A038333 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(19 + 56\,x + 28\,n x\right)\,P_{n}(x) = \left(361 + 2128\,x + 2352\,x^{2} + 532\,n x + 784\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 361 + 2128\,x + 2352\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	361	2128	2352															4841
1	6859	60648	134064	87808														289379
2	130321	1536416	5094432	6673408	3073280													16507857
3	2476099	36489880	161323680	316986880	291961600	103262208												912500347
4	47045881	831969264	4597724880	12045501440	16641811200	11771891712	3373232128											49309176505
5	893871739	18441985352	122299481808	400512922880	737786963200	782830798848	448639873024	107943428096										2619349324947
6	16983563041	400454539072	3098253539136	12175592855552	28035904601600	39663427141632	34096630349824	16407401070592	3400217985024									137294865645473
7	322687697779	8559715772664	75685907884608	347004396383232	958827937374720	1695611510304768	1943507929939968	1402832791535616	581437275439104	105784559534080								7119574711866539
8	6131066257801	180705110756240	1797540312259440	9418690758973440	30362884683532800	64433237391581184	92316626672148480	88846076797255680	55236541166714880	20099066311475200	3258164433649664							365955664704604809
9	116490258898219	3776736814805416	41742880584691440	246063296078181120	906548985551193600	2244424435806744576	3858834994895806464	4642207512656609280	3848145701281136640	2100352429549158400	680956366632779776	99522113609662464						18672691943719667395
10	2213314919066161	78281453979603168	951737677330964832	6233603500647255040	25836646088209017600	73104110194848251904	146635729806040645632	211684662577141383168	219344304973024788480	159626784645736038400	77629025796136894464	22691041903003041792	3018837446159761408					946836979377176712049
11	42052983462257059	1611293261080165208	21370836936431664864	153970006465987199488	709072398198625260800	2257089402265939777536	5174146466013148495872	8714351942758986940416	10835608665667424550912	9856953951874200371200	6391456457215270977536	2802343675020875661312	745652849201461067776	91029559914971267072				47754699557777865657051
12	799006685782884121	32969539034409534256	473720218757568571152	3723274701813872278528	18861325792083431937280	66709531222526664536064	172040369994937187487744	331145373824841503735808	480378650844589155090432	524389950239707459747840	425031854404815520006144	248474472518517641969664	99171828943794322014208	24213862937382357041152	2730886797449138012160			2397378870986936014846553
13	15181127029874798299	671165616057622661640	10385404795839003290640	88427774168079466615040	488679804613070736556800	1901221639842009939277824	5447945049839677603778560	11797053942509978570588160	19558273641529701314396160	24908522636386104338022400	24226815701074484640350208	17703806166944381990338560	9421323749660460591349760	3450475468576985878364160	778302737273004333465600	81562485683814255296512		119863482549640680159150323
14	288441413567621167681	13602289818767819275904	225511646995361214311040	2067849488238165988536320	12379888383531125326105600	52542852591997365594587136	165617529515126199154868224	398478266502559276162088960	743214398378128649947054080	1081741554494482245536972800	1227491995521107221777743872	1076391414950218425012584448	716020604974195004942581760	349648180815801235674234880	118302016065496658686771200	24794995647879533610139648	2426483949093474095071232	5971357433656082278164094785

Technical details

Node id: n-519543e978a17fc6 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-e740b38b8ddb66d7

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:             361,             2128,              2352;
  1:            6859,            60648,            134064,              87808;
  2:          130321,          1536416,           5094432,            6673408,            3073280;
  3:         2476099,         36489880,         161323680,          316986880,          291961600,           103262208;
  4:        47045881,        831969264,        4597724880,        12045501440,        16641811200,         11771891712,          3373232128;
  5:       893871739,      18441985352,      122299481808,       400512922880,       737786963200,        782830798848,        448639873024,        107943428096;
  6:     16983563041,     400454539072,     3098253539136,     12175592855552,     28035904601600,      39663427141632,      34096630349824,      16407401070592,       3400217985024;
  7:    322687697779,    8559715772664,    75685907884608,    347004396383232,    958827937374720,    1695611510304768,    1943507929939968,    1402832791535616,     581437275439104,     105784559534080;
  8:   6131066257801,  180705110756240,  1797540312259440,   9418690758973440,  30362884683532800,   64433237391581184,   92316626672148480,   88846076797255680,   55236541166714880,   20099066311475200,   3258164433649664;
  9: 116490258898219, 3776736814805416, 41742880584691440, 246063296078181120, 906548985551193600, 2244424435806744576, 3858834994895806464, 4642207512656609280, 3848145701281136640, 2100352429549158400, 680956366632779776, 99522113609662464;

%F Formula

The row polynomials satisfy (19+(28*n+56)*x)*P(n,x) = (361+(532*n+2128)*x+(784*n+2352)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 2128*x + 2352*x^2.

%t Mathematica

P[0, x_] := 361 + 2128*x + 2352*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((361 + (532*n + 2128)*x + (784*n + 2352)*x^2)*P[n - 1, x])/(19 + (28*n + 56)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 361 + 2128*x + 2352*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((361 + (532*n + 2128)*x + (784*n + 2352)*x^2)*v[i - 1])/(19 + (28*n + 56)*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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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

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.