A167884 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-23 + 16\,x + 8\,n\right)\,P_{n}(x) = \left(135 - 954\,x + 7959\,x^{2} - 9368\,x^{3} + 1920\,x^{4} - 72\,n + 672\,n x - 9560\,n x^{2} + 7808\,n x^{3} - 1024\,n x^{4} - 128\,n^{2} x + 3904\,n^{2} x^{2} - 1536\,n^{2} x^{3} - 512\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(120\,x - 3008\,x^{2} + 6792\,x^{3} - 4928\,x^{4} + 1024\,x^{5} - 64\,n x + 2496\,n x^{2} - 3968\,n x^{3} + 1536\,n x^{4} - 512\,n^{2} x^{2} + 512\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-222 + 1034\,x - 856\,x^{2} + 128\,x^{3} + 80\,n - 672\,n x + 320\,n x^{2} + 128\,n^{2} x\right)\,P_{n-1}(x) + \left(-184\,x + 312\,x^{2} - 128\,x^{3} + 64\,n x - 64\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	19	20	1						40
3	180	537	182	1					900
4	1637	13808	10994	1640	1				28080
5	14758	303481	577448	203390	14762	1			1113840
6	132855	6015124	24595435	18982640	3605405	132860	1		53464320
7	1195736	112088353	887976174	1387549775	554741500	62620719	1195742	1	3007368000

All 15 rows

Technical details

Node id: n-2bea1d1250f6acb3 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-21bb6cb63f1d0048

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:        2,           1;
  2:       19,          20,            1;
  3:      180,         537,          182,             1;
  4:     1637,       13808,        10994,          1640,             1;
  5:    14758,      303481,       577448,        203390,         14762,             1;
  6:   132855,     6015124,     24595435,      18982640,       3605405,        132860,            1;
  7:  1195736,   112088353,    887976174,    1387549775,     554741500,      62620719,      1195742,           1;
  8: 10761673,  2013118208,  28634320900,   82473675664,   64370894750,   15218117152,   1076509972,    10761680,        1;
  9: 96855114, 35363746833, 855635217056, 4207612448908, 5892417787764, 2658073173686, 402768044752, 18408286764, 96855122, 1;

%F Formula

The row polynomials satisfy (8*n-23+16*x)*P(n,x) = -(72*n-135+(128*n^2-672*n+954)*x+(512*n^3-3904*n^2+9560*n-7959)*x^2+(1536*n^2-7808*n+9368)*x^3+(1024*n-1920)*x^4)*P(n-2,x) - ((64*n-120)*x+(512*n^2-2496*n+3008)*x^2-(512*n^2-3968*n+6792)*x^3-(1536*n-4928)*x^4-1024*x^5)*d/dx P(n-2,x) + (80*n-222+(128*n^2-672*n+1034)*x+(320*n-856)*x^2+128*x^3)*P(n-1,x) + ((64*n-184)*x-(64*n-312)*x^2-128*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(72*n - 135 + (128*n^2 - 672*n + 954)*x + (512*n^3 - 3904*n^2 + 9560*n - 7959)*x^2 + (1536*n^2 - 7808*n + 9368)*x^3 + (1024*n - 1920)*x^4)*P[n - 2, x] - ((64*n - 120)*x + (512*n^2 - 2496*n + 3008)*x^2 - (512*n^2 - 3968*n + 6792)*x^3 - (1536*n - 4928)*x^4 - 1024*x^5)*D[P[n - 2, x], x] + (80*n - 222 + (128*n^2 - 672*n + 1034)*x + (320*n - 856)*x^2 + 128*x^3)*P[n - 1, x] + ((64*n - 184)*x - (64*n - 312)*x^2 - 128*x^3)*D[P[n - 1, x], x])/(8*n - 23 + 16*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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(72*n - 135 + (128*n^2 - 672*n + 954)*x + (512*n^3 - 3904*n^2 + 9560*n - 7959)*x^2 + (1536*n^2 - 7808*n + 9368)*x^3 + (1024*n - 1920)*x^4)*v[i - 2] - ((64*n - 120)*x + (512*n^2 - 2496*n + 3008)*x^2 - (512*n^2 - 3968*n + 6792)*x^3 - (1536*n - 4928)*x^4 - 1024*x^5)*deriv(v[i - 2], x) + (80*n - 222 + (128*n^2 - 672*n + 1034)*x + (320*n - 856)*x^2 + 128*x^3)*v[i - 1] + ((64*n - 184)*x - (64*n - 312)*x^2 - 128*x^3)*deriv(v[i - 1], x))/(8*n - 23 + 16*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) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n…?
  - 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) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n…?
    
    ↑ 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.