A038331 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(37 + 40\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1369 + 2960\,x + 1600\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1369	2960	1600															5929
1	50653	164280	177600	64000														456533
2	1874161	8104480	13142400	9472000	2560000													35153041
3	69343957	374832200	810448000	876160000	473600000	102400000												2706784157
4	2565726409	16642549680	44979864000	64835840000	52569600000	22732800000	4096000000											208422380089
5	94931877133	718403394520	2329956955200	4198120640000	4538508800000	2943897600000	1060864000000	163840000000										16048523266853
6	3512479453921	30378200682560	114944543123200	248528741888000	335849651200000	290464563200000	157007872000000	48496640000000	6553600000000									1235736291547681
7	129961739795077	1264492603411560	5468076122860800	13793345174784000	22367586769920000	24181174886400000	17427873792000000	8074690560000000	2182348800000000	262144000000000								95151694449171437
8	4808584372417849	51984695918030800	252898520682312000	729076816381440000	1379334517478400000	1789406941593600000	1612078325760000000	995878502400000000	403734528000000000	96993280000000000	10485760000000000							7326680472586200649
9	177917621779460413	2115777123863853560	11436633101966776000	37091783033405760000	80198449801958400000	121381437538099200000	131223175716864000000	101330637619200000000	54773317632000000000	19738132480000000000	4267704320000000000	419430400000000000						564154396389137449973
10	6582952005840035281	85400458454140998240	507786509727324854400	1829861296314684160000	4451013964008691200000	7699051180988006400000	9710515003047936000000	8998160620584960000000	6079838257152000000000	2921243607040000000000	947430359040000000000	186227097600000000000	16777216000000000000					43439888521963583647921
11	243569224216081305397	3423135043036818346120	22204119198076659542400	88016328352736308096000	237881968520908940800000	462905452256903884800000	667251102352293888000000	721352543083560960000000	584880440338022400000000	351279543746560000000000	151904667566080000000000	44787616972800000000000	8069840896000000000000	671088640000000000000				3344871416191195940889917
12	9012061295995008299689	136398765561005531022320	958477812050309136913600	4144768916974309781248000	12322285969383083133440000	26642780474341801369600000	43204508877311029248000000	53380088188183511040000000	50494678015849267200000000	36392560732143616000000000	19671654449807360000000000	7733328530636800000000000	2090088792064000000000000	347623915520000000000000	26843545600000000000000			257555099046722087448523609
13	333446267951815307088493	5407236777597004979813400	40919629668301659306696000	191695562410061827382720000	621715337546146467187200000	1478674316325969976012800000	2664278047434180136960000000	3703243618055231078400000000	4003506614113763328000000000	3366311867723284480000000000	2183553643928616960000000000	1072999333625856000000000000	386666426531840000000000000	96465636556800000000000000	14898167808000000000000000	1073741824000000000000000		19831742626597600733536317893
14	12337511914217166362274241	213405611489161796536635520	1730315768831041593540288000	8729520995904353985428480000	30671289985609892381235200000	79579563205906747799961600000	157725260408103464108032000000	243591135765410755379200000000	296259489444418486272000000000	284693803670312058880000000000	215443959534290206720000000000	127043121101301350400000000000	57226631126712320000000000000	19035885613875200000000000000	4409857671168000000000000000	635655159808000000000000000	42949672960000000000000000	1527044182248015256482296477761

Technical details

Node id: n-3de3dd57fdba145b · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-4f2c8cb4b6021dc1

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:               1369,                2960,                 1600;
  1:              50653,              164280,               177600,                64000;
  2:            1874161,             8104480,             13142400,              9472000,              2560000;
  3:           69343957,           374832200,            810448000,            876160000,            473600000,             102400000;
  4:         2565726409,         16642549680,          44979864000,          64835840000,          52569600000,           22732800000,            4096000000;
  5:        94931877133,        718403394520,        2329956955200,        4198120640000,        4538508800000,         2943897600000,         1060864000000,          163840000000;
  6:      3512479453921,      30378200682560,      114944543123200,      248528741888000,      335849651200000,       290464563200000,       157007872000000,        48496640000000,        6553600000000;
  7:    129961739795077,    1264492603411560,     5468076122860800,    13793345174784000,    22367586769920000,     24181174886400000,     17427873792000000,      8074690560000000,     2182348800000000,      262144000000000;
  8:   4808584372417849,   51984695918030800,   252898520682312000,   729076816381440000,  1379334517478400000,   1789406941593600000,   1612078325760000000,    995878502400000000,   403734528000000000,    96993280000000000,   10485760000000000;
  9: 177917621779460413, 2115777123863853560, 11436633101966776000, 37091783033405760000, 80198449801958400000, 121381437538099200000, 131223175716864000000, 101330637619200000000, 54773317632000000000, 19738132480000000000, 4267704320000000000, 419430400000000000;

%F Formula

The row polynomials satisfy P(n,x) = (37+40*x)*P(n-1,x) for n >= 1, with P(0,x) = 1369 + 2960*x + 1600*x^2.

%t Mathematica

P[0, x_] := 1369 + 2960*x + 1600*x^2;
P[n_, x_] := P[n, x] = Expand[(37 + 40*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] = 1369 + 2960*x + 1600*x^2; for(n = 1, N, my(i = n + 1); v[i] = (37 + 40*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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
    
    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
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.