A114656 → 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

\(\left(12 - 108\,x^{2} + 12\,n - 90\,n x + 378\,n x^{2} + 162\,n x^{3} - 90\,n^{2} x + 324\,n^{2} x^{2} - 648\,n^{2} x^{3} + 243\,n^{2} x^{4} - 162\,n^{3} x^{2} - 324\,n^{3} x^{3} - 243\,n^{3} x^{4} + 486\,n^{4} x^{3} - 243\,n^{4} x^{4} + 243\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-12 - 180\,x + 108\,x^{2} - 2106\,x^{3} - 1458\,x^{5} + 60\,n + 144\,n x + 1566\,n x^{2} - 81\,n x^{3} + 4860\,n x^{4} + 2187\,n x^{5} - 324\,n^{2} x + 324\,n^{2} x^{2} + 2592\,n^{2} x^{3} - 1701\,n^{2} x^{4} + 2187\,n^{2} x^{5} - 1134\,n^{3} x^{2} - 3159\,n^{3} x^{3} - 8991\,n^{3} x^{4} - 4374\,n^{3} x^{5} + 2106\,n^{4} x^{3} + 4617\,n^{4} x^{4} - 729\,n^{4} x^{5} + 1215\,n^{5} x^{4} + 2187\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(2 + 27\,x^{2} - 324\,x^{4} - 729\,x^{6} - 36\,n x + 45\,n x^{2} + 162\,n x^{3} - 486\,n x^{4} + 1458\,n x^{5} + 729\,n x^{6} - 54\,n^{2} x^{2} - 54\,n^{2} x^{3} + 324\,n^{2} x^{4} + 486\,n^{2} x^{5} + 1458\,n^{2} x^{6} + 324\,n^{3} x^{3} + 162\,n^{3} x^{4} - 2916\,n^{3} x^{5} - 1458\,n^{3} x^{6} + 162\,n^{4} x^{4} - 1458\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

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	12														15
3	11	96	81													188
4	45	612	1215	432												2304
5	197	3648	11826	10368	2025											28064
6	903	21180	96390	142560	70875	8748										340656
7	4279	121536	715635	1520640	1306125	419904	35721									4123840
8	20793	693588	5023053	14046480	17647875	9981468	2250423	139968								49803648
9	103049	3948288	33981444	118395648	198478350	166561920	67012596	11197440	531441							600210176
10	518859	22453740	224050860	937137600	1978688250	2240870184	1353254364	408146688	52612659	1968300						7219701504
11	2646723	127671840	1449731925	7087564800	18122028750	26086536000	21461534010	9808957440	2303796735	236196000	7144929					86693809152
12	13648869	726141636	9247925115	51793830000	155892701250	273796566120	288556738470	181108794240	64976633865	12232984500	1021724847	25509168				1039393198080
13	71039373	4132116288	58341769398	368512467840	1278173469375	2659126060800	3441627618660	2785777505280	1381871488635	400116024000	61775056134	4285540224	89813529			12443899969536
14	372693519	23529023388	364810310514	2566615997088	10090241743725	24317790678324	37487366728812	37476799835904	24114994715097	9710889516900	2319315402690	299120503968	17513638155	312487308		148789673275392
15	1968801519	134073072000	2264765143095	17567822039040	77255345041875	211983635799936	380343204332091	455098861780992	363497300907741	190848068611200	63717826633461	12776623921152	1397588324769	69997156992	1076168025	1776958157733888

Technical details

Node id: n-5809aa5bcc7f88f4 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-0136511bcf238556

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,       12;
   3:     11,       96,        81;
   4:     45,      612,      1215,       432;
   5:    197,     3648,     11826,     10368,       2025;
   6:    903,    21180,     96390,    142560,      70875,       8748;
   7:   4279,   121536,    715635,   1520640,    1306125,     419904,      35721;
   8:  20793,   693588,   5023053,  14046480,   17647875,    9981468,    2250423,    139968;
   9: 103049,  3948288,  33981444, 118395648,  198478350,  166561920,   67012596,  11197440,   531441;
  10: 518859, 22453740, 224050860, 937137600, 1978688250, 2240870184, 1353254364, 408146688, 52612659, 1968300;

%F Formula

The row polynomials satisfy (12*n+12-(90*n^2+90*n)*x-(162*n^3-324*n^2-378*n+108)*x^2+(486*n^4-324*n^3-648*n^2+162*n)*x^3+(243*n^5-243*n^4-243*n^3+243*n^2)*x^4)*P(n,x) = (60*n-12-(324*n^2-144*n+180)*x-(1134*n^3-324*n^2-1566*n-108)*x^2+(2106*n^4-3159*n^3+2592*n^2-81*n-2106)*x^3+(1215*n^5+4617*n^4-8991*n^3-1701*n^2+4860*n)*x^4+(2187*n^5-729*n^4-4374*n^3+2187*n^2+2187*n-1458)*x^5)*P(n-1,x) + (2-36*n*x-(54*n^2-45*n-27)*x^2+(324*n^3-54*n^2+162*n)*x^3+(162*n^4+162*n^3+324*n^2-486*n-324)*x^4-(2916*n^3-486*n^2-1458*n)*x^5-(1458*n^4+1458*n^3-1458*n^2-729*n+729)*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[((60*n - 12 - (324*n^2 - 144*n + 180)*x - (1134*n^3 - 324*n^2 - 1566*n - 108)*x^2 + (2106*n^4 - 3159*n^3 + 2592*n^2 - 81*n - 2106)*x^3 + (1215*n^5 + 4617*n^4 - 8991*n^3 - 1701*n^2 + 4860*n)*x^4 + (2187*n^5 - 729*n^4 - 4374*n^3 + 2187*n^2 + 2187*n - 1458)*x^5)*P[n - 1, x] + (2 - 36*n*x - (54*n^2 - 45*n - 27)*x^2 + (324*n^3 - 54*n^2 + 162*n)*x^3 + (162*n^4 + 162*n^3 + 324*n^2 - 486*n - 324)*x^4 - (2916*n^3 - 486*n^2 - 1458*n)*x^5 - (1458*n^4 + 1458*n^3 - 1458*n^2 - 729*n + 729)*x^6)*D[P[n - 1, x], x])/(12*n + 12 - (90*n^2 + 90*n)*x - (162*n^3 - 324*n^2 - 378*n + 108)*x^2 + (486*n^4 - 324*n^3 - 648*n^2 + 162*n)*x^3 + (243*n^5 - 243*n^4 - 243*n^3 + 243*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] = ((60*n - 12 - (324*n^2 - 144*n + 180)*x - (1134*n^3 - 324*n^2 - 1566*n - 108)*x^2 + (2106*n^4 - 3159*n^3 + 2592*n^2 - 81*n - 2106)*x^3 + (1215*n^5 + 4617*n^4 - 8991*n^3 - 1701*n^2 + 4860*n)*x^4 + (2187*n^5 - 729*n^4 - 4374*n^3 + 2187*n^2 + 2187*n - 1458)*x^5)*v[i - 1] + (2 - 36*n*x - (54*n^2 - 45*n - 27)*x^2 + (324*n^3 - 54*n^2 + 162*n)*x^3 + (162*n^4 + 162*n^3 + 324*n^2 - 486*n - 324)*x^4 - (2916*n^3 - 486*n^2 - 1458*n)*x^5 - (1458*n^4 + 1458*n^3 - 1458*n^2 - 729*n + 729)*x^6)*deriv(v[i - 1], x))/(12*n + 12 - (90*n^2 + 90*n)*x - (162*n^3 - 324*n^2 - 378*n + 108)*x^2 + (486*n^4 - 324*n^3 - 648*n^2 + 162*n)*x^3 + (243*n^5 - 243*n^4 - 243*n^3 + 243*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_A013610 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