A114656 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 16\,x^{2} + 4\,n - 20\,n x + 56\,n x^{2} + 16\,n x^{3} - 20\,n^{2} x + 48\,n^{2} x^{2} - 64\,n^{2} x^{3} + 16\,n^{2} x^{4} - 24\,n^{3} x^{2} - 32\,n^{3} x^{3} - 16\,n^{3} x^{4} + 48\,n^{4} x^{3} - 16\,n^{4} x^{4} + 16\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-4 - 40\,x + 16\,x^{2} - 208\,x^{3} - 64\,x^{5} + 20\,n + 32\,n x + 232\,n x^{2} - 8\,n x^{3} + 320\,n x^{4} + 96\,n x^{5} - 72\,n^{2} x + 48\,n^{2} x^{2} + 256\,n^{2} x^{3} - 112\,n^{2} x^{4} + 96\,n^{2} x^{5} - 168\,n^{3} x^{2} - 312\,n^{3} x^{3} - 592\,n^{3} x^{4} - 192\,n^{3} x^{5} + 208\,n^{4} x^{3} + 304\,n^{4} x^{4} - 32\,n^{4} x^{5} + 80\,n^{5} x^{4} + 96\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(1 + 6\,x^{2} - 32\,x^{4} - 32\,x^{6} - 12\,n x + 10\,n x^{2} + 24\,n x^{3} - 48\,n x^{4} + 96\,n x^{5} + 32\,n x^{6} - 12\,n^{2} x^{2} - 8\,n^{2} x^{3} + 32\,n^{2} x^{4} + 32\,n^{2} x^{5} + 64\,n^{2} x^{6} + 48\,n^{3} x^{3} + 16\,n^{3} x^{4} - 192\,n^{3} x^{5} - 64\,n^{3} x^{6} + 16\,n^{4} x^{4} - 64\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	8														11
3	11	64	36													111
4	45	408	540	128												1121
5	197	2432	5256	3072	400											11357
6	903	14120	42840	42240	14000	1152										115255
7	4279	81024	318060	450560	258000	55296	3136									1170355
8	20793	462392	2232468	4161920	3486000	1314432	197568	8192								11883765
9	103049	2632192	15102864	35080192	39205600	21934080	5883136	655360	20736							120617209
10	518859	14969160	99578160	277670400	390852000	295094016	118804224	23887872	2052864	51200						1223478755
11	2646723	85114560	644325300	2100019200	3579660000	3435264000	1884140160	574095360	89890560	6144000	123904					12401423767
12	13648869	484094424	4110188940	15346320000	30793620000	36055514880	25332827520	10599874560	2535287040	318208000	17718272	294912				125607597417
13	71039373	2754744192	25929675288	109188879360	252478710000	350172979200	302145634560	163045048320	53918472960	10407936000	1071273984	49545216	692224			1271234630677
14	372693519	15686015592	162137915784	760478813952	1993134171600	3202342805376	3291071976192	2193429528576	940929530112	252602521600	40220477440	3458138112	134983680	1605632		12856001177167
15	1968801519	89382048000	1006562285820	5205280604160	15260315070000	27915540516864	33390898597056	26635873026048	14183098465536	4964396236800	1104964596736	147710803968	10771697664	359661568	3686400	129917126098139

Technical details

Node id: n-95116ccff4ef466d · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d13aacdb3a9cb540

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:      1;
   2:      3,        8;
   3:     11,       64,       36;
   4:     45,      408,      540,       128;
   5:    197,     2432,     5256,      3072,       400;
   6:    903,    14120,    42840,     42240,     14000,      1152;
   7:   4279,    81024,   318060,    450560,    258000,     55296,      3136;
   8:  20793,   462392,  2232468,   4161920,   3486000,   1314432,    197568,     8192;
   9: 103049,  2632192, 15102864,  35080192,  39205600,  21934080,   5883136,   655360,   20736;
  10: 518859, 14969160, 99578160, 277670400, 390852000, 295094016, 118804224, 23887872, 2052864, 51200;

%F Formula

The row polynomials satisfy (4*n+4-(20*n^2+20*n)*x-(24*n^3-48*n^2-56*n+16)*x^2+(48*n^4-32*n^3-64*n^2+16*n)*x^3+(16*n^5-16*n^4-16*n^3+16*n^2)*x^4)*P(n,x) = (20*n-4-(72*n^2-32*n+40)*x-(168*n^3-48*n^2-232*n-16)*x^2+(208*n^4-312*n^3+256*n^2-8*n-208)*x^3+(80*n^5+304*n^4-592*n^3-112*n^2+320*n)*x^4+(96*n^5-32*n^4-192*n^3+96*n^2+96*n-64)*x^5)*P(n-1,x) + (1-12*n*x-(12*n^2-10*n-6)*x^2+(48*n^3-8*n^2+24*n)*x^3+(16*n^4+16*n^3+32*n^2-48*n-32)*x^4-(192*n^3-32*n^2-96*n)*x^5-(64*n^4+64*n^3-64*n^2-32*n+32)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((20*n - 4 - (72*n^2 - 32*n + 40)*x - (168*n^3 - 48*n^2 - 232*n - 16)*x^2 + (208*n^4 - 312*n^3 + 256*n^2 - 8*n - 208)*x^3 + (80*n^5 + 304*n^4 - 592*n^3 - 112*n^2 + 320*n)*x^4 + (96*n^5 - 32*n^4 - 192*n^3 + 96*n^2 + 96*n - 64)*x^5)*P[n - 1, x] + (1 - 12*n*x - (12*n^2 - 10*n - 6)*x^2 + (48*n^3 - 8*n^2 + 24*n)*x^3 + (16*n^4 + 16*n^3 + 32*n^2 - 48*n - 32)*x^4 - (192*n^3 - 32*n^2 - 96*n)*x^5 - (64*n^4 + 64*n^3 - 64*n^2 - 32*n + 32)*x^6)*D[P[n - 1, x], x])/(4*n + 4 - (20*n^2 + 20*n)*x - (24*n^3 - 48*n^2 - 56*n + 16)*x^2 + (48*n^4 - 32*n^3 - 64*n^2 + 16*n)*x^3 + (16*n^5 - 16*n^4 - 16*n^3 + 16*n^2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((20*n - 4 - (72*n^2 - 32*n + 40)*x - (168*n^3 - 48*n^2 - 232*n - 16)*x^2 + (208*n^4 - 312*n^3 + 256*n^2 - 8*n - 208)*x^3 + (80*n^5 + 304*n^4 - 592*n^3 - 112*n^2 + 320*n)*x^4 + (96*n^5 - 32*n^4 - 192*n^3 + 96*n^2 + 96*n - 64)*x^5)*v[i - 1] + (1 - 12*n*x - (12*n^2 - 10*n - 6)*x^2 + (48*n^3 - 8*n^2 + 24*n)*x^3 + (16*n^4 + 16*n^3 + 32*n^2 - 48*n - 32)*x^4 - (192*n^3 - 32*n^2 - 96*n)*x^5 - (64*n^4 + 64*n^3 - 64*n^2 - 32*n + 32)*x^6)*deriv(v[i - 1], x))/(4*n + 4 - (20*n^2 + 20*n)*x - (24*n^3 - 48*n^2 - 56*n + 16)*x^2 + (48*n^4 - 32*n^3 - 64*n^2 + 16*n)*x^3 + (16*n^5 - 16*n^4 - 16*n^3 + 16*n^2)*x^4)); 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 basis_map_A013609 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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 read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two c…?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Three-term recurrence criterion (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Three-term recurrence criterion (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (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