A103633 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-14 - 28\,x - 10\,x^{2} + 8\,x^{3} + 4\,x^{4} + n + 18\,n x + 24\,n x^{2} + 8\,n x^{3} + 9\,n^{2} + 12\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-56 - 116\,x - 42\,x^{2} + 58\,x^{3} + 52\,x^{4} + 12\,x^{5} + 36\,n + 147\,n x + 172\,n x^{2} + 60\,n x^{3} - 12\,n x^{4} - 8\,n x^{5} + 39\,n^{2} + 23\,n^{2} x - 56\,n^{2} x^{2} - 60\,n^{2} x^{3} - 16\,n^{2} x^{4} - 27\,n^{3} - 54\,n^{3} x - 36\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-8 + 12\,x + 112\,x^{2} + 196\,x^{3} + 152\,x^{4} + 56\,x^{5} + 8\,x^{6} + 36\,n + 174\,n x + 310\,n x^{2} + 260\,n x^{3} + 104\,n x^{4} + 16\,n x^{5} + 36\,n^{2} + 102\,n^{2} x + 106\,n^{2} x^{2} + 48\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	5	5	1						11
3	7	13	7	1					28
4	16	38	31	10	1				96
5	20	66	82	47	12	1			228
6	44	168	255	195	78	15	1		756
7	52	248	489	513	306	102	17	1	1728

All 15 rows

Technical details

Node id: n-44ba0a3b72230c8f · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-91a8a4d12dc5a97b

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:   5,   5,    1;
  3:   7,  13,    7,    1;
  4:  16,  38,   31,   10,    1;
  5:  20,  66,   82,   47,   12,    1;
  6:  44, 168,  255,  195,   78,   15,   1;
  7:  52, 248,  489,  513,  306,  102,  17,   1;
  8: 112, 592, 1328, 1644, 1221,  552, 146,  20,  1;
  9: 128, 800, 2168, 3332, 3186, 1953, 760, 178, 22, 1;

%F Formula

The row polynomials satisfy (9*n^2+n-14+(12*n^2+18*n-28)*x+(4*n^2+24*n-10)*x^2+(8*n+8)*x^3+4*x^4)*P(n,x) = -(27*n^3-39*n^2-36*n+56+(54*n^3-23*n^2-147*n+116)*x+(36*n^3+56*n^2-172*n+42)*x^2+(8*n^3+60*n^2-60*n-58)*x^3+(16*n^2+12*n-52)*x^4+(8*n-12)*x^5)*P(n-1,x) + (36*n^2+36*n-8+(102*n^2+174*n+12)*x+(106*n^2+310*n+112)*x^2+(48*n^2+260*n+196)*x^3+(8*n^2+104*n+152)*x^4+(16*n+56)*x^5+8*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[(-(27*n^3 - 39*n^2 - 36*n + 56 + (54*n^3 - 23*n^2 - 147*n + 116)*x + (36*n^3 + 56*n^2 - 172*n + 42)*x^2 + (8*n^3 + 60*n^2 - 60*n - 58)*x^3 + (16*n^2 + 12*n - 52)*x^4 + (8*n - 12)*x^5)*P[n - 1, x] + (36*n^2 + 36*n - 8 + (102*n^2 + 174*n + 12)*x + (106*n^2 + 310*n + 112)*x^2 + (48*n^2 + 260*n + 196)*x^3 + (8*n^2 + 104*n + 152)*x^4 + (16*n + 56)*x^5 + 8*x^6)*D[P[n - 1, x], x])/(9*n^2 + n - 14 + (12*n^2 + 18*n - 28)*x + (4*n^2 + 24*n - 10)*x^2 + (8*n + 8)*x^3 + 4*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] = (-(27*n^3 - 39*n^2 - 36*n + 56 + (54*n^3 - 23*n^2 - 147*n + 116)*x + (36*n^3 + 56*n^2 - 172*n + 42)*x^2 + (8*n^3 + 60*n^2 - 60*n - 58)*x^3 + (16*n^2 + 12*n - 52)*x^4 + (8*n - 12)*x^5)*v[i - 1] + (36*n^2 + 36*n - 8 + (102*n^2 + 174*n + 12)*x + (106*n^2 + 310*n + 112)*x^2 + (48*n^2 + 260*n + 196)*x^3 + (8*n^2 + 104*n + 152)*x^4 + (16*n + 56)*x^5 + 8*x^6)*deriv(v[i - 1], x))/(9*n^2 + n - 14 + (12*n^2 + 18*n - 28)*x + (4*n^2 + 24*n - 10)*x^2 + (8*n + 8)*x^3 + 4*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 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 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.