A038334 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(44 + 48\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1936 + 4224\,x + 2304\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1936	4224	2304															8464
1	85184	278784	304128	110592														778688
2	3748096	16355328	26763264	19464192	5308416													71639296
3	164916224	899543040	1962639360	2141061120	1167851520	254803968												6590815232
4	7256313856	47495872512	129534197760	188413378560	154156400640	67268247552	12230590464											606355001344
5	319277809664	2438121455616	7979306582016	14507830149120	15826723799040	10359310123008	3767021862912	587068342272										55784660123648
6	14048223625216	122602678910976	468119319478272	1021351242498048	1392751694315520	1215492387766272	662995847872512	206648056479744	28179280429056									5132188731375616
7	618121839509504	6068832606093312	26482178644770816	67409182004871168	110305934189789184	120333746388860928	87515451919171584	40916315182989312	11158995049906176	1352605460594688								472161363286556672
8	27197360938418176	296698482964561920	1456519825462394880	4237148583163330560	8089101840584540160	10589369682219761664	9626699711108874240	6001059560171765760	2454978910979358720	595146402661662720	64925062108545024							43438845422363213824
9	1196683881290399744	14360206575484796928	78328399502644346880	256347489281381498880	559303612977559633920	854209154365727440896	931864532035339026432	726128206780783656960	396069930971336540160	144025429444122378240	31423730060535791616	3116402981210161152						3996373778857415671808
10	52654090776777588736	689289915623270252544	4135739493739621515264	15039052704507714600960	36914038456518935838720	64431776215014869827584	82004078819109834326016	76679138636050754174976	52281230888216423301120	25348475582165538570240	8295864735981448986624	1645460774078965088256	149587343098087735296					367666387654882241806336
11	2316779994178213904384	32856152644709215371264	215058453674460318793728	860233814697841275174912	2346092221903203477749760	4606871999373563192672256	6700904726361546462068736	7310077883303505231347712	5980972813611958825648128	3624832008249672015544320	1581744876327129606782976	470601781386584015241216	85563960252106184589312	7180192468708211294208				33825307664249166246182912
12	101938319743841411792896	1556876156087759743746048	11039667288622296364744704	48173093623079111409795072	144519280869237334229385216	315314794623790547409567744	515969663929839077579292672	643286853730708460358598656	614046542197494439433207808	446579303416359592315060224	243588710954377959444578304	96630232444711917796196352	26353699757648704853508096	4422998560724258157232128	344649238497994142121984			3111928305110923294648827904
13	4485286068729022118887424	73395590215565816490885120	560475416191593507748577280	2649520149269351127538728960	8671156852154240053763112960	20810776445170176129031471104	37837775354854865689148129280	53071165432783447979584389120	57895816835763761432273879040	49123723375799555154656624640	32153709845977890646684336128	15943988353377466436372398080	5797813946682715067771781120	1459589525039005191886602240	227468497408676133800509440	16543163447903718821855232		286297404070204943107692167168
14	197352587024076973231046656	3444699700783888987305541632	28183906642777273532499886080	143481706545047937983635783680	508707868659715416487435960320	1331889692490891272258014150656	2663779384981782544516028301312	4151344496075505264180823326720	5094831881547211006040101355520	4940443036651840975554037678080	3772701955261405835877628772352	2244913560155547274241233649664	1020415254616157851927833477120	342517008542486551696055992320	80068911087853999097779322880	11646387067324218050586083328	794071845499378503449051136	26339361174458854765907679379456

Technical details

Node id: n-49a9936529864dac · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-60cc3cc9b4461a16

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:                1936,                 4224,                 2304;
  1:               85184,               278784,               304128,                110592;
  2:             3748096,             16355328,             26763264,              19464192,               5308416;
  3:           164916224,            899543040,           1962639360,            2141061120,            1167851520,             254803968;
  4:          7256313856,          47495872512,         129534197760,          188413378560,          154156400640,           67268247552,           12230590464;
  5:        319277809664,        2438121455616,        7979306582016,        14507830149120,        15826723799040,        10359310123008,         3767021862912,          587068342272;
  6:      14048223625216,      122602678910976,      468119319478272,      1021351242498048,      1392751694315520,      1215492387766272,       662995847872512,       206648056479744,        28179280429056;
  7:     618121839509504,     6068832606093312,    26482178644770816,     67409182004871168,    110305934189789184,    120333746388860928,     87515451919171584,     40916315182989312,     11158995049906176,      1352605460594688;
  8:   27197360938418176,   296698482964561920,  1456519825462394880,   4237148583163330560,   8089101840584540160,  10589369682219761664,   9626699711108874240,   6001059560171765760,   2454978910979358720,    595146402661662720,    64925062108545024;
  9: 1196683881290399744, 14360206575484796928, 78328399502644346880, 256347489281381498880, 559303612977559633920, 854209154365727440896, 931864532035339026432, 726128206780783656960, 396069930971336540160, 144025429444122378240, 31423730060535791616, 3116402981210161152;

%F Formula

The row polynomials satisfy P(n,x) = (44+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 1936 + 4224*x + 2304*x^2.

%t Mathematica

P[0, x_] := 1936 + 4224*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[(44 + 48*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] = 1936 + 4224*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (44 + 48*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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^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
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? 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
[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.