A144436 → D+1

nonnegative coefficients

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

Recurrence

\(\left(8\right)\,P_{n}(x) = \left(-64 - 224\,x - 224\,x^{2} + 32\,n + 16\,n x + 32\,n x^{3} + 24\,n^{2} + 12\,n^{2} x - 12\,n^{2} x^{3}\right)\,P_{n-4}(x) + \left(-32\,x - 96\,x^{2} + 16\,n x + 32\,n x^{3} + 12\,n^{2} x - 12\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(40 + 152\,x + 192\,x^{2} - 64\,x^{3} - 36\,n - 104\,n x - 80\,n x^{2} - 16\,n x^{3} - 7\,n^{2} + 4\,n^{2} x - 3\,n^{2} x^{2} + 14\,n^{2} x^{3} + 6\,n^{3} + 3\,n^{3} x - 3\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(-24 - 32\,x + 128\,x^{2} + 24\,x^{3} + 12\,n - 28\,n x - 20\,n x^{2} - 24\,n x^{3} - 8\,n x^{4} + 9\,n^{2} - 11\,n^{2} x - 6\,n^{2} x^{2} + 17\,n^{2} x^{3} + 3\,n^{2} x^{4} + 3\,n^{3} x - 3\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(-8\,x - 16\,x^{2} + 24\,x^{3} + 4\,n x - 4\,n x^{2} + 8\,n x^{3} - 8\,n x^{4} + 3\,n^{2} x - 3\,n^{2} x^{2} - 3\,n^{2} x^{3} + 3\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(108\,x - 36\,x^{2} + 16\,x^{3} - 16\,n + 38\,n x + 20\,n x^{2} - 16\,n x^{3} - 3\,n^{2} - 12\,n^{2} x - 5\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(8 + 44\,x - 32\,x^{2} - 4\,x^{3} + 16\,x^{4} - 4\,n - 6\,n x - 2\,n x^{2} - 3\,n^{2} + 3\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(8\,x^{2} - 16\,x^{3} + 8\,x^{4}\right)\,P_{n-2}''(x) + \left(20 - 44\,x + 16\,x^{2} + 6\,n + 16\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 9 + 10\,x + x^{2}\)

\(P_{3}(x) = 24 + 69\,x + 26\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	9	10	1						20
3	24	69	26	1					120
4	55	394	332	58	1				840
5	118	1753	3272	1286	122	1			6552
6	245	6754	24927	20244	4475	250	1		56896
7	500	23893	157026	242215	106928	14691	506	1	545760

All 15 rows

Technical details

Node id: n-3bd200d5c4e74de0 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a0a2dc4146e606e9

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:    9,     10,       1;
  3:   24,     69,      26,        1;
  4:   55,    394,     332,       58,        1;
  5:  118,   1753,    3272,     1286,      122,        1;
  6:  245,   6754,   24927,    20244,     4475,      250,       1;
  7:  500,  23893,  157026,   242215,   106928,    14691,     506,      1;
  8: 1011,  80130,  870482,  2330278,  1898196,   514334,   46630,   1018,    1;
  9: 2034, 259545, 4423568, 19120588, 26533476, 13011870, 2332896, 145020, 2042, 1;

%F Formula

The row polynomials satisfy 8*P(n,x) = (24*n^2+32*n-64+(12*n^2+16*n-224)*x-224*x^2-(12*n^2-32*n)*x^3)*P(n-4,x) + ((12*n^2+16*n-32)*x-96*x^2-(12*n^2-32*n)*x^3)*d/dx P(n-4,x) + (6*n^3-7*n^2-36*n+40+(3*n^3+4*n^2-104*n+152)*x-(3*n^2+80*n-192)*x^2-(3*n^3-14*n^2+16*n+64)*x^3)*P(n-3,x) + (9*n^2+12*n-24+(3*n^3-11*n^2-28*n-32)*x-(6*n^2+20*n-128)*x^2-(3*n^3-17*n^2+24*n-24)*x^3+(3*n^2-8*n)*x^4)*d/dx P(n-3,x) + ((3*n^2+4*n-8)*x-(3*n^2+4*n+16)*x^2-(3*n^2-8*n-24)*x^3+(3*n^2-8*n)*x^4)*d^2/dx^2 P(n-3,x) - (3*n^2+16*n+(12*n^2-38*n-108)*x+(5*n^2-20*n+36)*x^2+(16*n-16)*x^3)*P(n-2,x) - (3*n^2+4*n-8+(6*n-44)*x-(3*n^2-2*n-32)*x^2+4*x^3-16*x^4)*d/dx P(n-2,x) + (8*x^2-16*x^3+8*x^4)*d^2/dx^2 P(n-2,x) + (6*n+20+(16*n-44)*x+16*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 9 + 10*x + x^2, P(3,x) = 24 + 69*x + 26*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 9 + 10*x + x^2;
P[3, x_] := 24 + 69*x + 26*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((24*n^2 + 32*n - 64 + (12*n^2 + 16*n - 224)*x - 224*x^2 - (12*n^2 - 32*n)*x^3)*P[n - 4, x] + ((12*n^2 + 16*n - 32)*x - 96*x^2 - (12*n^2 - 32*n)*x^3)*D[P[n - 4, x], x] + (6*n^3 - 7*n^2 - 36*n + 40 + (3*n^3 + 4*n^2 - 104*n + 152)*x - (3*n^2 + 80*n - 192)*x^2 - (3*n^3 - 14*n^2 + 16*n + 64)*x^3)*P[n - 3, x] + (9*n^2 + 12*n - 24 + (3*n^3 - 11*n^2 - 28*n - 32)*x - (6*n^2 + 20*n - 128)*x^2 - (3*n^3 - 17*n^2 + 24*n - 24)*x^3 + (3*n^2 - 8*n)*x^4)*D[P[n - 3, x], x] + ((3*n^2 + 4*n - 8)*x - (3*n^2 + 4*n + 16)*x^2 - (3*n^2 - 8*n - 24)*x^3 + (3*n^2 - 8*n)*x^4)*D[P[n - 3, x], {x, 2}] - (3*n^2 + 16*n + (12*n^2 - 38*n - 108)*x + (5*n^2 - 20*n + 36)*x^2 + (16*n - 16)*x^3)*P[n - 2, x] - (3*n^2 + 4*n - 8 + (6*n - 44)*x - (3*n^2 - 2*n - 32)*x^2 + 4*x^3 - 16*x^4)*D[P[n - 2, x], x] + (8*x^2 - 16*x^3 + 8*x^4)*D[P[n - 2, x], {x, 2}] + (6*n + 20 + (16*n - 44)*x + 16*x^2)*P[n - 1, x])/(8)]];
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; v[3] = 9 + 10*x + x^2; v[4] = 24 + 69*x + 26*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((24*n^2 + 32*n - 64 + (12*n^2 + 16*n - 224)*x - 224*x^2 - (12*n^2 - 32*n)*x^3)*v[i - 4] + ((12*n^2 + 16*n - 32)*x - 96*x^2 - (12*n^2 - 32*n)*x^3)*deriv(v[i - 4], x) + (6*n^3 - 7*n^2 - 36*n + 40 + (3*n^3 + 4*n^2 - 104*n + 152)*x - (3*n^2 + 80*n - 192)*x^2 - (3*n^3 - 14*n^2 + 16*n + 64)*x^3)*v[i - 3] + (9*n^2 + 12*n - 24 + (3*n^3 - 11*n^2 - 28*n - 32)*x - (6*n^2 + 20*n - 128)*x^2 - (3*n^3 - 17*n^2 + 24*n - 24)*x^3 + (3*n^2 - 8*n)*x^4)*deriv(v[i - 3], x) + ((3*n^2 + 4*n - 8)*x - (3*n^2 + 4*n + 16)*x^2 - (3*n^2 - 8*n - 24)*x^3 + (3*n^2 - 8*n)*x^4)*deriv(deriv(v[i - 3], x), x) - (3*n^2 + 16*n + (12*n^2 - 38*n - 108)*x + (5*n^2 - 20*n + 36)*x^2 + (16*n - 16)*x^3)*v[i - 2] - (3*n^2 + 4*n - 8 + (6*n - 44)*x - (3*n^2 - 2*n - 32)*x^2 + 4*x^3 - 16*x^4)*deriv(v[i - 2], x) + (8*x^2 - 16*x^3 + 8*x^4)*deriv(deriv(v[i - 2], x), x) + (6*n + 20 + (16*n - 44)*x + 16*x^2)*v[i - 1])/(8)); 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 T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (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.