A391045 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-12\,n + 40\,n x - 20\,n x^{2} + 2\,n x^{3} - 16\,n^{2} + 38\,n^{2} x - 18\,n^{2} x^{2} + 2\,n^{2} x^{3} + 26\,n^{3} - 84\,n^{3} x + 42\,n^{3} x^{2} - 4\,n^{3} x^{3} + 12\,n^{4} + 2\,n^{4} x - 4\,n^{4} x^{2} - 14\,n^{5} + 4\,n^{5} x + 4\,n^{6}\right)\,P_{n}(x) = \left(-12\,n + 40\,n x - 72\,n x^{2} + 38\,n x^{3} - 4\,n x^{4} - 10\,n^{2} - 16\,n^{2} x + 140\,n^{2} x^{2} - 101\,n^{2} x^{3} + 11\,n^{2} x^{4} + 26\,n^{3} + 17\,n^{3} x - 75\,n^{3} x^{2} + 70\,n^{3} x^{3} - 7\,n^{3} x^{4} - 6\,n^{4} - 31\,n^{4} x - 7\,n^{4} x^{3} - 2\,n^{5} - 17\,n^{5} x + 7\,n^{5} x^{2} + 4\,n^{6} + 7\,n^{6} x\right)\,P_{n-1}(x) + \left(120\,x^{2} - 144\,x^{3} + 60\,x^{4} - 6\,x^{5} + 96\,n x - 234\,n x^{2} + 268\,n x^{3} - 136\,n x^{4} + 14\,n x^{5} - 50\,n^{2} x - 132\,n^{2} x^{2} + 90\,n^{2} x^{3} + 18\,n^{2} x^{4} - 2\,n^{2} x^{5} - 80\,n^{3} x + 80\,n^{3} x^{2} - 36\,n^{3} x^{3} - 2\,n^{3} x^{4} + 22\,n^{4} x - 16\,n^{4} x^{2} + 2\,n^{4} x^{3} + 12\,n^{5} x + 2\,n^{5} x^{2}\right)\,P_{n-1}'(x) + \left(48\,x^{3} - 68\,x^{4} + 22\,x^{5} - 2\,x^{6} + 48\,n x^{2} - 148\,n x^{3} + 182\,n x^{4} - 57\,n x^{5} + 5\,n x^{6} - 28\,n^{2} x^{2} + 3\,n^{2} x^{3} - 19\,n^{2} x^{4} + 5\,n^{2} x^{5} - 40\,n^{3} x^{2} + 30\,n^{3} x^{3} - 5\,n^{3} x^{4} + 20\,n^{4} x^{2} - 5\,n^{4} x^{3}\right)\,P_{n-1}''(x)\)

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

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	1														3
3	5	6	1													12
4	11	28	12	1												52
5	21	104	90	20	1											236
6	36	315	500	220	30	1										1102
7	57	812	2178	1700	455	42	1									5245
8	85	1848	7812	10010	4655	840	56	1								25307
9	121	3816	24024	47432	35490	10976	1428	72	1							123360
10	166	7293	65340	188760	214032	104958	23184	2280	90	1						606104
11	221	13090	160875	652080	1067430	783216	271320	45000	3465	110	1					2996808
12	287	22308	364936	2005575	4555980	4775232	2450448	632016	81675	5060	132	1				14893650
13	365	36400	773058	5599880	17089930	24639120	17907120	6786648	1354815	140360	7150	156	1			74335003
14	456	57239	1545336	14410968	57517460	110576466	109745064	58491576	17045028	2713425	230516	9828	182	1		372343545
15	561	87192	2939300	34590920	176534358	441012572	580041462	419613480	170945775	39507468	5135130	364364	13195	210	1	1870785988

Technical details

Node id: n-fdebe687ef8435e8 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-793e5ca69e3a2597

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,    1;
   3:   5,    6,     1;
   4:  11,   28,    12,      1;
   5:  21,  104,    90,     20,      1;
   6:  36,  315,   500,    220,     30,      1;
   7:  57,  812,  2178,   1700,    455,     42,     1;
   8:  85, 1848,  7812,  10010,   4655,    840,    56,    1;
   9: 121, 3816, 24024,  47432,  35490,  10976,  1428,   72,  1;
  10: 166, 7293, 65340, 188760, 214032, 104958, 23184, 2280, 90, 1;

%F Formula

The row polynomials satisfy (4*n^6-14*n^5+12*n^4+26*n^3-16*n^2-12*n+(4*n^5+2*n^4-84*n^3+38*n^2+40*n)*x-(4*n^4-42*n^3+18*n^2+20*n)*x^2-(4*n^3-2*n^2-2*n)*x^3)*P(n,x) = (4*n^6-2*n^5-6*n^4+26*n^3-10*n^2-12*n+(7*n^6-17*n^5-31*n^4+17*n^3-16*n^2+40*n)*x+(7*n^5-75*n^3+140*n^2-72*n)*x^2-(7*n^4-70*n^3+101*n^2-38*n)*x^3-(7*n^3-11*n^2+4*n)*x^4)*P(n-1,x) + ((12*n^5+22*n^4-80*n^3-50*n^2+96*n)*x+(2*n^5-16*n^4+80*n^3-132*n^2-234*n+120)*x^2+(2*n^4-36*n^3+90*n^2+268*n-144)*x^3-(2*n^3-18*n^2+136*n-60)*x^4-(2*n^2-14*n+6)*x^5)*d/dx P(n-1,x) + ((20*n^4-40*n^3-28*n^2+48*n)*x^2-(5*n^4-30*n^3-3*n^2+148*n-48)*x^3-(5*n^3+19*n^2-182*n+68)*x^4+(5*n^2-57*n+22)*x^5+(5*n-2)*x^6)*d^2/dx^2 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[((4*n^6 - 2*n^5 - 6*n^4 + 26*n^3 - 10*n^2 - 12*n + (7*n^6 - 17*n^5 - 31*n^4 + 17*n^3 - 16*n^2 + 40*n)*x + (7*n^5 - 75*n^3 + 140*n^2 - 72*n)*x^2 - (7*n^4 - 70*n^3 + 101*n^2 - 38*n)*x^3 - (7*n^3 - 11*n^2 + 4*n)*x^4)*P[n - 1, x] + ((12*n^5 + 22*n^4 - 80*n^3 - 50*n^2 + 96*n)*x + (2*n^5 - 16*n^4 + 80*n^3 - 132*n^2 - 234*n + 120)*x^2 + (2*n^4 - 36*n^3 + 90*n^2 + 268*n - 144)*x^3 - (2*n^3 - 18*n^2 + 136*n - 60)*x^4 - (2*n^2 - 14*n + 6)*x^5)*D[P[n - 1, x], x] + ((20*n^4 - 40*n^3 - 28*n^2 + 48*n)*x^2 - (5*n^4 - 30*n^3 - 3*n^2 + 148*n - 48)*x^3 - (5*n^3 + 19*n^2 - 182*n + 68)*x^4 + (5*n^2 - 57*n + 22)*x^5 + (5*n - 2)*x^6)*D[P[n - 1, x], {x, 2}])/(4*n^6 - 14*n^5 + 12*n^4 + 26*n^3 - 16*n^2 - 12*n + (4*n^5 + 2*n^4 - 84*n^3 + 38*n^2 + 40*n)*x - (4*n^4 - 42*n^3 + 18*n^2 + 20*n)*x^2 - (4*n^3 - 2*n^2 - 2*n)*x^3)]];
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] = ((4*n^6 - 2*n^5 - 6*n^4 + 26*n^3 - 10*n^2 - 12*n + (7*n^6 - 17*n^5 - 31*n^4 + 17*n^3 - 16*n^2 + 40*n)*x + (7*n^5 - 75*n^3 + 140*n^2 - 72*n)*x^2 - (7*n^4 - 70*n^3 + 101*n^2 - 38*n)*x^3 - (7*n^3 - 11*n^2 + 4*n)*x^4)*v[i - 1] + ((12*n^5 + 22*n^4 - 80*n^3 - 50*n^2 + 96*n)*x + (2*n^5 - 16*n^4 + 80*n^3 - 132*n^2 - 234*n + 120)*x^2 + (2*n^4 - 36*n^3 + 90*n^2 + 268*n - 144)*x^3 - (2*n^3 - 18*n^2 + 136*n - 60)*x^4 - (2*n^2 - 14*n + 6)*x^5)*deriv(v[i - 1], x) + ((20*n^4 - 40*n^3 - 28*n^2 + 48*n)*x^2 - (5*n^4 - 30*n^3 - 3*n^2 + 148*n - 48)*x^3 - (5*n^3 + 19*n^2 - 182*n + 68)*x^4 + (5*n^2 - 57*n + 22)*x^5 + (5*n - 2)*x^6)*deriv(deriv(v[i - 1], x), x))/(4*n^6 - 14*n^5 + 12*n^4 + 26*n^3 - 16*n^2 - 12*n + (4*n^5 + 2*n^4 - 84*n^3 + 38*n^2 + 40*n)*x - (4*n^4 - 42*n^3 + 18*n^2 + 20*n)*x^2 - (4*n^3 - 2*n^2 - 2*n)*x^3)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from reversal of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.?
    
    Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    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
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Even row Pascal-square read by antidiagonals.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.?
    
    ↑ Pólya–Schur theorem on multiplier sequences (above)
    
    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
    
    ↑ Row-hypergeometric rows (multiplier sequences) (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