A038333 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(33 + 42\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1089 + 2772\,x + 1764\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1089	2772	1764															5625
1	35937	137214	174636	74088														421875
2	1185921	6037416	11525976	9779616	3111696													31640625
3	39135393	249043410	633928680	806818320	513429840	130691232												2373046875
4	1291467969	9862119036	31379469660	53250009120	50829554160	25876863936	5489031744											177978515625
5	42618442977	379691582886	1449731498292	3075188026680	3913875670320	2988777784608	1267966332864	230539333248										13348388671875
6	1406408618241	14319796840272	63788185924848	162369927808704	258315794241120	263012445045504	167371555938048	60862383977472	9682651996416									1001129150390625
7	46411484401953	531622457695098	2706441602811408	8037311426530848	15343958177922528	19528674044628672	16569784037866752	9038064020654592	2875747642935552	406671383849472								75084686279296875
8	1531578985264449	19492823448820260	111640716115970580	378901824393597120	843917699785739040	1288892486945492352	1367007183124007040	994187042272005120	474498361084366080	134201556670325760	17080198121677824							5631351470947265625
9	50542106513726817	707589491192175438	4502842216677480060	17192670281859469320	43763160717460467360	77977995460202287296	99244721494802911104	90222474086184464640	57414301691208295680	24357582535664125440	6200111918169050112	717368321110468608						422351360321044921875
10	1667889514952984961	25473221682918315768	178312551780428210376	756477492401816650080	2166276455514293134320	4411326600320015109888	6550151618656992132864	7145619947625809599488	5684015867429621272320	3215200894707664558080	1227622159797471922176	284077855159745568768	30129469486639681536					31676352024078369140625
11	55040353993448503713	910667675164329788706	6954189519436700204664	32452884424037934288432	103259177712847972735920	236557388942160810267744	401430720629121374999808	510911826255245386363392	487688561425461505165056	344830295957397023854080	175549968851038484871168	60934699931765424500736	12925542409768423378944	1265437718438866624512				2375726401805877685546875
12	1816331681783800622529	32363728148147720183244	267736296498312957879564	1363021145809593240114144	4770574010333576340399504	12143279299030921593744192	23182624116331759406238912	33720180532846195499983872	37552019229760535897709312	31862319346463485004116992	20276021402294945002619904	9383943789491875373113344	2985800296656505800536064	584632225918756380524544	53148384174432398229504			178179480135440826416015625
13	59938945498865420543457	1144288959523794392193270	10194574366666531857721860	56224622264645721154708440	214675830465010935317977680	601092325302030618890337504	1275044326398246767343140160	2086436170469858346561502080	2655464216961637895623729920	2628641346083237512839651840	2007326118827199555259370496	1161263043949619577422776320	492657048948323457088450560	144696475914892204179824640	26308450166344037123604480	2232232135326160725639168		13363461010158061981201171875
14	1977985201462558877934081	40278971375237562605203104	384481090399994915776938720	2283584658133303136129696640	9445736540460481153991017920	28852431614497469706736200192	67322340433827429315717800448	122404255334231689664941455360	175260638319468101111166174720	198274661533135629539905167360	176644698456793560862824603648	122629377441079827375845179392	65030730461178696335675473920	25466579761021027935649136640	6945430843914825800631582720	1178618567452212863137480704	93753749683698750476845056	1002259575761854648590087890625

Technical details

Node id: n-cad214743f90c94e · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-64aa42accc9ceb8f

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:              1089,               2772,                1764;
  1:             35937,             137214,              174636,                74088;
  2:           1185921,            6037416,            11525976,              9779616,              3111696;
  3:          39135393,          249043410,           633928680,            806818320,            513429840,            130691232;
  4:        1291467969,         9862119036,         31379469660,          53250009120,          50829554160,          25876863936,           5489031744;
  5:       42618442977,       379691582886,       1449731498292,        3075188026680,        3913875670320,        2988777784608,        1267966332864,         230539333248;
  6:     1406408618241,     14319796840272,      63788185924848,      162369927808704,      258315794241120,      263012445045504,      167371555938048,       60862383977472,        9682651996416;
  7:    46411484401953,    531622457695098,    2706441602811408,     8037311426530848,    15343958177922528,    19528674044628672,    16569784037866752,     9038064020654592,     2875747642935552,      406671383849472;
  8:  1531578985264449,  19492823448820260,  111640716115970580,   378901824393597120,   843917699785739040,  1288892486945492352,  1367007183124007040,   994187042272005120,   474498361084366080,   134201556670325760,   17080198121677824;
  9: 50542106513726817, 707589491192175438, 4502842216677480060, 17192670281859469320, 43763160717460467360, 77977995460202287296, 99244721494802911104, 90222474086184464640, 57414301691208295680, 24357582535664125440, 6200111918169050112, 717368321110468608;

%F Formula

The row polynomials satisfy P(n,x) = (33+42*x)*P(n-1,x) for n >= 1, with P(0,x) = 1089 + 2772*x + 1764*x^2.

%t Mathematica

P[0, x_] := 1089 + 2772*x + 1764*x^2;
P[n_, x_] := P[n, x] = Expand[(33 + 42*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] = 1089 + 2772*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = (33 + 42*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_A013609 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

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.