A142461 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-17 + 12\,x + 6\,n\right)\,P_{n}(x) = \left(77 - 524\,x + 3329\,x^{2} - 3834\,x^{3} + 792\,x^{4} - 42\,n + 372\,n x - 3942\,n x^{2} + 3240\,n x^{3} - 432\,n x^{4} - 72\,n^{2} x + 1620\,n^{2} x^{2} - 648\,n^{2} x^{3} - 216\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(66\,x - 1260\,x^{2} + 2814\,x^{3} - 2052\,x^{4} + 432\,x^{5} - 36\,n x + 1044\,n x^{2} - 1656\,n x^{3} + 648\,n x^{4} - 216\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-130 + 572\,x - 474\,x^{2} + 72\,x^{3} + 48\,n - 372\,n x + 180\,n x^{2} + 72\,n^{2} x\right)\,P_{n-1}(x) + \left(-102\,x + 174\,x^{2} - 72\,x^{3} + 36\,n x - 36\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	15	16	1						32
3	112	333	114	1					560
4	797	6568	5274	800	1				13440
5	5598	110881	210568	75030	5602	1			407680
6	39211	1687752	6825459	5297480	1020321	39216	1		14909440
7	274508	24136573	187773930	293071495	118269464	13578075	274514	1	637378560

All 15 rows

Technical details

Node id: n-3277f0650054b6bf · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d5eaf77963b926a6

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:       15,         16,            1;
  3:      112,        333,          114,            1;
  4:      797,       6568,         5274,          800,            1;
  5:     5598,     110881,       210568,        75030,         5602,            1;
  6:    39211,    1687752,      6825459,      5297480,      1020321,        39216,           1;
  7:   274508,   24136573,    187773930,    293071495,    118269464,     13578075,      274514,          1;
  8:  1921593,  332441968,   4615861300,  13206393504,  10348383470,   2475094832,   178711332,    1921600,        1;
  9: 13451194, 4475472993, 105136312896, 511331441628, 715544511444, 325068022566, 49918843632, 2338620444, 13451202, 1;

%F Formula

The row polynomials satisfy (6*n-17+12*x)*P(n,x) = -(42*n-77+(72*n^2-372*n+524)*x+(216*n^3-1620*n^2+3942*n-3329)*x^2+(648*n^2-3240*n+3834)*x^3+(432*n-792)*x^4)*P(n-2,x) - ((36*n-66)*x+(216*n^2-1044*n+1260)*x^2-(216*n^2-1656*n+2814)*x^3-(648*n-2052)*x^4-432*x^5)*d/dx P(n-2,x) + (48*n-130+(72*n^2-372*n+572)*x+(180*n-474)*x^2+72*x^3)*P(n-1,x) + ((36*n-102)*x-(36*n-174)*x^2-72*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[(-(42*n - 77 + (72*n^2 - 372*n + 524)*x + (216*n^3 - 1620*n^2 + 3942*n - 3329)*x^2 + (648*n^2 - 3240*n + 3834)*x^3 + (432*n - 792)*x^4)*P[n - 2, x] - ((36*n - 66)*x + (216*n^2 - 1044*n + 1260)*x^2 - (216*n^2 - 1656*n + 2814)*x^3 - (648*n - 2052)*x^4 - 432*x^5)*D[P[n - 2, x], x] + (48*n - 130 + (72*n^2 - 372*n + 572)*x + (180*n - 474)*x^2 + 72*x^3)*P[n - 1, x] + ((36*n - 102)*x - (36*n - 174)*x^2 - 72*x^3)*D[P[n - 1, x], x])/(6*n - 17 + 12*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] = (-(42*n - 77 + (72*n^2 - 372*n + 524)*x + (216*n^3 - 1620*n^2 + 3942*n - 3329)*x^2 + (648*n^2 - 3240*n + 3834)*x^3 + (432*n - 792)*x^4)*v[i - 2] - ((36*n - 66)*x + (216*n^2 - 1044*n + 1260)*x^2 - (216*n^2 - 1656*n + 2814)*x^3 - (648*n - 2052)*x^4 - 432*x^5)*deriv(v[i - 2], x) + (48*n - 130 + (72*n^2 - 372*n + 572)*x + (180*n - 474)*x^2 + 72*x^3)*v[i - 1] + ((36*n - 102)*x - (36*n - 174)*x^2 - 72*x^3)*deriv(v[i - 1], x))/(6*n - 17 + 12*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) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) +…?
  - 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) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) +…?
    
    ↑ 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.