A375853 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 - x^{2} + n + 5\,n x - n x^{2} - 7\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{3} x + n^{3} x^{2} - 8\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(2 - 8\,x + 10\,x^{2} - 12\,x^{3} + n + 2\,n x + 7\,n x^{2} + 26\,n x^{3} - n^{2} x - 27\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 5\,n^{3} x^{2} - 25\,n^{3} x^{3} + 4\,n^{4} x^{2} + 12\,n^{4} x^{3} + 4\,n^{5} x^{2} + 12\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(2\,x - 6\,x^{2} + 10\,x^{3} - 6\,x^{4} - 4\,n x^{2} - 6\,n x^{3} + 10\,n x^{4} + 4\,n^{2} x^{2} - 6\,n^{2} x^{3} + 2\,n^{2} x^{4} + 16\,n^{3} x^{3} - 16\,n^{3} x^{4} + 8\,n^{4} x^{3} - 8\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

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	32														40
3	20	224	180													424
4	40	864	1944	640												3488
5	70	2464	10692	9856	1750											24832
6	112	5824	41184	73216	36400	4032										160768
7	168	12096	126360	366080	351000	108864	8232									972800
8	240	22848	330480	1414400	2210000	1321920	279888	15360								5595136
9	330	40128	767448	4547840	10497500	10232640	4178328	642048	26730							30932992
10	440	66528	1625184	12733952	40698000	58605120	38997728	11556864	1347192	44000						165675008
11	572	105248	3196908	31995392	135207800	269583552	265007288	127981568	28772172	2631200	69212					864550912
12	728	160160	5920200	73673600	397670000	1048380480	1426962320	1018035200	372972600	65780000	4844840	104832				4414504960
13	910	235872	10424700	157872000	1059581250	3568017600	6421330440	6343142400	3433043250	986700000	140154300	8491392	153790			22129147904
14	1120	337792	17589312	318489600	2601300000	10891843200	25037792640	32702423040	24506647200	10405200000	2408577600	281428992	14271712	219520		109186121728
15	1360	472192	28609776	610341888	5960370000	30370377600	86808662640	144825016320	143500034160	84362160000	28848190800	5493076992	537228016	23137408	306000	531367985152

Technical details

Node id: n-af8518402ee2134c · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-f6c2d99c4aa2ab10

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,    32;
   3:  20,   224,     180;
   4:  40,   864,    1944,      640;
   5:  70,  2464,   10692,     9856,     1750;
   6: 112,  5824,   41184,    73216,    36400,     4032;
   7: 168, 12096,  126360,   366080,   351000,   108864,     8232;
   8: 240, 22848,  330480,  1414400,  2210000,  1321920,   279888,    15360;
   9: 330, 40128,  767448,  4547840, 10497500, 10232640,  4178328,   642048,   26730;
  10: 440, 66528, 1625184, 12733952, 40698000, 58605120, 38997728, 11556864, 1347192, 44000;

%F Formula

The row polynomials satisfy (n-1+(2*n^3-7*n^2+5*n)*x+(4*n^5-8*n^4+n^3+5*n^2-n-1)*x^2)*P(n,x) = (n+2+(2*n^3-n^2+2*n-8)*x+(4*n^5+4*n^4-5*n^3-27*n^2+7*n+10)*x^2+(12*n^5+12*n^4-25*n^3-4*n^2+26*n-12)*x^3)*P(n-1,x) + (2*x+(4*n^2-4*n-6)*x^2+(8*n^4+16*n^3-6*n^2-6*n+10)*x^3-(8*n^4+16*n^3-2*n^2-10*n+6)*x^4)*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 - n^2 + 2*n - 8)*x + (4*n^5 + 4*n^4 - 5*n^3 - 27*n^2 + 7*n + 10)*x^2 + (12*n^5 + 12*n^4 - 25*n^3 - 4*n^2 + 26*n - 12)*x^3)*P[n - 1, x] + (2*x + (4*n^2 - 4*n - 6)*x^2 + (8*n^4 + 16*n^3 - 6*n^2 - 6*n + 10)*x^3 - (8*n^4 + 16*n^3 - 2*n^2 - 10*n + 6)*x^4)*D[P[n - 1, x], x])/(n - 1 + (2*n^3 - 7*n^2 + 5*n)*x + (4*n^5 - 8*n^4 + n^3 + 5*n^2 - n - 1)*x^2)]];
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 - n^2 + 2*n - 8)*x + (4*n^5 + 4*n^4 - 5*n^3 - 27*n^2 + 7*n + 10)*x^2 + (12*n^5 + 12*n^4 - 25*n^3 - 4*n^2 + 26*n - 12)*x^3)*v[i - 1] + (2*x + (4*n^2 - 4*n - 6)*x^2 + (8*n^4 + 16*n^3 - 6*n^2 - 6*n + 10)*x^3 - (8*n^4 + 16*n^3 - 2*n^2 - 10*n + 6)*x^4)*deriv(v[i - 1], x))/(n - 1 + (2*n^3 - 7*n^2 + 5*n)*x + (4*n^5 - 8*n^4 + n^3 + 5*n^2 - n - 1)*x^2)); 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.