A142462 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-20 + 14\,x + 7\,n\right)\,P_{n}(x) = \left(104 - 723\,x + 5312\,x^{2} - 6195\,x^{3} + 1274\,x^{4} - 56\,n + 511\,n x - 6342\,n x^{2} + 5194\,n x^{3} - 686\,n x^{4} - 98\,n^{2} x + 2597\,n^{2} x^{2} - 1029\,n^{2} x^{3} - 343\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(91\,x - 2009\,x^{2} + 4515\,x^{3} - 3283\,x^{4} + 686\,x^{5} - 49\,n x + 1666\,n x^{2} - 2646\,n x^{3} + 1029\,n x^{4} - 343\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-173 + 786\,x - 651\,x^{2} + 98\,x^{3} + 63\,n - 511\,n x + 245\,n x^{2} + 98\,n^{2} x\right)\,P_{n-1}(x) + \left(-140\,x + 238\,x^{2} - 98\,x^{3} + 49\,n x - 49\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	17	18	1						36
3	144	429	146	1					720
4	1167	9746	7788	1170	1				19872
5	9358	189361	360008	127430	9362	1			695520
6	74893	3317658	13501647	10445620	1995843	74898	1		29410560
7	599180	54632893	429442986	670710215	269226328	30611931	599186	1	1455822720

All 15 rows

Technical details

Node id: n-87a630b6538ab782 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-49ff540d672629c6

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:       17,          18,            1;
  3:      144,         429,          146,             1;
  4:     1167,        9746,         7788,          1170,             1;
  5:     9358,      189361,       360008,        127430,          9362,            1;
  6:    74893,     3317658,     13501647,      10445620,       1995843,        74898,            1;
  7:   599180,    54632893,    429442986,     670710215,     269226328,     30611931,       599186,          1;
  8:  4793483,   866828586,  12202212946,   35044685774,   27398897484,   6509837110,    464571926,    4793490,        1;
  9: 38347914, 13448222673, 321272500256, 1572380898508, 2201287566324, 996016386806, 151790415952, 7011720684, 38347922, 1;

%F Formula

The row polynomials satisfy (7*n-20+14*x)*P(n,x) = -(56*n-104+(98*n^2-511*n+723)*x+(343*n^3-2597*n^2+6342*n-5312)*x^2+(1029*n^2-5194*n+6195)*x^3+(686*n-1274)*x^4)*P(n-2,x) - ((49*n-91)*x+(343*n^2-1666*n+2009)*x^2-(343*n^2-2646*n+4515)*x^3-(1029*n-3283)*x^4-686*x^5)*d/dx P(n-2,x) + (63*n-173+(98*n^2-511*n+786)*x+(245*n-651)*x^2+98*x^3)*P(n-1,x) + ((49*n-140)*x-(49*n-238)*x^2-98*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[(-(56*n - 104 + (98*n^2 - 511*n + 723)*x + (343*n^3 - 2597*n^2 + 6342*n - 5312)*x^2 + (1029*n^2 - 5194*n + 6195)*x^3 + (686*n - 1274)*x^4)*P[n - 2, x] - ((49*n - 91)*x + (343*n^2 - 1666*n + 2009)*x^2 - (343*n^2 - 2646*n + 4515)*x^3 - (1029*n - 3283)*x^4 - 686*x^5)*D[P[n - 2, x], x] + (63*n - 173 + (98*n^2 - 511*n + 786)*x + (245*n - 651)*x^2 + 98*x^3)*P[n - 1, x] + ((49*n - 140)*x - (49*n - 238)*x^2 - 98*x^3)*D[P[n - 1, x], x])/(7*n - 20 + 14*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] = (-(56*n - 104 + (98*n^2 - 511*n + 723)*x + (343*n^3 - 2597*n^2 + 6342*n - 5312)*x^2 + (1029*n^2 - 5194*n + 6195)*x^3 + (686*n - 1274)*x^4)*v[i - 2] - ((49*n - 91)*x + (343*n^2 - 1666*n + 2009)*x^2 - (343*n^2 - 2646*n + 4515)*x^3 - (1029*n - 3283)*x^4 - 686*x^5)*deriv(v[i - 2], x) + (63*n - 173 + (98*n^2 - 511*n + 786)*x + (245*n - 651)*x^2 + 98*x^3)*v[i - 1] + ((49*n - 140)*x - (49*n - 238)*x^2 - 98*x^3)*deriv(v[i - 1], x))/(7*n - 20 + 14*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) + (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) (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) + (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.