A103633 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(32 + 52\,x + 20\,x^{2} + 24\,n x + 29\,n x^{2} + 2\,n x^{3} - 4\,n x^{4} + 9\,n^{2} x^{2} + 12\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n}(x) = \left(128 + 344\,x + 304\,x^{2} + 74\,x^{3} - 22\,x^{4} - 8\,x^{5} - 96\,n - 96\,n x + 128\,n x^{2} + 201\,n x^{3} + 74\,n x^{4} + 4\,n x^{5} - 108\,n^{2} x - 195\,n^{2} x^{2} - 91\,n^{2} x^{3} + 12\,n^{2} x^{4} + 12\,n^{2} x^{5} - 27\,n^{3} x^{2} - 54\,n^{3} x^{3} - 36\,n^{3} x^{4} - 8\,n^{3} x^{5}\right)\,P_{n-1}(x) + \left(64 + 208\,x + 232\,x^{2} + 80\,x^{3} - 28\,x^{4} - 24\,x^{5} - 4\,x^{6} + 96\,n x + 292\,n x^{2} + 326\,n x^{3} + 158\,n x^{4} + 28\,n x^{5} + 36\,n^{2} x^{2} + 102\,n^{2} x^{3} + 106\,n^{2} x^{4} + 48\,n^{2} x^{5} + 8\,n^{2} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	2	6	3						11
3	2	10	12	4					28
4	4	24	39	24	5				96
5	4	32	75	76	35	6			228
6	8	72	198	252	165	54	7		756
7	8	88	306	516	480	252	70	8	1728

All 15 rows

Technical details

Node id: n-8ca3f60c018179d4 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-91efc7c4f6d258e7

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:  1,   2;
  2:  2,   6,    3;
  3:  2,  10,   12,    4;
  4:  4,  24,   39,   24,    5;
  5:  4,  32,   75,   76,   35,    6;
  6:  8,  72,  198,  252,  165,   54,    7;
  7:  8,  88,  306,  516,  480,  252,   70,   8;
  8: 16, 192,  744, 1440, 1605, 1080,  434,  96,   9;
  9: 16, 224, 1032, 2432, 3405, 3006, 1694, 592, 117, 10;

%F Formula

The row polynomials satisfy (32+(24*n+52)*x+(9*n^2+29*n+20)*x^2+(12*n^2+2*n)*x^3+(4*n^2-4*n)*x^4)*P(n,x) = -(96*n-128+(108*n^2+96*n-344)*x+(27*n^3+195*n^2-128*n-304)*x^2+(54*n^3+91*n^2-201*n-74)*x^3+(36*n^3-12*n^2-74*n+22)*x^4+(8*n^3-12*n^2-4*n+8)*x^5)*P(n-1,x) + (64+(96*n+208)*x+(36*n^2+292*n+232)*x^2+(102*n^2+326*n+80)*x^3+(106*n^2+158*n-28)*x^4+(48*n^2+28*n-24)*x^5+(8*n^2-4)*x^6)*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[(-(96*n - 128 + (108*n^2 + 96*n - 344)*x + (27*n^3 + 195*n^2 - 128*n - 304)*x^2 + (54*n^3 + 91*n^2 - 201*n - 74)*x^3 + (36*n^3 - 12*n^2 - 74*n + 22)*x^4 + (8*n^3 - 12*n^2 - 4*n + 8)*x^5)*P[n - 1, x] + (64 + (96*n + 208)*x + (36*n^2 + 292*n + 232)*x^2 + (102*n^2 + 326*n + 80)*x^3 + (106*n^2 + 158*n - 28)*x^4 + (48*n^2 + 28*n - 24)*x^5 + (8*n^2 - 4)*x^6)*D[P[n - 1, x], x])/(32 + (24*n + 52)*x + (9*n^2 + 29*n + 20)*x^2 + (12*n^2 + 2*n)*x^3 + (4*n^2 - 4*n)*x^4)]];
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] = (-(96*n - 128 + (108*n^2 + 96*n - 344)*x + (27*n^3 + 195*n^2 - 128*n - 304)*x^2 + (54*n^3 + 91*n^2 - 201*n - 74)*x^3 + (36*n^3 - 12*n^2 - 74*n + 22)*x^4 + (8*n^3 - 12*n^2 - 4*n + 8)*x^5)*v[i - 1] + (64 + (96*n + 208)*x + (36*n^2 + 292*n + 232)*x^2 + (102*n^2 + 326*n + 80)*x^3 + (106*n^2 + 158*n - 28)*x^4 + (48*n^2 + 28*n - 24)*x^5 + (8*n^2 - 4)*x^6)*deriv(v[i - 1], x))/(32 + (24*n + 52)*x + (9*n^2 + 29*n + 20)*x^2 + (12*n^2 + 2*n)*x^3 + (4*n^2 - 4*n)*x^4)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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: triangle of repeated stepped binomial coefficients.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

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.