961285193728	8077313654784	2403328942080	532102348800	82475864064	7981535232	362797056							584318301411328
12	16777216	603979776	10267656192	109622329344	824658886656	4648983330816	20394736091136	71371610652672	202654004281344	472424902557696	911552022577152	1463533858455552	1961099504123904	2195300787683328	2050992050798592	1594434046132224	1025935896674304	541978168393728	232308790788096	79432238628864	21135105294336	4214250602496	592084795392	52242776064	2176782336					12855002631049216
13	67108864	2617245696	48419045376	565325070336	4675709435904	29149573939200	142314697654272	558117175689216	1789443761504256	4749777325522944	10531230946689024	19627309120094208	30876116453425152	41095600350953472	46314174680137728	44161445520211968	35542904445075456	24045747710459904	13588588563922944	6357303454334976	2431580029452288	747071697715200	179749978177536	32599492263936	4188129214464	339578044416	13060694016			282810057883082752
14	268435456	11274289152	225485783040	2858032300032	25777252859904	176098759409664	947107934502912	4115142518243328	14709069340213248	43821137494278144	109858914262056960	233382671793782784	422219560934965248	652659653565480960	863660601652543488	978989480348221440	949994012103671808	787666517304016896	556160753451663360	332766762847174656	167545492953366528	70311067870298112	24273340462006272	6769827893477376	1486446293090304	247212816334848	29255954595840	2194196594688	78364164096	6221821273427820544

Technical details

Node id: n-ec0bc4323ea11152 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-3c4a6b4bc3495597

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:      1;
  1:      4,      12,        6;
  2:     16,      96,      192,       144,         36;
  3:     64,     576,     2016,      3456,       3024,        1296,         216;
  4:    256,    3072,    15360,     41472,      65664,       62208,       34560,       10368,         1296;
  5:   1024,   15360,    99840,    368640,     852480,     1285632,     1278720,      829440,       336960,        77760,         7776;
  6:   4096,   73728,   589824,   2764800,    8432640,    17584128,    25657344,    26376192,     18973440,      9331200,      2985984,       559872,        46656;
  7:  16384,  344064,  3268608,  18579456,   70447104,   188116992,   364234752,   518897664,    546352128,    423263232,    237758976,     94058496,     24820992,     3919104,      279936;
  8:  65536, 1572864, 17301504, 115605504,  524353536,  1709309952,  4137025536,  7575109632,  10597588992,  11362664448,   9308307456,   5768921088,   2654539776,   877879296,   197074944,   26873856,    1679616;
  9: 262144, 7077888, 88473600, 679477248, 3588489216, 13823115264, 40205942784, 90200604672, 158113824768, 218072383488, 237170737152, 202951360512, 135695056896, 69979521024, 27250089984, 7739670528, 1511654400, 181398528, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (4+12*x+6*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[(4 + 12*x + 6*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] = (4 + 12*x + 6*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

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.