A272867 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 36\,x - 36\,x^{2} - 6\,x^{3} + x^{4} + 108\,n x + 48\,n x^{2} - 6\,n x^{3} - 4\,n x^{4} + 36\,n^{2} x^{2} + 24\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n}(x) = \left(216 + 648\,x + 756\,x^{2} + 432\,x^{3} + 126\,x^{4} + 18\,x^{5} + x^{6} + 648\,n x + 1368\,n x^{2} + 1080\,n x^{3} + 396\,n x^{4} + 66\,n x^{5} + 4\,n x^{6} + 216\,n^{2} x^{2} + 360\,n^{2} x^{3} + 204\,n^{2} x^{4} + 48\,n^{2} x^{5} + 4\,n^{2} x^{6}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	6	24	9																											39
2	36	288	432	192	25																									973
3	216	2592	6804	6912	3150	648	49																							20371
4	1296	20736	77760	124416	102600	46656	11760	1536	81																					386841
5	7776	155520	758160	1658880	1998000	1446336	652680	184320	31590	3000	121																			6896383
6	46656	1119744	6718464	18662400	29646000	29673216	19643904	8792064	2668140	540000	69696	5184	169																	117585637
7	279936	7838208	55847232	188116992	371498400	476171136	418300848	259448832	115246152	36741600	8324316	1306368	134862	8232	225															1939263339
8	1679616	53747712	443418624	1755758592	4147718400	6490036224	7126672896	5681332224	3353143392	1479513600	488847744	120185856	21634704	2765952	237600	12288	289													31166705713
9	10077696	362797056	3401222400	15479341056	42578265600	78726961152	103891527936	101475680256	75042303552	42592262400	18683371872	6342230016	1658887776	330728832	49280400	5308416	390150	17496	361											490630654423
10	60466176	2418647040	25395793920	130606940160	410036256000	875066499072	1349739417600	1562983464960	1392573824640	969978240000	533420287488	232794777600	80708153760	22160355840	4782240000	800243712	101595060	9447840	606480	24000	441									7593637281789
11	362797056	15963070464	185570694144	1064204697600	3755201472000	9086977861632	16048483976448	21511124287488	22455550529280	18577392768000	12322828349568	6602422689792	2868537177792	1011435828480	288780228000	66392358912	12172492728	1752889248	193705380	15840000	902286	31944	529							115875364648771
12	2176782336	104485552128	1332190789632	8428501204992	33023602022400	89361268457472	177861417947136	270989084196864	324612529807104	311412899635200	242353552877568	154357086633984	80914505907456	35016190949376	12518262028800	3690819551232	893661609456	176425185024	28085707584	3546892800	346827096	25299648	1294992	41472	625					1747080667202377
13	13060694016	679156088832	9423290732544	65198984527872	280859340902400	840455659929600	1861678460049408	3178651727167488	4299514350303744	4696435099699200	4199893982279424	3105101638139904	1910911992417792	983244344334336	424018933228800	153338352500736	46440517327776	11741087749248	2464253226864	425809612800	59853345552	6727406400	589716504	38817792	1803750	52728	729			26070557252056239
14	78364164096	4388393189376	65825897840640	494425632669696	2322572332953600	7616056380162048	18584276291223552	35155533935149056	53012441129776128	64993508368588800	65718214030909440	55382801997938688	39196530284550912	23423088152862720	11860573447987200	5098903543480320	1861899614108352	576904187478528	151287338442240	33432874675200	6186198786768	949730386176	119452499712	12091742208	960277500	57578976	2449440	65856	841	385560010691938069

Technical details

Node id: n-cdbfc3bc09489b9a · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-19a74c9c27488ed0

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        6,        24,          9;
  2:       36,       288,        432,         192,          25;
  3:      216,      2592,       6804,        6912,        3150,         648,           49;
  4:     1296,     20736,      77760,      124416,      102600,       46656,        11760,         1536,          81;
  5:     7776,    155520,     758160,     1658880,     1998000,     1446336,       652680,       184320,       31590,        3000,         121;
  6:    46656,   1119744,    6718464,    18662400,    29646000,    29673216,     19643904,      8792064,     2668140,      540000,       69696,       5184,        169;
  7:   279936,   7838208,   55847232,   188116992,   371498400,   476171136,    418300848,    259448832,   115246152,    36741600,     8324316,    1306368,     134862,      8232,      225;
  8:  1679616,  53747712,  443418624,  1755758592,  4147718400,  6490036224,   7126672896,   5681332224,  3353143392,  1479513600,   488847744,  120185856,   21634704,   2765952,   237600,   12288,    289;
  9: 10077696, 362797056, 3401222400, 15479341056, 42578265600, 78726961152, 103891527936, 101475680256, 75042303552, 42592262400, 18683371872, 6342230016, 1658887776, 330728832, 49280400, 5308416, 390150, 17496, 361;

%F Formula

The row polynomials satisfy (36+(108*n-36)*x+(36*n^2+48*n-36)*x^2+(24*n^2-6*n-6)*x^3+(4*n^2-4*n+1)*x^4)*P(n,x) = (216+(648*n+648)*x+(216*n^2+1368*n+756)*x^2+(360*n^2+1080*n+432)*x^3+(204*n^2+396*n+126)*x^4+(48*n^2+66*n+18)*x^5+(4*n^2+4*n+1)*x^6)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((216 + (648*n + 648)*x + (216*n^2 + 1368*n + 756)*x^2 + (360*n^2 + 1080*n + 432)*x^3 + (204*n^2 + 396*n + 126)*x^4 + (48*n^2 + 66*n + 18)*x^5 + (4*n^2 + 4*n + 1)*x^6)*P[n - 1, x])/(36 + (108*n - 36)*x + (36*n^2 + 48*n - 36)*x^2 + (24*n^2 - 6*n - 6)*x^3 + (4*n^2 - 4*n + 1)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((216 + (648*n + 648)*x + (216*n^2 + 1368*n + 756)*x^2 + (360*n^2 + 1080*n + 432)*x^3 + (204*n^2 + 396*n + 126)*x^4 + (48*n^2 + 66*n + 18)*x^5 + (4*n^2 + 4*n + 1)*x^6)*v[i - 1])/(36 + (108*n - 36)*x + (36*n^2 + 48*n - 36)*x^2 + (24*n^2 - 6*n - 6)*x^3 + (4*n^2 - 4*n + 1)*x^4)); 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A007318 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Parent-row provers of polytool (above)

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.