A197208 → D+1

nonnegative coefficients

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

Recurrence

\(\left(30 - 30\,x + 5\,x^{2} + 5\,x^{3} + 46\,n - 21\,n x + 21\,n x^{2} + 6\,n x^{3} + 33\,n^{2} + 27\,n^{2} x + 19\,n^{2} x^{2} + n^{2} x^{3} + 25\,n^{3} + 21\,n^{3} x + 3\,n^{3} x^{2} + 9\,n^{4} + 3\,n^{4} x + n^{5}\right)\,P_{n}(x) = \left(240 - 120\,x - 80\,x^{2} + 60\,x^{3} + 20\,x^{4} + 328\,n + 22\,n x + 3\,n x^{2} + 129\,n x^{3} + 30\,n x^{4} + 231\,n^{2} + 165\,n^{2} x + 164\,n^{2} x^{2} + 111\,n^{2} x^{3} + 10\,n^{2} x^{4} + 85\,n^{3} + 148\,n^{3} x + 135\,n^{3} x^{2} + 30\,n^{3} x^{3} + 15\,n^{4} + 63\,n^{4} x + 30\,n^{4} x^{2} + n^{5} + 10\,n^{5} x\right)\,P_{n-1}(x) + \left(192\,x + 43\,x^{2} + 94\,x^{3} - 18\,x^{4} - 15\,x^{5} + 258\,n x + 189\,n x^{2} + 36\,n x^{3} - 87\,n x^{4} - 15\,n x^{5} + 183\,n^{2} x + 14\,n^{2} x^{2} - 127\,n^{2} x^{3} - 45\,n^{2} x^{4} + 48\,n^{3} x - 63\,n^{3} x^{2} - 45\,n^{3} x^{3} + 3\,n^{4} x - 15\,n^{4} x^{2}\right)\,P_{n-1}'(x) + \left(29\,x^{2} + 35\,x^{3} + 30\,x^{4} + 30\,x^{5} + 6\,x^{6} + 39\,n x^{2} + 87\,n x^{3} + 66\,n x^{4} + 18\,n x^{5} + 28\,n^{2} x^{2} + 46\,n^{2} x^{3} + 18\,n^{2} x^{4} + 6\,n^{3} x^{2} + 6\,n^{3} x^{3}\right)\,P_{n-1}''(x)\)

\(P_{1}(x) = 150 + 180\,x + 30\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	150	180	30								360
2	660	1800	780	60							3300
3	2205	12600	11550	2520	105						28980
4	6048	64680	117600	52920	6720	168					248136
5	14364	261072	864360	740880	194040	15624	252				2090592
6	30600	876960	4868640	7408800	3598560	604800	32760	360			17421480
7	59895	2554200	22203720	55883520	47151720	14469840	1663200	63360	495		144049950
8	109560	6642900	85595400	334802160	461039040	241496640	50311800	4138200	114840	660	1184251200

All 15 rows

Technical details

Node id: n-3725039f3886dd63 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-379b71f8e1339420

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:    150,      180,        30;
   2:    660,     1800,       780,         60;
   3:   2205,    12600,     11550,       2520,         105;
   4:   6048,    64680,    117600,      52920,        6720,         168;
   5:  14364,   261072,    864360,     740880,      194040,       15624,         252;
   6:  30600,   876960,   4868640,    7408800,     3598560,      604800,       32760,        360;
   7:  59895,  2554200,  22203720,   55883520,    47151720,    14469840,     1663200,      63360,       495;
   8: 109560,  6642900,  85595400,  334802160,   461039040,   241496640,    50311800,    4138200,    114840,      660;
   9: 189618, 15761460, 288094950, 1663164360,  3531888360,  2982483504,  1046485440,  155727000,   9485190,   197340,    858;
  10: 313404, 34666632, 867446580, 7085578500, 22107004920, 28569052512, 15985065096, 3967923960, 438017580, 20300280, 324324, 1092;

%F Formula

The row polynomials satisfy (n^5+9*n^4+25*n^3+33*n^2+46*n+30+(3*n^4+21*n^3+27*n^2-21*n-30)*x+(3*n^3+19*n^2+21*n+5)*x^2+(n^2+6*n+5)*x^3)*P(n,x) = (n^5+15*n^4+85*n^3+231*n^2+328*n+240+(10*n^5+63*n^4+148*n^3+165*n^2+22*n-120)*x+(30*n^4+135*n^3+164*n^2+3*n-80)*x^2+(30*n^3+111*n^2+129*n+60)*x^3+(10*n^2+30*n+20)*x^4)*P(n-1,x) + ((3*n^4+48*n^3+183*n^2+258*n+192)*x-(15*n^4+63*n^3-14*n^2-189*n-43)*x^2-(45*n^3+127*n^2-36*n-94)*x^3-(45*n^2+87*n+18)*x^4-(15*n+15)*x^5)*d/dx P(n-1,x) + ((6*n^3+28*n^2+39*n+29)*x^2+(6*n^3+46*n^2+87*n+35)*x^3+(18*n^2+66*n+30)*x^4+(18*n+30)*x^5+6*x^6)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 150 + 180*x + 30*x^2.

%t Mathematica

P[1, x_] := 150 + 180*x + 30*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^5 + 15*n^4 + 85*n^3 + 231*n^2 + 328*n + 240 + (10*n^5 + 63*n^4 + 148*n^3 + 165*n^2 + 22*n - 120)*x + (30*n^4 + 135*n^3 + 164*n^2 + 3*n - 80)*x^2 + (30*n^3 + 111*n^2 + 129*n + 60)*x^3 + (10*n^2 + 30*n + 20)*x^4)*P[n - 1, x] + ((3*n^4 + 48*n^3 + 183*n^2 + 258*n + 192)*x - (15*n^4 + 63*n^3 - 14*n^2 - 189*n - 43)*x^2 - (45*n^3 + 127*n^2 - 36*n - 94)*x^3 - (45*n^2 + 87*n + 18)*x^4 - (15*n + 15)*x^5)*D[P[n - 1, x], x] + ((6*n^3 + 28*n^2 + 39*n + 29)*x^2 + (6*n^3 + 46*n^2 + 87*n + 35)*x^3 + (18*n^2 + 66*n + 30)*x^4 + (18*n + 30)*x^5 + 6*x^6)*D[P[n - 1, x], {x, 2}])/(n^5 + 9*n^4 + 25*n^3 + 33*n^2 + 46*n + 30 + (3*n^4 + 21*n^3 + 27*n^2 - 21*n - 30)*x + (3*n^3 + 19*n^2 + 21*n + 5)*x^2 + (n^2 + 6*n + 5)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 150 + 180*x + 30*x^2; for(n = 2, N, my(i = n); v[i] = ((n^5 + 15*n^4 + 85*n^3 + 231*n^2 + 328*n + 240 + (10*n^5 + 63*n^4 + 148*n^3 + 165*n^2 + 22*n - 120)*x + (30*n^4 + 135*n^3 + 164*n^2 + 3*n - 80)*x^2 + (30*n^3 + 111*n^2 + 129*n + 60)*x^3 + (10*n^2 + 30*n + 20)*x^4)*v[i - 1] + ((3*n^4 + 48*n^3 + 183*n^2 + 258*n + 192)*x - (15*n^4 + 63*n^3 - 14*n^2 - 189*n - 43)*x^2 - (45*n^3 + 127*n^2 - 36*n - 94)*x^3 - (45*n^2 + 87*n + 18)*x^4 - (15*n + 15)*x^5)*deriv(v[i - 1], x) + ((6*n^3 + 28*n^2 + 39*n + 29)*x^2 + (6*n^3 + 46*n^2 + 87*n + 35)*x^3 + (18*n^2 + 66*n + 30)*x^4 + (18*n + 30)*x^5 + 6*x^6)*deriv(deriv(v[i - 1], x), x))/(n^5 + 9*n^4 + 25*n^3 + 33*n^2 + 46*n + 30 + (3*n^4 + 21*n^3 + 27*n^2 - 21*n - 30)*x + (3*n^3 + 19*n^2 + 21*n + 5)*x^2 + (n^2 + 6*n + 5)*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 Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,…?
    
    Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
    
    from Triangle of Dyck paths counted by number of long interior inclines.?
    
    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
    
    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 of Dyck paths counted by number of long interior inclines.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of Dyck paths counted by number of long interior inclines.?
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of Dyck paths counted by number of long interior inclines.?
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (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
  - ↑ Pólya–Schur theorem on multiplier sequences (above)
  - ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

A056939 → D+1

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