A257621 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 8\,n + 16\,n^{2}\right)\,P_{n}(x) = \left(-9 - 500\,x - 95\,x^{2} + 24\,n + 880\,n x + 656\,n x^{2} + 48\,n^{2} + 576\,n^{2} x - 544\,n^{2} x^{2} - 768\,n^{3} x - 2816\,n^{3} x^{2} + 2304\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(184\,x - 540\,x^{2} - 148\,x^{3} - 192\,n x + 528\,n x^{2} + 928\,n x^{3} - 384\,n^{2} x + 1216\,n^{2} x^{2} - 256\,n^{2} x^{3} + 768\,n^{3} x^{2} - 4608\,n^{3} x^{3} + 1024\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(112\,x^{2} - 80\,x^{3} - 32\,x^{4} - 128\,n x^{2} - 64\,n x^{3} + 192\,n x^{4} - 256\,n^{2} x^{2} + 256\,n^{2} x^{3} + 1024\,n^{3} x^{3} - 1024\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(6 - 4\,x - 32\,n + 48\,n x + 32\,n^{2} - 192\,n^{2} x + 256\,n^{3} x\right)\,P_{n-1}(x) + \left(-24\,x - 12\,x^{2} + 80\,n x^{2} + 128\,n^{2} x - 64\,n^{2} x^{2} - 256\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	12														15
2	9	168	81													258
3	27	1572	3537	432												5568
4	81	12624	77814	50496	2025											143040
5	243	94524	1282014	2279136	590775	8748										4255440
6	729	684024	18143055	68374080	50397375	6156216	35721									143791200
7	2187	4866900	234971847	1638443280	2560067625	939887388	59619525	139968								5437998720
8	6561	34339488	2880354492	34184386944	97282569750	76914870624	15681930012	549431808	531441							227528421120
9	19683	241294956	34079078700	651022819776	3076870231050	4430693132712	1993757385564	242340115200	4886222859	1968300						10433892268800
10	59049	1692135240	393871843485	11643478189440	85925559599250	203811884104752	168414096814530	46573912757760	3544846591365	42303381000	7144929					520351652620800
11	177147	11855103108	4481075145645	199159754214000	2196566607186750	8028459091628136	10927624874716074	5623210514398080	1008246255708375	49789723840500	358616869017	25509168				28037908394496000
12	531441	83019025392	50438807827794	3298084189974720	52626314595126375	282709292452263648	590091743234667996	502594297692913152	170509259288209455	20613026187342000	678121749684786	2988684914112	89813529			1623173649992294400
13	1594323	581241591708	563606148040938	53327029492456608	1200876982633985625	9154679981996433204	27861238601641271484	36390189193980436224	20598029959491974709	4803507930535942500	403285660536703098	9017698368655008	24557457249663	312487308		100474741783838822400
14	4782969	4069041893016	6266365101098811	846997867200936384	26399572464070155225	277998890930642596392	1188372224719977745203	2256191730026787045888	1964452044945269333907	772219141474007212200	127773930726099551289	7622980804808427456	117668411342855451	199383052757784	1076168025	6622001652188482560000

Technical details

Node id: n-6be60b87a855006c · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-7fcfcfdc25807408

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:     3,        12;
  2:     9,       168,          81;
  3:    27,      1572,        3537,          432;
  4:    81,     12624,       77814,        50496,          2025;
  5:   243,     94524,     1282014,      2279136,        590775,          8748;
  6:   729,    684024,    18143055,     68374080,      50397375,       6156216,         35721;
  7:  2187,   4866900,   234971847,   1638443280,    2560067625,     939887388,      59619525,       139968;
  8:  6561,  34339488,  2880354492,  34184386944,   97282569750,   76914870624,   15681930012,    549431808,     531441;
  9: 19683, 241294956, 34079078700, 651022819776, 3076870231050, 4430693132712, 1993757385564, 242340115200, 4886222859, 1968300;

%F Formula

The row polynomials satisfy (16*n^2-8*n+1)*P(n,x) = (48*n^2+24*n-9-(768*n^3-576*n^2-880*n+500)*x+(2304*n^4-2816*n^3-544*n^2+656*n-95)*x^2)*P(n-2,x) - ((384*n^2+192*n-184)*x-(768*n^3+1216*n^2+528*n-540)*x^2-(1024*n^4-4608*n^3-256*n^2+928*n-148)*x^3)*d/dx P(n-2,x) - ((256*n^2+128*n-112)*x^2-(1024*n^3+256*n^2-64*n-80)*x^3+(1024*n^3-192*n+32)*x^4)*d^2/dx^2 P(n-2,x) + (32*n^2-32*n+6+(256*n^3-192*n^2+48*n-4)*x)*P(n-1,x) + ((128*n^2-24)*x-(256*n^3+64*n^2-80*n+12)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[((48*n^2 + 24*n - 9 - (768*n^3 - 576*n^2 - 880*n + 500)*x + (2304*n^4 - 2816*n^3 - 544*n^2 + 656*n - 95)*x^2)*P[n - 2, x] - ((384*n^2 + 192*n - 184)*x - (768*n^3 + 1216*n^2 + 528*n - 540)*x^2 - (1024*n^4 - 4608*n^3 - 256*n^2 + 928*n - 148)*x^3)*D[P[n - 2, x], x] - ((256*n^2 + 128*n - 112)*x^2 - (1024*n^3 + 256*n^2 - 64*n - 80)*x^3 + (1024*n^3 - 192*n + 32)*x^4)*D[P[n - 2, x], {x, 2}] + (32*n^2 - 32*n + 6 + (256*n^3 - 192*n^2 + 48*n - 4)*x)*P[n - 1, x] + ((128*n^2 - 24)*x - (256*n^3 + 64*n^2 - 80*n + 12)*x^2)*D[P[n - 1, x], x])/(16*n^2 - 8*n + 1)]];
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] = 3 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = ((48*n^2 + 24*n - 9 - (768*n^3 - 576*n^2 - 880*n + 500)*x + (2304*n^4 - 2816*n^3 - 544*n^2 + 656*n - 95)*x^2)*v[i - 2] - ((384*n^2 + 192*n - 184)*x - (768*n^3 + 1216*n^2 + 528*n - 540)*x^2 - (1024*n^4 - 4608*n^3 - 256*n^2 + 928*n - 148)*x^3)*deriv(v[i - 2], x) - ((256*n^2 + 128*n - 112)*x^2 - (1024*n^3 + 256*n^2 - 64*n - 80)*x^3 + (1024*n^3 - 192*n + 32)*x^4)*deriv(deriv(v[i - 2], x), x) + (32*n^2 - 32*n + 6 + (256*n^3 - 192*n^2 + 48*n - 4)*x)*v[i - 1] + ((128*n^2 - 24)*x - (256*n^3 + 64*n^2 - 80*n + 12)*x^2)*deriv(v[i - 1], x))/(16*n^2 - 8*n + 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 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 read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

References

[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.