A078812 → 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

\(\left(-400 - 120\,x + 656\,x^{2} + 480\,x^{3} - 64\,x^{4} + 400\,n + 120\,n x - 2256\,n x^{2} - 1440\,n x^{3} + 320\,n x^{4} + 2400\,n^{2} x^{2} + 1440\,n^{2} x^{3} - 640\,n^{2} x^{4} - 800\,n^{3} x^{2} - 480\,n^{3} x^{3} + 640\,n^{3} x^{4} - 320\,n^{4} x^{4} + 64\,n^{5} x^{4}\right)\,P_{n}(x) = \left(600\,n + 380\,n x - 264\,n x^{2} - 304\,n x^{3} - 96\,n x^{4} + 700\,n^{2} x + 2280\,n^{2} x^{2} + 2000\,n^{2} x^{3} + 480\,n^{2} x^{4} - 1200\,n^{3} x^{2} - 1440\,n^{3} x^{3} - 288\,n^{3} x^{4} + 64\,n^{3} x^{5} - 160\,n^{4} x^{3} - 384\,n^{4} x^{4} - 128\,n^{4} x^{5} + 96\,n^{5} x^{4} + 64\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(125 + 450\,x + 440\,x^{2} - 48\,x^{3} - 304\,x^{4} - 96\,x^{5} + 700\,n x^{2} + 2280\,n x^{3} + 2000\,n x^{4} + 480\,n x^{5} - 700\,n^{2} x^{2} - 2280\,n^{2} x^{3} - 1920\,n^{2} x^{4} - 288\,n^{2} x^{5} + 64\,n^{2} x^{6} - 160\,n^{3} x^{4} - 384\,n^{3} x^{5} - 128\,n^{3} x^{6} + 80\,n^{4} x^{4} + 192\,n^{4} x^{5} + 64\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

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	8	48	36													92
4	21	200	324	128												673
5	55	720	1836	1536	400											4547
6	144	2400	8424	11008	6000	1152										29128
7	377	7632	34236	61440	52000	20736	3136									179557
8	987	23512	128628	295040	342000	210816	65856	8192								1075031
9	2584	70800	457452	1281024	1896000	1596672	768320	196608	20736							6290196
10	6765	209560	1562328	5174528	9349200	10039680	6585600	2588672	559872	51200						36127405
11	17711	612000	5172552	19797504	42322000	55448064	46318720	24772608	8211456	1536000	123904					204332519
12	46368	1768128	16709328	72609280	179484000	278191872	283312512	192724992	86780160	24832000	4088832	294912				1140842384
13	121393	5063328	52912008	257441280	723034000	1296041472	1560918912	1293484032	739964160	287232000	72236032	10616832	692224			6299757673
14	317811	14393000	164811672	887846144	2794122000	5692270464	7927557120	7765229568	5407243776	2663936000	907720704	203489280	26996736	1605632		34457539907
15	832040	40658160	506291508	2992094208	10433851200	23827841280	37717377600	42710925312	35137504512	21055795200	9097651200	2760376320	557932544	67436544	3686400	186910254028

Technical details

Node id: n-f9a60fb52c66bdee · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e8c880dcea1ddd79

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:    8,     48,      36;
   4:   21,    200,     324,     128;
   5:   55,    720,    1836,    1536,     400;
   6:  144,   2400,    8424,   11008,    6000,     1152;
   7:  377,   7632,   34236,   61440,   52000,    20736,    3136;
   8:  987,  23512,  128628,  295040,  342000,   210816,   65856,    8192;
   9: 2584,  70800,  457452, 1281024, 1896000,  1596672,  768320,  196608,  20736;
  10: 6765, 209560, 1562328, 5174528, 9349200, 10039680, 6585600, 2588672, 559872, 51200;

%F Formula

The row polynomials satisfy (400*n-400+(120*n-120)*x-(800*n^3-2400*n^2+2256*n-656)*x^2-(480*n^3-1440*n^2+1440*n-480)*x^3+(64*n^5-320*n^4+640*n^3-640*n^2+320*n-64)*x^4)*P(n,x) = (600*n+(700*n^2+380*n)*x-(1200*n^3-2280*n^2+264*n)*x^2-(160*n^4+1440*n^3-2000*n^2+304*n)*x^3+(96*n^5-384*n^4-288*n^3+480*n^2-96*n)*x^4+(64*n^5-128*n^4+64*n^3)*x^5)*P(n-1,x) + (125+450*x-(700*n^2-700*n-440)*x^2-(2280*n^2-2280*n+48)*x^3+(80*n^4-160*n^3-1920*n^2+2000*n-304)*x^4+(192*n^4-384*n^3-288*n^2+480*n-96)*x^5+(64*n^4-128*n^3+64*n^2)*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[((600*n + (700*n^2 + 380*n)*x - (1200*n^3 - 2280*n^2 + 264*n)*x^2 - (160*n^4 + 1440*n^3 - 2000*n^2 + 304*n)*x^3 + (96*n^5 - 384*n^4 - 288*n^3 + 480*n^2 - 96*n)*x^4 + (64*n^5 - 128*n^4 + 64*n^3)*x^5)*P[n - 1, x] + (125 + 450*x - (700*n^2 - 700*n - 440)*x^2 - (2280*n^2 - 2280*n + 48)*x^3 + (80*n^4 - 160*n^3 - 1920*n^2 + 2000*n - 304)*x^4 + (192*n^4 - 384*n^3 - 288*n^2 + 480*n - 96)*x^5 + (64*n^4 - 128*n^3 + 64*n^2)*x^6)*D[P[n - 1, x], x])/(400*n - 400 + (120*n - 120)*x - (800*n^3 - 2400*n^2 + 2256*n - 656)*x^2 - (480*n^3 - 1440*n^2 + 1440*n - 480)*x^3 + (64*n^5 - 320*n^4 + 640*n^3 - 640*n^2 + 320*n - 64)*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] = ((600*n + (700*n^2 + 380*n)*x - (1200*n^3 - 2280*n^2 + 264*n)*x^2 - (160*n^4 + 1440*n^3 - 2000*n^2 + 304*n)*x^3 + (96*n^5 - 384*n^4 - 288*n^3 + 480*n^2 - 96*n)*x^4 + (64*n^5 - 128*n^4 + 64*n^3)*x^5)*v[i - 1] + (125 + 450*x - (700*n^2 - 700*n - 440)*x^2 - (2280*n^2 - 2280*n + 48)*x^3 + (80*n^4 - 160*n^3 - 1920*n^2 + 2000*n - 304)*x^4 + (192*n^4 - 384*n^3 - 288*n^2 + 480*n - 96)*x^5 + (64*n^4 - 128*n^3 + 64*n^2)*x^6)*deriv(v[i - 1], x))/(400*n - 400 + (120*n - 120)*x - (800*n^3 - 2400*n^2 + 2256*n - 656)*x^2 - (480*n^3 - 1440*n^2 + 1440*n - 480)*x^3 + (64*n^5 - 320*n^4 + 640*n^3 - 640*n^2 + 320*n - 64)*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) = binomial(n+k-1, 2*k-1).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    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 read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ 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