A104978 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 2\,n x^{2} + 4\,n^{2} + 2\,n^{2} x - 4\,n^{2} x^{2} - 4\,n^{3} + 4\,n^{3} x + 8\,n^{4}\right)\,P_{n}(x) = \left(1 - 4\,x^{2} - 3\,x^{3} - 10\,n + 4\,n x + 27\,n x^{2} + 15\,n x^{3} + 13\,n^{2} - 18\,n^{2} x - 47\,n^{2} x^{2} - 18\,n^{2} x^{3} - 8\,n^{3} + 2\,n^{3} x + 18\,n^{3} x^{2} + 28\,n^{4} + 36\,n^{4} x\right)\,P_{n-1}(x) + \left(-2 - x + 8\,x^{2} + 10\,x^{3} + 3\,x^{4} + 4\,n + 2\,n x - 24\,n x^{2} - 31\,n x^{3} - 9\,n x^{4} - 4\,n^{2} - 9\,n^{2} x + 4\,n^{2} x^{2} + 9\,n^{2} x^{3} + 2\,n^{3} + 20\,n^{3} x + 18\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	7	11	3						21
3	26	77	64	12					179
4	98	444	675	385	55				1657
5	372	2310	5280	5434	2366	273			16035
6	1419	11297	35035	53690	41860	14756	1428		159485
7	5434	53053	209664	433160	499800	313956	93024	7752	1615843

All 15 rows

Technical details

Node id: n-7f226db9395183ec · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-0e9e89899f02b109

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:     7,      11,       3;
  3:    26,      77,      64,       12;
  4:    98,     444,     675,      385,       55;
  5:   372,    2310,    5280,     5434,     2366,      273;
  6:  1419,   11297,   35035,    53690,    41860,    14756,     1428;
  7:  5434,   53053,  209664,   433160,   499800,   313956,    93024,     7752;
  8: 20878,  242216, 1169532,  3065916,  4748100,  4395384,  2313003,   591261,   43263;
  9: 80444, 1083342, 6201600, 19806360, 38798760, 48050772, 37148496, 16824500, 3782350, 246675;

%F Formula

The row polynomials satisfy (8*n^4-4*n^3+4*n^2+4*n+(4*n^3+2*n^2)*x-(4*n^2+2*n)*x^2)*P(n,x) = (28*n^4-8*n^3+13*n^2-10*n+1+(36*n^4+2*n^3-18*n^2+4*n)*x+(18*n^3-47*n^2+27*n-4)*x^2-(18*n^2-15*n+3)*x^3)*P(n-1,x) + (2*n^3-4*n^2+4*n-2+(20*n^3-9*n^2+2*n-1)*x+(18*n^3+4*n^2-24*n+8)*x^2+(9*n^2-31*n+10)*x^3-(9*n-3)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((28*n^4 - 8*n^3 + 13*n^2 - 10*n + 1 + (36*n^4 + 2*n^3 - 18*n^2 + 4*n)*x + (18*n^3 - 47*n^2 + 27*n - 4)*x^2 - (18*n^2 - 15*n + 3)*x^3)*P[n - 1, x] + (2*n^3 - 4*n^2 + 4*n - 2 + (20*n^3 - 9*n^2 + 2*n - 1)*x + (18*n^3 + 4*n^2 - 24*n + 8)*x^2 + (9*n^2 - 31*n + 10)*x^3 - (9*n - 3)*x^4)*D[P[n - 1, x], x])/(8*n^4 - 4*n^3 + 4*n^2 + 4*n + (4*n^3 + 2*n^2)*x - (4*n^2 + 2*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((28*n^4 - 8*n^3 + 13*n^2 - 10*n + 1 + (36*n^4 + 2*n^3 - 18*n^2 + 4*n)*x + (18*n^3 - 47*n^2 + 27*n - 4)*x^2 - (18*n^2 - 15*n + 3)*x^3)*v[i - 1] + (2*n^3 - 4*n^2 + 4*n - 2 + (20*n^3 - 9*n^2 + 2*n - 1)*x + (18*n^3 + 4*n^2 - 24*n + 8)*x^2 + (9*n^2 - 31*n + 10)*x^3 - (9*n - 3)*x^4)*deriv(v[i - 1], x))/(8*n^4 - 4*n^3 + 4*n^2 + 4*n + (4*n^3 + 2*n^2)*x - (4*n^2 + 2*n)*x^2)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch th…?
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - ↑ Iterated real-rootedness preservers (above)
  - ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

A330798 → D+1

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.