A128813 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 - 4\,n + 8\,n x + 9\,n^{2} x^{2} - 8\,n^{3} x - 9\,n^{3} x^{2} - 9\,n^{4} x^{2} + 9\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(-8\,n x^{2} + 4\,n^{2} x^{3} + 8\,n^{3} x^{2} - 4\,n^{3} x^{3} - 4\,n^{4} x^{3} + 4\,n^{5} x^{3}\right)\,P_{n-2}'(x) + \left(8\,n - 8\,n x + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(8\,n x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	1	16	27													44
3	1	40	243	288												572
4	1	80	1143	4608	4500											10332
5	1	140	3843	35088	112500	97200										248772
6	1	224	10458	178560	1263825	3499200	2778300									7730568
7	1	336	24570	699136	9075825	54456624	136136700	101606400								301999592
8	1	480	51786	2271616	48402225	524947392	2804511276	6502809600	4629441600							14512435976
9	1	660	100386	6414496	208125225	3661411572	34957539036	171339390720	374984769600	257191200000						842348951696
10	1	880	182061	16230016	759370975	20144929392	309054877551	2682574848000	12301812670500	25719120000000	17116074360000					58149557469376
11	1	1144	312741	37591840	2433091375	92315546856	2118741034599	29324366741376	236380642942500	1028089041300000	2071044997560000	1344389840640000				4711442416762432
12	1	1456	513513	80958592	7014596875	365600370096	11919574925811	245176514592768	3131245472192712	23790669472800000	99102088715454000	193592137052160000	123067686661920000			442941296160485824
13	1	1820	811629	164033584	18525896675	1284793144596	57203242988535	1661903134346304	31368684363806664	375572568200623200	2718692704365462000	10926102637277856000	20798439045864480000	12988374315396480000		47840270365329931008
14	1	2240	1241604	315537664	45437656550	4085908721856	240821596570380	9506919315631104	252220030576545969	4441883504249635200	50435187494254639560	350650348009226496000	1367212961952083610000	2545721365817710080000	1565562975516540000000	5884286694197216368128

Technical details

Node id: n-bcf1d9f70b071bdc · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-2cb6ab531ea3ff6a

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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: 1,   4;
  2: 1,  16,     27;
  3: 1,  40,    243,     288;
  4: 1,  80,   1143,    4608,      4500;
  5: 1, 140,   3843,   35088,    112500,      97200;
  6: 1, 224,  10458,  178560,   1263825,    3499200,     2778300;
  7: 1, 336,  24570,  699136,   9075825,   54456624,   136136700,    101606400;
  8: 1, 480,  51786, 2271616,  48402225,  524947392,  2804511276,   6502809600,   4629441600;
  9: 1, 660, 100386, 6414496, 208125225, 3661411572, 34957539036, 171339390720, 374984769600, 257191200000;

%F Formula

The row polynomials satisfy (4*n-4)*P(n,x) = -(4*n+4+(8*n^3-8*n)*x-(9*n^5-9*n^4-9*n^3+9*n^2)*x^2)*P(n-2,x) + ((8*n^3-8*n)*x^2+(4*n^5-4*n^4-4*n^3+4*n^2)*x^3)*d/dx P(n-2,x) + (8*n+(8*n^3-8*n)*x)*P(n-1,x) - (8*n^3-8*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n + 4 + (8*n^3 - 8*n)*x - (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + ((8*n^3 - 8*n)*x^2 + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^3)*D[P[n - 2, x], x] + (8*n + (8*n^3 - 8*n)*x)*P[n - 1, x] - (8*n^3 - 8*n)*x^2*D[P[n - 1, x], x])/(4*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + 4 + (8*n^3 - 8*n)*x - (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*v[i - 2] + ((8*n^3 - 8*n)*x^2 + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^3)*deriv(v[i - 2], x) + (8*n + (8*n^3 - 8*n)*x)*v[i - 1] - (8*n^3 - 8*n)*x^2*deriv(v[i - 1], x))/(4*n - 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
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 Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - 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.