A232535 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 6\,x + 9\,x^{2} - 20\,n - 2\,n x + 3\,n x^{2} + 32\,n^{2} - 88\,n^{2} x - 57\,n^{2} x^{2} - 16\,n^{3} + 172\,n^{3} x + 27\,n^{3} x^{2} + 9\,n^{3} x^{3} - 72\,n^{4} x + 120\,n^{4} x^{2} + 36\,n^{4} x^{3} - 36\,n^{5} x^{2} + 36\,n^{5} x^{3}\right)\,P_{n}(x) = \left(16 + 152\,x + 456\,x^{2} + 342\,x^{3} - 27\,x^{4} - 80\,n - 664\,n x - 1668\,n x^{2} - 1098\,n x^{3} - 135\,n x^{4} + 128\,n^{2} + 736\,n^{2} x + 966\,n^{2} x^{2} - 135\,n^{2} x^{4} - 64\,n^{3} + 112\,n^{3} x + 1572\,n^{3} x^{2} + 2007\,n^{3} x^{3} + 837\,n^{3} x^{4} - 288\,n^{4} x - 600\,n^{4} x^{2} + 756\,n^{4} x^{3} + 1080\,n^{4} x^{4} - 144\,n^{5} x^{2} - 180\,n^{5} x^{3} + 324\,n^{5} x^{4}\right)\,P_{n-1}(x) + \left(8\,x + 76\,x^{2} + 210\,x^{3} + 144\,x^{4} - 54\,x^{5} - 32\,n x - 256\,n x^{2} - 678\,n x^{3} - 720\,n x^{4} - 378\,n x^{5} + 32\,n^{2} x + 112\,n^{2} x^{2} - 186\,n^{2} x^{3} - 1044\,n^{2} x^{4} - 1026\,n^{2} x^{5} + 144\,n^{3} x^{2} + 480\,n^{3} x^{3} - 144\,n^{3} x^{4} - 864\,n^{3} x^{5} + 72\,n^{4} x^{3} + 144\,n^{4} x^{4} - 216\,n^{4} x^{5}\right)\,P_{n-1}'(x)\)

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

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	24														27
2	12	168	243													423
3	48	960	2997	1728												5733
4	192	4992	23976	32832	10125											72117
5	768	24576	159408	352512	273375	52488										863127
6	3072	116736	953856	2930688	3798900	1907064	250047									9960363
7	12288	540672	5329152	21012480	38685600	33312384	11752209	1119744								111764529
8	49152	2457600	28366848	136691712	327888000	408146688	252047376	66064896	4782969							1226495241
9	196608	11010048	145649664	829882368	2459808000	4040036352	3658973472	1706489856	345968091	19683000						13217697459
10	786432	48758784	727252992	4784652288	16910208000	34644879360	41845579776	28970016768	10592682012	1712421000	78594219					140315831631
11	3145728	213909504	3551330304	26499612672	108855705600	267878596608	406924487424	379172192256	207925228368	61316481600	8095204557	306110016				1470742004637
12	12582912	931135488	17029398528	142123991040	665675366400	1915138473984	3517077086208	4151329947648	3088480000320	1378061942400	335125749816	36835238592	1167575877			15248988489213
13	50331648	4026531840	80433119232	742442139648	3906330624000	12875371905024	27788391235584	39881662267392	37824041968128	23057768448000	8549793519696	1745643384576	162293046903	4374822312		156622623343983
14	201326592	17314086912	375071440896	3794200952832	22159982592000	82374173392896	204559922036736	346654683168768	401822046922752	314031085056000	160075675229184	50172656062464	8728797256452	695596747608	16142520375	1595477548792467

Technical details

Node id: n-c91b23b261d98e03 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-b44bae31b520ea3b

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,       24;
  2:     12,      168,       243;
  3:     48,      960,      2997,      1728;
  4:    192,     4992,     23976,     32832,      10125;
  5:    768,    24576,    159408,    352512,     273375,      52488;
  6:   3072,   116736,    953856,   2930688,    3798900,    1907064,     250047;
  7:  12288,   540672,   5329152,  21012480,   38685600,   33312384,   11752209,    1119744;
  8:  49152,  2457600,  28366848, 136691712,  327888000,  408146688,  252047376,   66064896,   4782969;
  9: 196608, 11010048, 145649664, 829882368, 2459808000, 4040036352, 3658973472, 1706489856, 345968091, 19683000;

%F Formula

The row polynomials satisfy -(16*n^3-32*n^2+20*n-4+(72*n^4-172*n^3+88*n^2+2*n-6)*x+(36*n^5-120*n^4-27*n^3+57*n^2-3*n-9)*x^2-(36*n^5+36*n^4+9*n^3)*x^3)*P(n,x) = -(64*n^3-128*n^2+80*n-16+(288*n^4-112*n^3-736*n^2+664*n-152)*x+(144*n^5+600*n^4-1572*n^3-966*n^2+1668*n-456)*x^2+(180*n^5-756*n^4-2007*n^3+1098*n-342)*x^3-(324*n^5+1080*n^4+837*n^3-135*n^2-135*n-27)*x^4)*P(n-1,x) + ((32*n^2-32*n+8)*x+(144*n^3+112*n^2-256*n+76)*x^2+(72*n^4+480*n^3-186*n^2-678*n+210)*x^3+(144*n^4-144*n^3-1044*n^2-720*n+144)*x^4-(216*n^4+864*n^3+1026*n^2+378*n+54)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(64*n^3 - 128*n^2 + 80*n - 16 + (288*n^4 - 112*n^3 - 736*n^2 + 664*n - 152)*x + (144*n^5 + 600*n^4 - 1572*n^3 - 966*n^2 + 1668*n - 456)*x^2 + (180*n^5 - 756*n^4 - 2007*n^3 + 1098*n - 342)*x^3 - (324*n^5 + 1080*n^4 + 837*n^3 - 135*n^2 - 135*n - 27)*x^4)*P[n - 1, x] + ((32*n^2 - 32*n + 8)*x + (144*n^3 + 112*n^2 - 256*n + 76)*x^2 + (72*n^4 + 480*n^3 - 186*n^2 - 678*n + 210)*x^3 + (144*n^4 - 144*n^3 - 1044*n^2 - 720*n + 144)*x^4 - (216*n^4 + 864*n^3 + 1026*n^2 + 378*n + 54)*x^5)*D[P[n - 1, x], x])/(-16*n^3 + 32*n^2 - 20*n + 4 - (72*n^4 - 172*n^3 + 88*n^2 + 2*n - 6)*x - (36*n^5 - 120*n^4 - 27*n^3 + 57*n^2 - 3*n - 9)*x^2 + (36*n^5 + 36*n^4 + 9*n^3)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(64*n^3 - 128*n^2 + 80*n - 16 + (288*n^4 - 112*n^3 - 736*n^2 + 664*n - 152)*x + (144*n^5 + 600*n^4 - 1572*n^3 - 966*n^2 + 1668*n - 456)*x^2 + (180*n^5 - 756*n^4 - 2007*n^3 + 1098*n - 342)*x^3 - (324*n^5 + 1080*n^4 + 837*n^3 - 135*n^2 - 135*n - 27)*x^4)*v[i - 1] + ((32*n^2 - 32*n + 8)*x + (144*n^3 + 112*n^2 - 256*n + 76)*x^2 + (72*n^4 + 480*n^3 - 186*n^2 - 678*n + 210)*x^3 + (144*n^4 - 144*n^3 - 1044*n^2 - 720*n + 144)*x^4 - (216*n^4 + 864*n^3 + 1026*n^2 + 378*n + 54)*x^5)*deriv(v[i - 1], x))/(-16*n^3 + 32*n^2 - 20*n + 4 - (72*n^4 - 172*n^3 + 88*n^2 + 2*n - 6)*x - (36*n^5 - 120*n^4 - 27*n^3 + 57*n^2 - 3*n - 9)*x^2 + (36*n^5 + 36*n^4 + 9*n^3)*x^3)); 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 basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (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
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023