A111125 → 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

\(\left(-900 - 1080\,x + 8424\,x^{2} + 9720\,x^{3} - 2916\,x^{4} + 600\,n + 720\,n x - 17766\,n x^{2} - 19602\,n x^{3} + 10692\,n x^{4} + 12150\,n^{2} x^{2} + 13122\,n^{2} x^{3} - 15309\,n^{2} x^{4} - 2700\,n^{3} x^{2} - 2916\,n^{3} x^{3} + 10692\,n^{3} x^{4} - 3645\,n^{4} x^{4} + 486\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-600 - 1170\,x + 1296\,x^{2} + 3726\,x^{3} + 1944\,x^{4} + 900\,n - 45\,n x - 12204\,n x^{2} - 22599\,n x^{3} - 10692\,n x^{4} + 1575\,n^{2} x + 14850\,n^{2} x^{2} + 25758\,n^{2} x^{3} + 8262\,n^{2} x^{4} - 2916\,n^{2} x^{5} - 4050\,n^{3} x^{2} - 6399\,n^{3} x^{3} + 4617\,n^{3} x^{4} + 5832\,n^{3} x^{5} - 810\,n^{4} x^{3} - 4860\,n^{4} x^{4} - 3645\,n^{4} x^{5} + 729\,n^{5} x^{4} + 729\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(125 + 900\,x + 1440\,x^{2} - 1728\,x^{3} - 4617\,x^{4} - 1944\,x^{5} + 3150\,n x^{2} + 16200\,n x^{3} + 23166\,n x^{4} + 8748\,n x^{5} - 1575\,n^{2} x^{2} - 8100\,n^{2} x^{3} - 9963\,n^{2} x^{4} + 1458\,n^{2} x^{5} + 2916\,n^{2} x^{6} - 1620\,n^{3} x^{4} - 5832\,n^{3} x^{5} - 2916\,n^{3} x^{6} + 405\,n^{4} x^{4} + 1458\,n^{4} x^{5} + 729\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	12														16
3	11	84	81													176
4	29	372	810	432												1643
5	76	1380	4860	5616	2025											13957
6	199	4680	23085	42336	32400	8748										111448
7	521	15048	95985	244512	293625	166212	35721									851624
8	1364	46716	366444	1203120	1994625	1758348	785862	139968								6296447
9	3571	141468	1318680	5319216	11329875	13725612	9501786	3499200	531441							45370849
10	9349	420540	4544505	21787488	56928825	88363548	83765745	47589120	14880348	1968300						320257768
11	24476	1232340	15153480	84280608	261585450	497297556	602613270	467493120	224799543	61017300	7144929					2222642072
12	64079	3570192	49234230	311872896	1122892875	2533578264	3754705752	3716150400	2434531221	1013674500	242927586	25509168				15208711163
13	167761	10247184	156648006	1113920640	4568997375	11954334456	21006948564	25393274496	21188552670	11996788500	4401276264	943839216	89813529			102824808661
14	439204	29184492	489878280	3865345056	17805602250	53067493764	108079672302	154776614400	157546215891	113452812000	56509243461	18519655968	3592541160	312487308		688047185536
15	1149851	82576740	1509982155	13094798688	66966614325	224168348556	519924334545	862432427520	1039117552803	911866150800	576960161679	256367138400	75892432005	13436954244	1076168025	4562896790336

Technical details

Node id: n-79f9fc35447e5f9f · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-144f627d80fdb61a

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:    4,     12;
   3:   11,     84,      81;
   4:   29,    372,     810,      432;
   5:   76,   1380,    4860,     5616,     2025;
   6:  199,   4680,   23085,    42336,    32400,     8748;
   7:  521,  15048,   95985,   244512,   293625,   166212,    35721;
   8: 1364,  46716,  366444,  1203120,  1994625,  1758348,   785862,   139968;
   9: 3571, 141468, 1318680,  5319216, 11329875, 13725612,  9501786,  3499200,   531441;
  10: 9349, 420540, 4544505, 21787488, 56928825, 88363548, 83765745, 47589120, 14880348, 1968300;

%F Formula

The row polynomials satisfy (600*n-900+(720*n-1080)*x-(2700*n^3-12150*n^2+17766*n-8424)*x^2-(2916*n^3-13122*n^2+19602*n-9720)*x^3+(486*n^5-3645*n^4+10692*n^3-15309*n^2+10692*n-2916)*x^4)*P(n,x) = (900*n-600+(1575*n^2-45*n-1170)*x-(4050*n^3-14850*n^2+12204*n-1296)*x^2-(810*n^4+6399*n^3-25758*n^2+22599*n-3726)*x^3+(729*n^5-4860*n^4+4617*n^3+8262*n^2-10692*n+1944)*x^4+(729*n^5-3645*n^4+5832*n^3-2916*n^2)*x^5)*P(n-1,x) + (125+900*x-(1575*n^2-3150*n-1440)*x^2-(8100*n^2-16200*n+1728)*x^3+(405*n^4-1620*n^3-9963*n^2+23166*n-4617)*x^4+(1458*n^4-5832*n^3+1458*n^2+8748*n-1944)*x^5+(729*n^4-2916*n^3+2916*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[((900*n - 600 + (1575*n^2 - 45*n - 1170)*x - (4050*n^3 - 14850*n^2 + 12204*n - 1296)*x^2 - (810*n^4 + 6399*n^3 - 25758*n^2 + 22599*n - 3726)*x^3 + (729*n^5 - 4860*n^4 + 4617*n^3 + 8262*n^2 - 10692*n + 1944)*x^4 + (729*n^5 - 3645*n^4 + 5832*n^3 - 2916*n^2)*x^5)*P[n - 1, x] + (125 + 900*x - (1575*n^2 - 3150*n - 1440)*x^2 - (8100*n^2 - 16200*n + 1728)*x^3 + (405*n^4 - 1620*n^3 - 9963*n^2 + 23166*n - 4617)*x^4 + (1458*n^4 - 5832*n^3 + 1458*n^2 + 8748*n - 1944)*x^5 + (729*n^4 - 2916*n^3 + 2916*n^2)*x^6)*D[P[n - 1, x], x])/(600*n - 900 + (720*n - 1080)*x - (2700*n^3 - 12150*n^2 + 17766*n - 8424)*x^2 - (2916*n^3 - 13122*n^2 + 19602*n - 9720)*x^3 + (486*n^5 - 3645*n^4 + 10692*n^3 - 15309*n^2 + 10692*n - 2916)*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] = ((900*n - 600 + (1575*n^2 - 45*n - 1170)*x - (4050*n^3 - 14850*n^2 + 12204*n - 1296)*x^2 - (810*n^4 + 6399*n^3 - 25758*n^2 + 22599*n - 3726)*x^3 + (729*n^5 - 4860*n^4 + 4617*n^3 + 8262*n^2 - 10692*n + 1944)*x^4 + (729*n^5 - 3645*n^4 + 5832*n^3 - 2916*n^2)*x^5)*v[i - 1] + (125 + 900*x - (1575*n^2 - 3150*n - 1440)*x^2 - (8100*n^2 - 16200*n + 1728)*x^3 + (405*n^4 - 1620*n^3 - 9963*n^2 + 23166*n - 4617)*x^4 + (1458*n^4 - 5832*n^3 + 1458*n^2 + 8748*n - 1944)*x^5 + (729*n^4 - 2916*n^3 + 2916*n^2)*x^6)*deriv(v[i - 1], x))/(600*n - 900 + (720*n - 1080)*x - (2700*n^3 - 12150*n^2 + 17766*n - 8424)*x^2 - (2916*n^3 - 13122*n^2 + 19602*n - 9720)*x^3 + (486*n^5 - 3645*n^4 + 10692*n^3 - 15309*n^2 + 10692*n - 2916)*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(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    
    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(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    
    ↑ 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(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    
    ↑ 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(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    
    ↑ 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