A038330 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(160 + 168\,x + 36\,x^{2} + 48\,n + 36\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 640 + 672\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	640	672	144															1456
1	13312	23040	12096	1728														50176
2	262144	638976	552960	193536	20736													1668352
3	4980736	15728640	19169280	11059200	2903040	248832												54089728
4	92274688	358612992	566231040	460062720	199065600	41803776	2985984											1721036800
5	1677721600	7751073792	15061745664	15854469120	9661317120	3344302080	585252864	35831808										53971714048
6	30064771072	161061273600	372051542016	481975861248	380507258880	185497288704	53508833280	8026324992	429981696									1673123135488
7	532575944704	3246995275776	8697308774400	13393855512576	13013348253696	8218956791808	3338951196672	825564856320	108355387392	5159780352								51381071773696
8	9345848836096	63909113364480	194819716546560	347892350976000	401815665377280	312320358088704	164379135836160	57239163371520	12383472844800	1444738498560	61917364224							1565611481104384
9	162727720910848	1233652046364672	4218001482055680	8572067528048640	11480447582208000	10607933565960192	6871047877951488	3099720847196160	944446195630080	181624268390400	19070548180992	743008370688						47391482671267840
10	2814749767106560	23432791811162112	88822947338256384	202464071138672640	308594431009751040	330636890367590400	254590405583044608	141347270632144896	55794975249530880	15111139130081280	2615389464821760	249650812551168	8916100448256					1426483628405161984
11	48413695994232832	439100963668623360	1827757761270644736	4618793261589331968	7896098774408232960	9628146247504232448	8596559149557350400	5673729038707851264	2756271777326825472	967112904325201920	235733770429267968	37090977864744960	3245460563165184	106993205379072				42728160776415084544
12	828662331436171264	8133500927031115776	36884480948164362240	102354434631156105216	193989316986751942656	265308918820116627456	269588094930118508544	206317419589376409600	119148309812864876544	51450406510100742144	16247496792663392256	3600297584737910784	519273690106429440	41941336508596224	1283918464548864			1274413838809597739008
13	14123288431433875456	149159219658510827520	732015083432800419840	2213068856889861734400	4605949558402024734720	6983615411523069935616	7959267564603498823680	6932265298203047362560	4642141940760969216000	2382966196257297530880	926107317181813358592	265868129334491873280	54004463771068661760	7189943401473638400	539245755110522880	15407021574586368		37868296924628047101952
14	239807672958224171008	2711671378835304087552	14319285087217039441920	46848965339699226869760	106227305130713363251200	176868463042637749813248	223475693168738237939712	218311338914838824878080	166374367156873136701440	99032361402900676608000	45752950968140112592896	16164782263537105895424	4253890069351869972480	797604388003475619840	98604938077352755200	6902345665414692864	184884258895036416	1121484178152446010327040

Technical details

Node id: n-8c5ad656fba85f85 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-87f28ed98a2aba0a

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:             640,              672,              144;
  1:           13312,            23040,            12096,             1728;
  2:          262144,           638976,           552960,           193536,             20736;
  3:         4980736,         15728640,         19169280,         11059200,           2903040,            248832;
  4:        92274688,        358612992,        566231040,        460062720,         199065600,          41803776,          2985984;
  5:      1677721600,       7751073792,      15061745664,      15854469120,        9661317120,        3344302080,        585252864,         35831808;
  6:     30064771072,     161061273600,     372051542016,     481975861248,      380507258880,      185497288704,      53508833280,       8026324992,       429981696;
  7:    532575944704,    3246995275776,    8697308774400,   13393855512576,    13013348253696,     8218956791808,    3338951196672,     825564856320,    108355387392,      5159780352;
  8:   9345848836096,   63909113364480,  194819716546560,  347892350976000,   401815665377280,   312320358088704,  164379135836160,   57239163371520,  12383472844800,   1444738498560,    61917364224;
  9: 162727720910848, 1233652046364672, 4218001482055680, 8572067528048640, 11480447582208000, 10607933565960192, 6871047877951488, 3099720847196160, 944446195630080, 181624268390400, 19070548180992, 743008370688;

%F Formula

The row polynomials satisfy (3*n+7+3*x)*P(n,x) = (48*n+160+(36*n+168)*x+36*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 640 + 672*x + 144*x^2.

%t Mathematica

P[0, x_] := 640 + 672*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((48*n + 160 + (36*n + 168)*x + 36*x^2)*P[n - 1, x])/(3*n + 7 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 640 + 672*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((48*n + 160 + (36*n + 168)*x + 36*x^2)*v[i - 1])/(3*n + 7 + 3*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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^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

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.