A085478 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-12 + 150\,x - 108\,x^{2} + 32\,n - 466\,n x + 432\,n x^{2} - 28\,n^{2} + 532\,n^{2} x - 684\,n^{2} x^{2} + 8\,n^{3} - 264\,n^{3} x + 536\,n^{3} x^{2} + 48\,n^{4} x - 208\,n^{4} x^{2} + 32\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-12 - 26\,x + 92\,x^{2} + 16\,x^{3} + 32\,n - 18\,n x - 616\,n x^{2} - 136\,n x^{3} - 28\,n^{2} + 164\,n^{2} x + 1040\,n^{2} x^{2} - 76\,n^{2} x^{3} + 8\,n^{3} - 168\,n^{3} x - 568\,n^{3} x^{2} + 488\,n^{3} x^{3} + 48\,n^{4} x + 32\,n^{4} x^{2} - 400\,n^{4} x^{3} + 32\,n^{5} x^{2} + 96\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x - 9\,x^{2} + 30\,x^{3} + 8\,x^{4} + 8\,n x - 10\,n x^{2} - 184\,n x^{3} - 64\,n x^{4} - 4\,n^{2} x + 44\,n^{2} x^{2} + 236\,n^{2} x^{3} - 16\,n^{2} x^{4} - 24\,n^{3} x^{2} - 64\,n^{3} x^{3} + 128\,n^{3} x^{4} - 16\,n^{4} x^{3} - 64\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	4														5
3	1	12	9													22
4	1	20	54	16												91
5	1	28	135	160	25											349
6	1	36	252	560	375	36										1260
7	1	44	405	1344	1750	756	49									4349
8	1	52	594	2640	5250	4536	1372	64								14509
9	1	60	819	4576	12375	16632	10290	2304	81							47138
10	1	68	1080	7280	25025	46332	45276	21120	3645	100						149927
11	1	76	1377	10880	45500	108108	147147	109824	40095	5500	121					468629
12	1	84	1710	15504	76500	222768	392392	411840	243243	71500	7986	144				1443672
13	1	92	2079	21280	121125	418608	909636	1244672	1042470	500500	121121	11232	169			4392985
14	1	100	2484	28336	182875	732564	1899240	3224832	3544398	2431000	968968	196560	15379	196		13226933
15	1	108	2925	36800	265650	1211364	3656037	7441920	10203570	9237800	5294718	1782144	307580	20580	225	39461422

Technical details

Node id: n-39c55baab4418287 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-6414dcf212314547

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: 1,  4;
   3: 1, 12,    9;
   4: 1, 20,   54,   16;
   5: 1, 28,  135,  160,    25;
   6: 1, 36,  252,  560,   375,    36;
   7: 1, 44,  405, 1344,  1750,   756,    49;
   8: 1, 52,  594, 2640,  5250,  4536,  1372,    64;
   9: 1, 60,  819, 4576, 12375, 16632, 10290,  2304,   81;
  10: 1, 68, 1080, 7280, 25025, 46332, 45276, 21120, 3645, 100;

%F Formula

The row polynomials satisfy (8*n^3-28*n^2+32*n-12+(48*n^4-264*n^3+532*n^2-466*n+150)*x+(32*n^5-208*n^4+536*n^3-684*n^2+432*n-108)*x^2)*P(n,x) = (8*n^3-28*n^2+32*n-12+(48*n^4-168*n^3+164*n^2-18*n-26)*x+(32*n^5+32*n^4-568*n^3+1040*n^2-616*n+92)*x^2+(96*n^5-400*n^4+488*n^3-76*n^2-136*n+16)*x^3)*P(n-1,x) - ((4*n^2-8*n+4)*x+(24*n^3-44*n^2+10*n+9)*x^2+(16*n^4+64*n^3-236*n^2+184*n-30)*x^3+(64*n^4-128*n^3+16*n^2+64*n-8)*x^4)*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[((8*n^3 - 28*n^2 + 32*n - 12 + (48*n^4 - 168*n^3 + 164*n^2 - 18*n - 26)*x + (32*n^5 + 32*n^4 - 568*n^3 + 1040*n^2 - 616*n + 92)*x^2 + (96*n^5 - 400*n^4 + 488*n^3 - 76*n^2 - 136*n + 16)*x^3)*P[n - 1, x] - ((4*n^2 - 8*n + 4)*x + (24*n^3 - 44*n^2 + 10*n + 9)*x^2 + (16*n^4 + 64*n^3 - 236*n^2 + 184*n - 30)*x^3 + (64*n^4 - 128*n^3 + 16*n^2 + 64*n - 8)*x^4)*D[P[n - 1, x], x])/(8*n^3 - 28*n^2 + 32*n - 12 + (48*n^4 - 264*n^3 + 532*n^2 - 466*n + 150)*x + (32*n^5 - 208*n^4 + 536*n^3 - 684*n^2 + 432*n - 108)*x^2)]];
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] = ((8*n^3 - 28*n^2 + 32*n - 12 + (48*n^4 - 168*n^3 + 164*n^2 - 18*n - 26)*x + (32*n^5 + 32*n^4 - 568*n^3 + 1040*n^2 - 616*n + 92)*x^2 + (96*n^5 - 400*n^4 + 488*n^3 - 76*n^2 - 136*n + 16)*x^3)*v[i - 1] - ((4*n^2 - 8*n + 4)*x + (24*n^3 - 44*n^2 + 10*n + 9)*x^2 + (16*n^4 + 64*n^3 - 236*n^2 + 184*n - 30)*x^3 + (64*n^4 - 128*n^3 + 16*n^2 + 64*n - 8)*x^4)*deriv(v[i - 1], x))/(8*n^3 - 28*n^2 + 32*n - 12 + (48*n^4 - 264*n^3 + 532*n^2 - 466*n + 150)*x + (32*n^5 - 208*n^4 + 536*n^3 - 684*n^2 + 432*n - 108)*x^2)); 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from reversal of Triangle read by rows: T(n, k) = binomial(n + k, 2*k).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n + k, 2*k).?
    
    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
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n + k, 2*k).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Three-term recurrence criterion (above)

Related ?

Same fingerprint ?

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