A257614 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-13 + 10\,x + 5\,n\right)\,P_{n}(x) = \left(112 - 636\,x + 1852\,x^{2} - 1860\,x^{3} + 400\,x^{4} - 70\,n + 470\,n x - 2010\,n x^{2} + 1700\,n x^{3} - 250\,n x^{4} - 100\,n^{2} x + 850\,n^{2} x^{2} - 375\,n^{2} x^{3} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(80\,x - 700\,x^{2} + 1470\,x^{3} - 1100\,x^{4} + 250\,x^{5} - 50\,n x + 575\,n x^{2} - 900\,n x^{3} + 375\,n x^{4} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-107 + 363\,x - 300\,x^{2} + 50\,x^{3} + 45\,n - 235\,n x + 125\,n x^{2} + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-65\,x + 115\,x^{2} - 50\,x^{3} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 4 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	2							6
2	32	36	4						72
3	252	732	268	8					1260
4	1860	13556	11388	1908	16				28728
5	13292	216500	406480	154660	13420	32			804384
6	93748	3116004	11876400	9499720	1979580	94068	64		26659584
7	657964	41984612	296609028	459909400	195298820	24610404	658732	128	1019729088

All 15 rows

Technical details

Node id: n-358c98927e7dd542 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-103989f75d6354bd

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 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        4,          2;
  2:       32,         36,            4;
  3:      252,        732,          268,            8;
  4:     1860,      13556,        11388,         1908,           16;
  5:    13292,     216500,       406480,       154660,        13420,           32;
  6:    93748,    3116004,     11876400,      9499720,      1979580,        94068,          64;
  7:   657964,   41984612,    296609028,    459909400,    195298820,     24610404,      658732,        128;
  8:  4609844,  542632404,   6633821716,  18310414316,  14655219900,   3735671884,   301278524,    4611636,      256;
  9: 32278380, 6829661172, 137425562056, 629962633916, 877930515672, 414375493420, 68466465032, 3656914596, 32282476, 512;

%F Formula

The row polynomials satisfy (5*n-13+10*x)*P(n,x) = -(70*n-112+(100*n^2-470*n+636)*x+(125*n^3-850*n^2+2010*n-1852)*x^2+(375*n^2-1700*n+1860)*x^3+(250*n-400)*x^4)*P(n-2,x) - ((50*n-80)*x+(125*n^2-575*n+700)*x^2-(125*n^2-900*n+1470)*x^3-(375*n-1100)*x^4-250*x^5)*d/dx P(n-2,x) + (45*n-107+(50*n^2-235*n+363)*x+(125*n-300)*x^2+50*x^3)*P(n-1,x) + ((25*n-65)*x-(25*n-115)*x^2-50*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(70*n - 112 + (100*n^2 - 470*n + 636)*x + (125*n^3 - 850*n^2 + 2010*n - 1852)*x^2 + (375*n^2 - 1700*n + 1860)*x^3 + (250*n - 400)*x^4)*P[n - 2, x] - ((50*n - 80)*x + (125*n^2 - 575*n + 700)*x^2 - (125*n^2 - 900*n + 1470)*x^3 - (375*n - 1100)*x^4 - 250*x^5)*D[P[n - 2, x], x] + (45*n - 107 + (50*n^2 - 235*n + 363)*x + (125*n - 300)*x^2 + 50*x^3)*P[n - 1, x] + ((25*n - 65)*x - (25*n - 115)*x^2 - 50*x^3)*D[P[n - 1, x], x])/(5*n - 13 + 10*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(70*n - 112 + (100*n^2 - 470*n + 636)*x + (125*n^3 - 850*n^2 + 2010*n - 1852)*x^2 + (375*n^2 - 1700*n + 1860)*x^3 + (250*n - 400)*x^4)*v[i - 2] - ((50*n - 80)*x + (125*n^2 - 575*n + 700)*x^2 - (125*n^2 - 900*n + 1470)*x^3 - (375*n - 1100)*x^4 - 250*x^5)*deriv(v[i - 2], x) + (45*n - 107 + (50*n^2 - 235*n + 363)*x + (125*n - 300)*x^2 + 50*x^3)*v[i - 1] + ((25*n - 65)*x - (25*n - 115)*x^2 - 50*x^3)*deriv(v[i - 1], x))/(5*n - 13 + 10*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

References

[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.