A062145 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-16 + 12\,x - 2\,x^{2} + 16\,n - 10\,n x + n x^{2} + 2\,n^{2} - 4\,n^{2} x + n^{2} x^{2} - 3\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(20 - 40\,x + 24\,x^{2} - 4\,x^{3} + 12\,n - 20\,n x + 6\,n x^{2} - 2\,n x^{3} + n^{2} + 22\,n^{2} x - 18\,n^{2} x^{2} + 3\,n^{2} x^{3} + 2\,n^{3} - 10\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(3\,x - 7\,x^{2} + 5\,x^{3} - x^{4} + 8\,n x - 14\,n x^{2} + 8\,n x^{3} - 2\,n x^{4} - 4\,n^{2} x + 8\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	5	1														6
3	20	12	1													33
4	65	81	21	1												168
5	175	392	210	32	1											810
6	406	1470	1400	440	45	1										3762
7	840	4536	6930	3840	810	60	1									17017
8	1590	12054	27342	24150	8925	1365	77	1								75504
9	2805	28512	90552	118272	69300	18480	2156	96	1							330174
10	4675	61479	261360	476784	411642	173250	35112	3240	117	1						1427660
11	7436	122980	675675	1647360	1981980	1225224	390390	62400	4680	140	1					6118266
12	11375	231231	1596595	5025735	8069490	6936930	3237234	810810	105105	6545	165	1				26031216
13	16835	412776	3501498	13842400	28667925	32844240	21273252	7783776	1576575	169400	8910	192	1			110097780
14	24220	705068	7211568	35008688	90998050	134349930	115823136	58714656	17333316	2902900	263120	11856	221	1		463346730
15	34000	1159536	14077700	82374656	262810548	486686200	540221682	364066560	148660512	36212176	5105100	396032	15470	252	1	1941820425

Technical details

Node id: n-cd8f32920984c05e · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-609112ad990f7757

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:    5,     1;
   3:   20,    12,      1;
   4:   65,    81,     21,      1;
   5:  175,   392,    210,     32,      1;
   6:  406,  1470,   1400,    440,     45,      1;
   7:  840,  4536,   6930,   3840,    810,     60,     1;
   8: 1590, 12054,  27342,  24150,   8925,   1365,    77,    1;
   9: 2805, 28512,  90552, 118272,  69300,  18480,  2156,   96,   1;
  10: 4675, 61479, 261360, 476784, 411642, 173250, 35112, 3240, 117, 1;

%F Formula

The row polynomials satisfy (n^4-3*n^3+2*n^2+16*n-16+(2*n^3-4*n^2-10*n+12)*x+(n^2+n-2)*x^2)*P(n,x) = (n^4+2*n^3+n^2+12*n+20+(3*n^4-10*n^3+22*n^2-20*n-40)*x+(6*n^3-18*n^2+6*n+24)*x^2+(3*n^2-2*n-4)*x^3)*P(n-1,x) + ((2*n^3-4*n^2+8*n+3)*x-(2*n^3-8*n^2+14*n+7)*x^2-(4*n^2-8*n-5)*x^3-(2*n+1)*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^4 + 2*n^3 + n^2 + 12*n + 20 + (3*n^4 - 10*n^3 + 22*n^2 - 20*n - 40)*x + (6*n^3 - 18*n^2 + 6*n + 24)*x^2 + (3*n^2 - 2*n - 4)*x^3)*P[n - 1, x] + ((2*n^3 - 4*n^2 + 8*n + 3)*x - (2*n^3 - 8*n^2 + 14*n + 7)*x^2 - (4*n^2 - 8*n - 5)*x^3 - (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 3*n^3 + 2*n^2 + 16*n - 16 + (2*n^3 - 4*n^2 - 10*n + 12)*x + (n^2 + n - 2)*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^4 + 2*n^3 + n^2 + 12*n + 20 + (3*n^4 - 10*n^3 + 22*n^2 - 20*n - 40)*x + (6*n^3 - 18*n^2 + 6*n + 24)*x^2 + (3*n^2 - 2*n - 4)*x^3)*v[i - 1] + ((2*n^3 - 4*n^2 + 8*n + 3)*x - (2*n^3 - 8*n^2 + 14*n + 7)*x^2 - (4*n^2 - 8*n - 5)*x^3 - (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 3*n^3 + 2*n^2 + 16*n - 16 + (2*n^3 - 4*n^2 - 10*n + 12)*x + (n^2 + n - 2)*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 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by 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) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n -…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n -…?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ 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
[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.