A200536 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(28 + 38\,x + 8\,x^{2}\right)\,P_{n-1}(x) + \left(38\,x + 16\,x^{2}\right)\,P_{n-1}'(x) + \left(8\,x^{2}\right)\,P_{n-1}''(x)\)

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

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	28	38	8																											74
2	784	3572	3236	864	64																									8520
3	21952	265544	587480	419256	117856	13440	512																							1426040
4	614656	18360080	85012592	122906592	73284112	20455680	2782720	176128	4096																					323596656
5	17210368	1235122208	11197872608	29450014560	31476482560	16190479200	4380272256	647490560	51875840	2088960	32768																			94648941888
6	481890304	82172059712	1410578995136	6389234826624	11309793419584	9566202475200	4334903445056	1124399368704	172628917248	15767470080	832159744	23199744	262144																	34407018489280
7	13492928512	5441667772544	173710625763200	1313352523195776	3682261810855936	4820737343036160	3369402348178048	1365778283722624	337989688319488	52567621908480	5189387083776	321670250496	12025724928	245891072	2097152															15126778736728192
8	377801998336	359662804271360	21139233094729472	261740256552801792	1130642261797180672	2208336240560540160	2269528665661426176	1350506132145400832	495453918233018624	116761484856238080	18124853324398592	1874574370013184	128872146665472	5774505934848	160922861568	2516582400	16777216													7874602471310838784
9	10578455953408	23752101557846528	2556132785214176768	51213272364097574400	334736040184320655360	952170531687619023360	1395315032648588567552	1172602394374717953024	607186205368856060928	203505764157845521920	45659056139621091328	7008498451718307840	744620847642836992	54858064197844992	2775926122217472	94090611916800	2029875363840	25065160704	134217728											4772779041826262290944
10	296196766695424	1568040684144100352	308018632337472551936	9909641060955994281984	96811937576466605691904	394473243255633478272000	807110267254313539505152	933329087084484130643968	660658119605104129017856	302690182139549713474560	93398954045171324494848	19961808175886216454144	3011807223154655576064	324429556201784606720	25048317636202987520	1380495571550208000	53552960520585216	1420943960309760	24433330749440	244276264960	1073741824									3322015547757930870213632
11	8293509467471872	103501940630645049344	37046912447014621214720	1904405650170735145015296	27574788051405698459631616	159064980711916696521646080	447649030762915998876186624	699305886621348084425566208	661769578492688348546564096	403480792855137187409049600	165728898269192134914701312	47383871999285442975971328	9657201705659190504284160	1426650441209819268022272	154401536965461480701952	12302864582290833408000	721045677071197536256	30838225252336533504	946125637028937728	20184364566446080	283119412314112	2338609692672	8589934592							2625150693235656035835879424
12	232218265089212416	6831443234982337187840	4451218664783885118205952	364431185419255746198331392	7774716763402232114299359232	62954426335549258409224765440	241024185286823751219819855872	501820198539375370609688887296	624657346463602452521164419072	497141027163716829823095091200	265669858175225203080602619904	98873182471193637598521237504	26338070420715775252773081088	5125398649237277464192548864	739487120705259409062887424	79892331733087980591513600	6498401400194524435709952	398299287686417863409664	18317972257805348896768	625505454568757002240	15555838888294481920	272621086313545728	3178765425311744	22058952032256	68719476736					2332573594617036882545889071104
13	6502111422497947648	450884077802907644469248	534515139566501381106556928	69553565187739343594654883840	2177198518394390328800625295360	24592709714188001400315082629120	127038815943966610954233565708288	348947601318925260214833657806848	564433953504810808567720138817536	578003092798149131096038760939520	395152525737159311722892377202688	187685600092807088304183240400896	63844848362139533959998910660608	15920697265684158520546790039552	2962358666419899426884332716032	416743546506650604675484876800	44735879166337738334618517504	3684821635428184939548377088	233344782719010635656462336	11338565272162573557432320	420014394789348962729984	11714225825867982962688	240902167727043510272	3529879018939613184	34754119444987904	205471235440640	549755813888			2311294899816491575579015616798720
14	182059119829942534144	29758596215225959456980992	64166164540198414125571751936	13252597052512752802923042668544	606892673804113846309161753985024	9518674949360794257605522253004800	65930610042628778321167433343909888	237056931845127943814415022884257792	493616934459142952602680625474371584	643278542879399074320812880154951680	555444083548817331951443417086164992	331735330561404935861247977545236480	141647005103452873689276674880749568	44365434084295510098601033745154048	10399365385757936512333316390535168	1852845347669636995458824041267200	253898482409367322698144388218880	26988038507926826985494597861376	2237551253705512846611366019072	145075210788581678161602805760	7352362688085313071223930880	290109424818970893798604800	8839749502024735736725504	205138553164241917968384	3546491287253186248704	44092483355677818880	371446638822031360	1893359023030272	4398046511104	2535749244472740093657711086955298816

Technical details

Node id: n-269219ae86b544b7 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-d1f860823afc60e5

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:             28,                38,                   8;
  2:            784,              3572,                3236,                  864,                    64;
  3:          21952,            265544,              587480,               419256,                117856,                 13440,                    512;
  4:         614656,          18360080,            85012592,            122906592,              73284112,              20455680,                2782720,                 176128,                  4096;
  5:       17210368,        1235122208,         11197872608,          29450014560,           31476482560,           16190479200,             4380272256,              647490560,              51875840,               2088960,                32768;
  6:      481890304,       82172059712,       1410578995136,        6389234826624,        11309793419584,         9566202475200,          4334903445056,          1124399368704,          172628917248,           15767470080,            832159744,            23199744,             262144;
  7:    13492928512,     5441667772544,     173710625763200,     1313352523195776,      3682261810855936,      4820737343036160,       3369402348178048,       1365778283722624,       337989688319488,        52567621908480,        5189387083776,        321670250496,        12025724928,         245891072,          2097152;
  8:   377801998336,   359662804271360,   21139233094729472,   261740256552801792,   1130642261797180672,   2208336240560540160,    2269528665661426176,    1350506132145400832,    495453918233018624,    116761484856238080,    18124853324398592,    1874574370013184,    128872146665472,     5774505934848,     160922861568,     2516582400,      16777216;
  9: 10578455953408, 23752101557846528, 2556132785214176768, 51213272364097574400, 334736040184320655360, 952170531687619023360, 1395315032648588567552, 1172602394374717953024, 607186205368856060928, 203505764157845521920, 45659056139621091328, 7008498451718307840, 744620847642836992, 54858064197844992, 2775926122217472, 94090611916800, 2029875363840, 25065160704, 134217728;

%F Formula

The row polynomials satisfy P(n,x) = (28+38*x+8*x^2)*P(n-1,x) + (38*x+16*x^2)*d/dx P(n-1,x) + 8*x^2*d^2/dx^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[(28 + 38*x + 8*x^2)*P[n - 1, x] + (38*x + 16*x^2)*D[P[n - 1, x], x] + 8*x^2*D[P[n - 1, x], {x, 2}]];
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] = (28 + 38*x + 8*x^2)*v[i - 1] + (38*x + 16*x^2)*deriv(v[i - 1], x) + 8*x^2*deriv(deriv(v[i - 1], x), 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)

polytool c48629374597

Properties

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

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from basis_map_A013609 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
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.