A056242 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(32 + 28\,x - 32\,n - 20\,n x + 9\,n x^{2} - 8\,n^{2} x - 10\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-32 - 116\,x - 80\,x^{2} - 32\,n - 48\,n x - 57\,n x^{2} - 36\,n x^{3} - 12\,n^{2} x - 8\,n^{2} x^{2} + 4\,n^{2} x^{3} + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-64 - 176\,x - 152\,x^{2} - 40\,x^{3} - 32\,n x - 84\,n x^{2} - 70\,n x^{3} - 18\,n x^{4} + 4\,n^{2} x^{2} + 6\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	4							7
3	10	26	12						48
4	34	130	120	32					316
5	116	584	807	432	80				2019
6	396	2472	4545	3656	1360	192			12621
7	1352	10072	23106	24804	13880	3936	448		77598
8	4616	39960	109842	147460	110240	47040	10752	1024	470934

All 15 rows

Technical details

Node id: n-e26a2c1b1e22c73e · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-748cfd7c85ef5e44

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     3,      4;
   3:    10,     26,      12;
   4:    34,    130,     120,      32;
   5:   116,    584,     807,     432,      80;
   6:   396,   2472,    4545,    3656,    1360,     192;
   7:  1352,  10072,   23106,   24804,   13880,    3936,     448;
   8:  4616,  39960,  109842,  147460,  110240,   47040,   10752,   1024;
   9: 15760, 155456,  497928,  802336,  750845,  428208,  147280,  28160,  2304;
  10: 53808, 595712, 2178456, 4095776, 4604415, 3288492, 1511888, 434944, 71424, 5120;

%F Formula

The row polynomials satisfy -(32*n-32+(8*n^2+20*n-28)*x-(n^3-10*n^2+9*n)*x^2)*P(n,x) = -(32*n+32+(12*n^2+48*n+116)*x-(n^3-8*n^2-57*n-80)*x^2-(4*n^2-36*n)*x^3)*P(n-1,x) - (64+(32*n+176)*x-(4*n^2-84*n-152)*x^2-(6*n^2-70*n-40)*x^3-(2*n^2-18*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(32*n + 32 + (12*n^2 + 48*n + 116)*x - (n^3 - 8*n^2 - 57*n - 80)*x^2 - (4*n^2 - 36*n)*x^3)*P[n - 1, x] - (64 + (32*n + 176)*x - (4*n^2 - 84*n - 152)*x^2 - (6*n^2 - 70*n - 40)*x^3 - (2*n^2 - 18*n)*x^4)*D[P[n - 1, x], x])/(-32*n + 32 - (8*n^2 + 20*n - 28)*x + (n^3 - 10*n^2 + 9*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(32*n + 32 + (12*n^2 + 48*n + 116)*x - (n^3 - 8*n^2 - 57*n - 80)*x^2 - (4*n^2 - 36*n)*x^3)*v[i - 1] - (64 + (32*n + 176)*x - (4*n^2 - 84*n - 152)*x^2 - (6*n^2 - 70*n - 40)*x^3 - (2*n^2 - 18*n)*x^4)*deriv(v[i - 1], x))/(-32*n + 32 - (8*n^2 + 20*n - 28)*x + (n^3 - 10*n^2 + 9*n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from shift_x_plus_one of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
    no recorded property covers this requirement
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    ↑ Affine-current basis map theorem (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    ↑ Affine-current basis map theorem (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[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.