A038333 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(40 + 42\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1600 + 3360\,x + 1764\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1600	3360	1764															6724
1	64000	201600	211680	74088														551368
2	2560000	10752000	16934400	11854080	3111696													45212176
3	102400000	537600000	1128960000	1185408000	622339200	130691232												3707398432
4	4096000000	25804800000	67737600000	94832640000	74680704000	31365895680	5489031744											304006671424
5	163840000000	1204224000000	3793305600000	6638284800000	6970199040000	4391225395200	1536928888320	230539333248										24928547056768
6	6553600000000	55050240000000	202309632000000	424850227200000	557615923200000	468397375488000	245908622131200	73772586639360	9682651996416									2044140858654976
7	262144000000000	2477260800000000	10404495360000000	25491013632000000	40148346470400000	42155763793920000	29509034655744000	13279065595084800	3485754718709760	406671383849472								167619550409708032
8	10485760000000000	110100480000000000	520224768000000000	1456629350400000000	2676556431360000000	3372461103513600000	2950903465574400000	1770542079344640000	697150943741952000	162668553539788800	17080198121677824							13744803133596058624
9	419430400000000000	4844421120000000000	25433210880000000000	80114614272000000000	168240689971200000000	247313814257664000000	259679504970547200000	194759628727910400000	102248805082152960000	35787081778753536000	7515287173538242560	717368321110468608						1127073856954876807168
10	16777216000000000000	211392921600000000000	1220794122240000000000	4272779427840000000000	10094441398272000000000	16958661549096960000000	20774360397643776000000	18696924357879398400000	12269856609858355200000	5725933084600565760000	1803668921649178214400	344336794133024931840	30129469486639681536					92420056270299898187776
11	671088640000000000000	9160359936000000000000	57710267596800000000000	222184530247680000000000	583234391900160000000000	1102313000691302400000000	1543238200967823360000000	1620400111016214528000000	1276065087425268940800000	744371300998073548800000	312635946419190890496000	89527566474586482278400	15667324133052634398720	1265437718438866624512				7578444614164591651397632
12	26843545600000000000000	394600120320000000000000	2693145821184000000000000	11311212448972800000000000	32661125946408960000000000	68588364487458816000000000	108026674067747635200000000	129632008881297162240000000	119099408159691767808000000	83369585711784237465600000	43769032498686724669440000	16711812408589476691968000	4386850757254737631641600	708645122325765309726720	53148384174432398229504			621432458361496515414605824
13	1073741824000000000000000	16911433728000000000000000	124299037900800000000000000	565560622448640000000000000	1781515960713216000000000000	4115301869247528960000000000	7201778271183175680000000000	9722400666097287168000000000	10208520699402151526400000000	8336958571178423746560000000	5252283899842406960332800000	2506771861288421503795200000	877370151450947526328320000	212593536697729592918016000	31889030504659438937702400	2232232135326160725639168		50957461585642714263997677568
14	42949672960000000000000000	721554505728000000000000000	5682241732608000000000000000	27842984489779200000000000000	95014184571371520000000000000	239435745119856230400000000000	460913809355723243520000000000	691370714033584865280000000000	816681655952172122112000000000	762236212222027313971200000000	560243615983190075768832000000	320866798244917952485785600000	140379224232151604212531200000	45353287828848979822510080000	10204489761491020460064768000	1428628566608742864409067520	93753749683698750476845056	4178511850022702569647809560576

Technical details

Node id: n-509590a43fbc65c5 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-640a8dc8403e6c62

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:               1600,                3360,                 1764;
  1:              64000,              201600,               211680,                74088;
  2:            2560000,            10752000,             16934400,             11854080,               3111696;
  3:          102400000,           537600000,           1128960000,           1185408000,             622339200,             130691232;
  4:         4096000000,         25804800000,          67737600000,          94832640000,           74680704000,           31365895680,            5489031744;
  5:       163840000000,       1204224000000,        3793305600000,        6638284800000,         6970199040000,         4391225395200,         1536928888320,          230539333248;
  6:      6553600000000,      55050240000000,      202309632000000,      424850227200000,       557615923200000,       468397375488000,       245908622131200,        73772586639360,         9682651996416;
  7:    262144000000000,    2477260800000000,    10404495360000000,    25491013632000000,     40148346470400000,     42155763793920000,     29509034655744000,     13279065595084800,      3485754718709760,      406671383849472;
  8:  10485760000000000,  110100480000000000,   520224768000000000,  1456629350400000000,   2676556431360000000,   3372461103513600000,   2950903465574400000,   1770542079344640000,    697150943741952000,   162668553539788800,   17080198121677824;
  9: 419430400000000000, 4844421120000000000, 25433210880000000000, 80114614272000000000, 168240689971200000000, 247313814257664000000, 259679504970547200000, 194759628727910400000, 102248805082152960000, 35787081778753536000, 7515287173538242560, 717368321110468608;

%F Formula

The row polynomials satisfy P(n,x) = (40+42*x)*P(n-1,x) for n >= 1, with P(0,x) = 1600 + 3360*x + 1764*x^2.

%t Mathematica

P[0, x_] := 1600 + 3360*x + 1764*x^2;
P[n_, x_] := P[n, x] = Expand[(40 + 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] = 1600 + 3360*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = (40 + 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_A013610 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.