A375853 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + x + n - 3\,n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(2 - 6\,x + 4\,x^{2} + n + 3\,n x - 8\,n x^{2} + 3\,n^{2} x + n^{2} x^{2} + 2\,n^{3} x + 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2} + 2\,x^{3} + 4\,n x^{2} - 4\,n x^{3} + 4\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	2															2
2	8	16														24
3	20	112	60													192
4	40	432	648	160												1280
5	70	1232	3564	2464	350											7680
6	112	2912	13728	18304	7280	672										43008
7	168	6048	42120	91520	70200	18144	1176									229376
8	240	11424	110160	353600	442000	220320	39984	1920								1179648
9	330	20064	255816	1136960	2099500	1705440	596904	80256	2970							5898240
10	440	33264	541728	3183488	8139600	9767520	5571104	1444608	149688	4400						28835840
11	572	52624	1065636	7998848	27041560	44930592	37858184	15997696	3196908	263120	6292					138412032
12	728	80080	1973400	18418400	79534000	174730080	203851760	127254400	41441400	6578000	440440	8736				654311424
13	910	117936	3474900	39468000	211916250	594669600	917332920	792892800	381449250	98670000	12741300	707616	11830			3053453312
14	1120	168896	5863104	79622400	520260000	1815307200	3576827520	4087802880	2722960800	1040520000	218961600	23452416	1097824	15680		14092861440
15	1360	236096	9536592	152585472	1192074000	5061729600	12401237520	18103127040	15944448240	8436216000	2622562800	457756416	41325232	1652672	20400	64424509440

Technical details

Node id: n-e58e6b720d3c95de · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-550bf3d279916e6c

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   2;
   2:   8,    16;
   3:  20,   112,     60;
   4:  40,   432,    648,     160;
   5:  70,  1232,   3564,    2464,     350;
   6: 112,  2912,  13728,   18304,    7280,     672;
   7: 168,  6048,  42120,   91520,   70200,   18144,    1176;
   8: 240, 11424, 110160,  353600,  442000,  220320,   39984,    1920;
   9: 330, 20064, 255816, 1136960, 2099500, 1705440,  596904,   80256,   2970;
  10: 440, 33264, 541728, 3183488, 8139600, 9767520, 5571104, 1444608, 149688, 4400;

%F Formula

The row polynomials satisfy (n-1+(2*n^3-3*n^2+1)*x)*P(n,x) = (n+2+(2*n^3+3*n^2+3*n-6)*x+(6*n^3+n^2-8*n+4)*x^2)*P(n-1,x) + (2*x+(4*n^2+4*n-4)*x^2-(4*n^2+4*n-2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.

%t Mathematica

P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (2*n^3 + 3*n^2 + 3*n - 6)*x + (6*n^3 + n^2 - 8*n + 4)*x^2)*P[n - 1, x] + (2*x + (4*n^2 + 4*n - 4)*x^2 - (4*n^2 + 4*n - 2)*x^3)*D[P[n - 1, x], x])/(n - 1 + (2*n^3 - 3*n^2 + 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((n + 2 + (2*n^3 + 3*n^2 + 3*n - 6)*x + (6*n^3 + n^2 - 8*n + 4)*x^2)*v[i - 1] + (2*x + (4*n^2 + 4*n - 4)*x^2 - (4*n^2 + 4*n - 2)*x^3)*deriv(v[i - 1], x))/(n - 1 + (2*n^3 - 3*n^2 + 1)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
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) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.?
  - 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) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.?
    
    ↑ 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.