A049403 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 76\,x + 16\,x^{2} + 60\,n - 90\,n x + 16\,n x^{2} + 37\,n^{2} - 34\,n^{2} x + 4\,n^{2} x^{2} + 10\,n^{3} - 4\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(144\,x - 204\,x^{2} + 36\,x^{3} - 36\,n + 288\,n x - 206\,n x^{2} + 24\,n x^{3} - 60\,n^{2} + 191\,n^{2} x - 70\,n^{2} x^{2} + 4\,n^{2} x^{3} - 37\,n^{3} + 52\,n^{3} x - 8\,n^{3} x^{2} - 10\,n^{4} + 5\,n^{4} x - n^{5}\right)\,P_{n-1}(x) + \left(72\,x - 240\,x^{2} + 72\,x^{3} + 120\,n x - 236\,n x^{2} + 48\,n x^{3} + 74\,n^{2} x - 76\,n^{2} x^{2} + 8\,n^{2} x^{3} + 20\,n^{3} x - 8\,n^{3} x^{2} + 2\,n^{4} x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 12\,x + 9\,x^{2}\)

Rows ?

n	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	12	9															21
1	12	54	16														82
2	0	135	160	25													320
3	0	135	720	375	36												1266
4	0	0	1680	2625	756	49											5110
5	0	0	1680	10500	7560	1372	64										21176
6	0	0	0	23625	45360	18522	2304	81									89892
7	0	0	0	23625	170100	154350	40320	3645	100								392140
8	0	0	0	0	374220	848925	443520	80190	5500	121							1752476
9	0	0	0	0	374220	3056130	3326400	1122660	148500	7986	144						8036040
10	0	0	0	0	0	6621615	17297280	10945935	2574000	259545	11232	169					37709776
11	0	0	0	0	0	6621615	60540480	76621545	31531500	5450445	432432	15379	196				181213592
12	0	0	0	0	0	0	129729600	383107725	283783500	81756675	10810800	692055	20580	225			889901160
13	0	0	0	0	0	0	129729600	1313512200	1891890000	915674760	194594400	20300280	1070160	27000	256		4466798656
14	0	0	0	0	0	0	0	2791213425	9189180000	7783235460	2646483840	431380950	36385440	1606500	34816	289	22879520720

Technical details

Node id: n-407234a194bc3378 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-f6ea1d516eaa5cd6

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 0, 12,   9;
  1: 0, 12,  54,   16;
  2: 0,  0, 135,  160,    25;
  3: 0,  0, 135,  720,   375,     36;
  4: 0,  0,   0, 1680,  2625,    756,      49;
  5: 0,  0,   0, 1680, 10500,   7560,    1372,      64;
  6: 0,  0,   0,    0, 23625,  45360,   18522,    2304,      81;
  7: 0,  0,   0,    0, 23625, 170100,  154350,   40320,    3645,    100;
  8: 0,  0,   0,    0,     0, 374220,  848925,  443520,   80190,   5500,  121;
  9: 0,  0,   0,    0,     0, 374220, 3056130, 3326400, 1122660, 148500, 7986, 144;

%F Formula

The row polynomials satisfy (n^4+10*n^3+37*n^2+60*n+36-(4*n^3+34*n^2+90*n+76)*x+(4*n^2+16*n+16)*x^2)*P(n,x) = -(n^5+10*n^4+37*n^3+60*n^2+36*n-(5*n^4+52*n^3+191*n^2+288*n+144)*x+(8*n^3+70*n^2+206*n+204)*x^2-(4*n^2+24*n+36)*x^3)*P(n-1,x) + ((2*n^4+20*n^3+74*n^2+120*n+72)*x-(8*n^3+76*n^2+236*n+240)*x^2+(8*n^2+48*n+72)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 12*x + 9*x^2.

%t Mathematica

P[0, x_] := 12*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n - (5*n^4 + 52*n^3 + 191*n^2 + 288*n + 144)*x + (8*n^3 + 70*n^2 + 206*n + 204)*x^2 - (4*n^2 + 24*n + 36)*x^3)*P[n - 1, x] + ((2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x - (8*n^3 + 76*n^2 + 236*n + 240)*x^2 + (8*n^2 + 48*n + 72)*x^3)*D[P[n - 1, x], x])/(n^4 + 10*n^3 + 37*n^2 + 60*n + 36 - (4*n^3 + 34*n^2 + 90*n + 76)*x + (4*n^2 + 16*n + 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 12*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n - (5*n^4 + 52*n^3 + 191*n^2 + 288*n + 144)*x + (8*n^3 + 70*n^2 + 206*n + 204)*x^2 - (4*n^2 + 24*n + 36)*x^3)*v[i - 1] + ((2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x - (8*n^3 + 76*n^2 + 236*n + 240)*x^2 + (8*n^2 + 48*n + 72)*x^3)*deriv(v[i - 1], x))/(n^4 + 10*n^3 + 37*n^2 + 60*n + 36 - (4*n^3 + 34*n^2 + 90*n + 76)*x + (4*n^2 + 16*n + 16)*x^2)); 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 A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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.