A257611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 4\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-9 + 52\,x - 23\,x^{2} - 24\,n - 8\,n x - 64\,n x^{2} - 12\,n^{2} - 144\,n^{2} x + 56\,n^{2} x^{2} - 96\,n^{3} x + 256\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-56\,x + 132\,x^{2} - 76\,x^{3} - 96\,n x + 312\,n x^{2} - 344\,n x^{3} - 48\,n^{2} x + 304\,n^{2} x^{2} - 448\,n^{2} x^{3} + 96\,n^{3} x^{2} - 96\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-20\,x^{2} + 40\,x^{3} - 20\,x^{4} - 32\,n x^{2} + 128\,n x^{3} - 96\,n x^{4} - 16\,n^{2} x^{2} + 160\,n^{2} x^{3} - 144\,n^{2} x^{4} + 64\,n^{3} x^{3} - 64\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(6 + 4\,x + 20\,n + 24\,n x + 16\,n^{2} + 48\,n^{2} x + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(12\,x - 12\,x^{2} + 24\,n x - 56\,n x^{2} + 16\,n^{2} x - 80\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	120	81													210
3	27	852	1917	432												3228
4	81	5232	26838	20928	2025											55104
5	243	29724	293814	522336	185775	8748										1040640
6	729	161256	2792367	9402048	7756575	1451304	35721									21600000
7	2187	850020	24263307	139224720	217538625	97053228	10412745	139968								489484800
8	6561	4398816	198531324	1813774464	4785849750	4080992544	1080892764	70381056	531441							12035358720
9	19683	22492716	1557973980	21618138816	89486641050	128860763112	66205550124	11078926080	455477499	1968300						319287951360
10	59049	114116952	11867381469	241648685184	1491028180050	3350379840912	2922415232898	966594740736	106806433221	2852923800	7144929					9093714739200
11	177147	576017268	88463696265	2575691940720	22820103180750	75761146620936	103119338456274	58419464142720	13039440449895	982929958500	17424522357	25509168				276824604672000
12	531441	2897801040	649054767474	26483872127040	327486989478375	1543531031936928	3093580153688220	2744055167887872	1061057845909935	165524200794000	8726180762706	104320837440	89813529			8971201806336000
13	1594323	14546400828	4706384680818	264893977413408	4471358938430625	29025947447308404	82119295349863884	107257855150842624	65308381756443909	17885435753722500	2003260704188898	75302154893088	614585434983	312487308		308417067063705600
14	4782969	72916945608	33828386616027	2593218449201472	58704575636031225	512551211113151352	1982432682288072819	3646767075128649216	3277082597251712211	1423753364203198200	284130146078391129	23338966042813248	635221926456507	3572930334792	1076168025	11212026533432524800

Technical details

Node id: n-c71b694d77c566c2 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-5be2da89e97eeb88

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,      120,         81;
  3:    27,      852,       1917,         432;
  4:    81,     5232,      26838,       20928,        2025;
  5:   243,    29724,     293814,      522336,      185775,         8748;
  6:   729,   161256,    2792367,     9402048,     7756575,      1451304,       35721;
  7:  2187,   850020,   24263307,   139224720,   217538625,     97053228,    10412745,      139968;
  8:  6561,  4398816,  198531324,  1813774464,  4785849750,   4080992544,  1080892764,    70381056,    531441;
  9: 19683, 22492716, 1557973980, 21618138816, 89486641050, 128860763112, 66205550124, 11078926080, 455477499, 1968300;

%F Formula

The row polynomials satisfy (4*n^2+4*n+1)*P(n,x) = -(12*n^2+24*n+9+(96*n^3+144*n^2+8*n-52)*x-(144*n^4+256*n^3+56*n^2-64*n-23)*x^2)*P(n-2,x) - ((48*n^2+96*n+56)*x-(96*n^3+304*n^2+312*n+132)*x^2-(64*n^4-96*n^3-448*n^2-344*n-76)*x^3)*d/dx P(n-2,x) - ((16*n^2+32*n+20)*x^2-(64*n^3+160*n^2+128*n+40)*x^3+(64*n^3+144*n^2+96*n+20)*x^4)*d^2/dx^2 P(n-2,x) + (16*n^2+20*n+6+(32*n^3+48*n^2+24*n+4)*x)*P(n-1,x) + ((16*n^2+24*n+12)*x-(32*n^3+80*n^2+56*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[(-(12*n^2 + 24*n + 9 + (96*n^3 + 144*n^2 + 8*n - 52)*x - (144*n^4 + 256*n^3 + 56*n^2 - 64*n - 23)*x^2)*P[n - 2, x] - ((48*n^2 + 96*n + 56)*x - (96*n^3 + 304*n^2 + 312*n + 132)*x^2 - (64*n^4 - 96*n^3 - 448*n^2 - 344*n - 76)*x^3)*D[P[n - 2, x], x] - ((16*n^2 + 32*n + 20)*x^2 - (64*n^3 + 160*n^2 + 128*n + 40)*x^3 + (64*n^3 + 144*n^2 + 96*n + 20)*x^4)*D[P[n - 2, x], {x, 2}] + (16*n^2 + 20*n + 6 + (32*n^3 + 48*n^2 + 24*n + 4)*x)*P[n - 1, x] + ((16*n^2 + 24*n + 12)*x - (32*n^3 + 80*n^2 + 56*n + 12)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 4*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] = (-(12*n^2 + 24*n + 9 + (96*n^3 + 144*n^2 + 8*n - 52)*x - (144*n^4 + 256*n^3 + 56*n^2 - 64*n - 23)*x^2)*v[i - 2] - ((48*n^2 + 96*n + 56)*x - (96*n^3 + 304*n^2 + 312*n + 132)*x^2 - (64*n^4 - 96*n^3 - 448*n^2 - 344*n - 76)*x^3)*deriv(v[i - 2], x) - ((16*n^2 + 32*n + 20)*x^2 - (64*n^3 + 160*n^2 + 128*n + 40)*x^3 + (64*n^3 + 144*n^2 + 96*n + 20)*x^4)*deriv(deriv(v[i - 2], x), x) + (16*n^2 + 20*n + 6 + (32*n^3 + 48*n^2 + 24*n + 4)*x)*v[i - 1] + ((16*n^2 + 24*n + 12)*x - (32*n^3 + 80*n^2 + 56*n + 12)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 4*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(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.?
  - 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(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.?
    
    ↑ 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.