A053117 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 5\,x - x^{2} + n - 15\,n x + 5\,n x^{2} + 15\,n^{2} x - 10\,n^{2} x^{2} - 5\,n^{3} x + 10\,n^{3} x^{2} - 5\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(n - n x - n x^{2} + 10\,n^{2} x + 5\,n^{2} x^{2} - 5\,n^{3} x - 3\,n^{3} x^{2} + n^{3} x^{3} - 4\,n^{4} x^{2} - 2\,n^{4} x^{3} + n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(2\,x - x^{2} - x^{3} + 10\,n x^{2} + 5\,n x^{3} - 10\,n^{2} x^{2} - 3\,n^{2} x^{3} + n^{2} x^{4} - 4\,n^{3} x^{3} - 2\,n^{3} x^{4} + 2\,n^{4} x^{3} + n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	8														10
3	3	32	36													71
4	4	80	216	128												428
5	5	160	756	1024	400											2345
6	6	280	2016	4608	4000	1152										12062
7	7	448	4536	15360	22000	13824	3136									59311
8	8	672	9072	42240	88000	89856	43904	8192								281944
9	9	960	16632	101376	286000	419328	329280	131072	20736							1305393
10	10	1320	28512	219648	800800	1572480	1756160	1114112	373248	51200						5917490
11	11	1760	46332	439296	2002000	5031936	7463680	6684672	3545856	1024000	123904					26363447
12	12	2288	72072	823680	4576000	14257152	26869248	31752192	23639040	10752000	2725888	294912				115764484
13	13	2912	108108	1464320	9724000	36661248	85085952	127008768	124104960	78848000	31347712	7077888	692224			502126105
14	14	3640	157248	2489344	19448000	87070464	243102720	444530688	546061824	453376000	250781696	88473600	17997824	1605632		2155098694
15	15	4480	222768	4073472	36951200	193489920	638144640	1397096448	2093236992	2176204800	1567385600	766771200	242970624	44957696	3686400	9165196255

Technical details

Node id: n-c8a1244763a9c604 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-b38c31b44d930868

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:  2,    8;
   3:  3,   32,    36;
   4:  4,   80,   216,    128;
   5:  5,  160,   756,   1024,    400;
   6:  6,  280,  2016,   4608,   4000,    1152;
   7:  7,  448,  4536,  15360,  22000,   13824,    3136;
   8:  8,  672,  9072,  42240,  88000,   89856,   43904,    8192;
   9:  9,  960, 16632, 101376, 286000,  419328,  329280,  131072,  20736;
  10: 10, 1320, 28512, 219648, 800800, 1572480, 1756160, 1114112, 373248, 51200;

%F Formula

The row polynomials satisfy (n-1-(5*n^3-15*n^2+15*n-5)*x+(n^5-5*n^4+10*n^3-10*n^2+5*n-1)*x^2)*P(n,x) = (n-(5*n^3-10*n^2+n)*x+(n^5-4*n^4-3*n^3+5*n^2-n)*x^2+(n^5-2*n^4+n^3)*x^3)*P(n-1,x) + (2*x-(10*n^2-10*n+1)*x^2+(2*n^4-4*n^3-3*n^2+5*n-1)*x^3+(n^4-2*n^3+n^2)*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[((n - (5*n^3 - 10*n^2 + n)*x + (n^5 - 4*n^4 - 3*n^3 + 5*n^2 - n)*x^2 + (n^5 - 2*n^4 + n^3)*x^3)*P[n - 1, x] + (2*x - (10*n^2 - 10*n + 1)*x^2 + (2*n^4 - 4*n^3 - 3*n^2 + 5*n - 1)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*D[P[n - 1, x], x])/(n - 1 - (5*n^3 - 15*n^2 + 15*n - 5)*x + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*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] = ((n - (5*n^3 - 10*n^2 + n)*x + (n^5 - 4*n^4 - 3*n^3 + 5*n^2 - n)*x^2 + (n^5 - 2*n^4 + n^3)*x^3)*v[i - 1] + (2*x - (10*n^2 - 10*n + 1)*x^2 + (2*n^4 - 4*n^3 - 3*n^2 + 5*n - 1)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*deriv(v[i - 1], x))/(n - 1 - (5*n^3 - 15*n^2 + 15*n - 5)*x + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*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 shift_x_plus_one of Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing or…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

chebyshev_u → x ↦ x+1 → (θ+1)²

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.
[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