A038328 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(22 + 8\,x\right)\,P_{n-1}(x) + \left(8\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 484 + 416\,x + 64\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	484	416	64															964
1	10648	16352	5760	512														33272
2	234256	575744	349696	69632	4096													1233424
3	5153632	19146368	17894400	6000640	778240	32768												49006048
4	113379904	615620096	833158144	419184640	90030080	8257536	262144											2079892544
5	2494357888	19375642112	36584970240	25947758592	8215101440	1232207872	84410368	2097152										93936545664
6	54875873536	601224126464	1545234006016	1486276657152	651197546496	142117699584	15766388736	838860800	16777216									4497547936000
7	1207269217792	18475730782208	63528685240320	80730598277120	47054880768000	14020877746176	2240588808192	191562252288	8153726976	134217728								227458481036800
8	26559922791424	563930077208576	2561895885389824	4221837002670080	3186808347688960	1245733466406912	269008238542848	32866566144000	2233718538240	77846282240	1073741824							12110952145404928
9	584318301411328	17130381698588672	101863484262481920	214699669205942272	205862346796564480	102729941698740224	28796444429254656	4715658067574784	455032323440640	25187298836480	732291923968	8589934592						676863204964694016
10	12855002631049216	518585997368950784	4007855455563022336	10691092657573199872	12834164080662020096	8016155159694409728	2837590643637747712	598192884704870400	76858044356493312	6007864678154240	276192166936576	6803228196864	68719476736					39599640854984527872
11	282810057883082752	15660419942116917248	156447195290346455040	523853105892871372800	778573601616334684160	599674932546349563904	262760586332197617664	69359770156081872896	11395334892297388032	1179603634598445056	76234518452764672	2957892437147648	62534723829760	549755813888				2419266615856446955520
12	6221821273427820544	472075078726572179456	6070276780570100629504	25348820433394854789120	46233799334425043927040	43408434630804350435328	23190640503624629747712	7512146762831967027200	1534876961986230353920	202045420790632939520	17212749958969556992	935244315708358656	31042236909092864	569547023187968	4398046511104			153997517338884466556928
13	136880068015412051968	14212026931984587948032	234447118291476401356800	1214607954180724125335552	2699415727526111210373120	3061193341785270078406656	1970612312300158885822464	771472571529890466103296	192096592833471546654720	31271285250209339146240	3372063862139958329344	240578874399675908096	11144938480939827200	320100820194426880	5145714417991680	35184372088832		10193089923654051611770880
14	3011361496339065143296	427455848503660934856704	9022686710531979955142656	57747542838645120976289792	155485312919855798362832896	211389313010895634544197632	162432408595293282634825728	75939759077732727442636800	22692087555917676741132288	4476273556187450252984320	594120795939950462763008	53440187081084016525312	3239733735948306874368	129492211192016404480	3250332293566300160	46161896180547584	281474976710656	700266784980803510436167680

Technical details

Node id: n-2003ecbc129eb9c3 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-ee32e1d26c1bc310

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:             484,               416,                 64;
  1:           10648,             16352,               5760,                512;
  2:          234256,            575744,             349696,              69632,               4096;
  3:         5153632,          19146368,           17894400,            6000640,             778240,              32768;
  4:       113379904,         615620096,          833158144,          419184640,           90030080,            8257536,            262144;
  5:      2494357888,       19375642112,        36584970240,        25947758592,         8215101440,         1232207872,          84410368,          2097152;
  6:     54875873536,      601224126464,      1545234006016,      1486276657152,       651197546496,       142117699584,       15766388736,        838860800,        16777216;
  7:   1207269217792,    18475730782208,     63528685240320,     80730598277120,     47054880768000,     14020877746176,     2240588808192,     191562252288,      8153726976,      134217728;
  8:  26559922791424,   563930077208576,   2561895885389824,   4221837002670080,   3186808347688960,   1245733466406912,   269008238542848,   32866566144000,   2233718538240,    77846282240,   1073741824;
  9: 584318301411328, 17130381698588672, 101863484262481920, 214699669205942272, 205862346796564480, 102729941698740224, 28796444429254656, 4715658067574784, 455032323440640, 25187298836480, 732291923968, 8589934592;

%F Formula

The row polynomials satisfy P(n,x) = (22+8*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 484 + 416*x + 64*x^2.

%t Mathematica

P[0, x_] := 484 + 416*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 8*x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 484 + 416*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 8*x)*v[i - 1] + 8*x*deriv(v[i - 1], 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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    
    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
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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

Related ?

Same fingerprint ?

A038255 → basis A013611 → basis A008277

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
[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.