A103633 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(8 + 26\,x + 20\,x^{2} + 12\,n x + 29\,n x^{2} + 4\,n x^{3} - 16\,n x^{4} + 9\,n^{2} x^{2} + 24\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n}(x) = \left(32 + 172\,x + 304\,x^{2} + 148\,x^{3} - 88\,x^{4} - 64\,x^{5} - 24\,n - 48\,n x + 128\,n x^{2} + 402\,n x^{3} + 296\,n x^{4} + 32\,n x^{5} - 54\,n^{2} x - 195\,n^{2} x^{2} - 182\,n^{2} x^{3} + 48\,n^{2} x^{4} + 96\,n^{2} x^{5} - 27\,n^{3} x^{2} - 108\,n^{3} x^{3} - 144\,n^{3} x^{4} - 64\,n^{3} x^{5}\right)\,P_{n-1}(x) + \left(8 + 52\,x + 116\,x^{2} + 80\,x^{3} - 56\,x^{4} - 96\,x^{5} - 32\,x^{6} + 24\,n x + 146\,n x^{2} + 326\,n x^{3} + 316\,n x^{4} + 112\,n x^{5} + 18\,n^{2} x^{2} + 102\,n^{2} x^{3} + 212\,n^{2} x^{4} + 192\,n^{2} x^{5} + 64\,n^{2} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	4							5
2	2	12	12						26
3	2	20	48	32					102
4	4	48	156	192	80				480
5	4	64	300	608	560	192			1728
6	8	144	792	2016	2640	1728	448		7776
7	8	176	1224	4128	7680	8064	4480	1024	26784

All 15 rows

Technical details

Node id: n-75ad3144101df727 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c7f8d68687ddb27c

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,   4;
  2:  2,  12,   12;
  3:  2,  20,   48,    32;
  4:  4,  48,  156,   192,    80;
  5:  4,  64,  300,   608,   560,   192;
  6:  8, 144,  792,  2016,  2640,  1728,    448;
  7:  8, 176, 1224,  4128,  7680,  8064,   4480,  1024;
  8: 16, 384, 2976, 11520, 25680, 34560,  27776, 12288,  2304;
  9: 16, 448, 4128, 19456, 54480, 96192, 108416, 75776, 29952, 5120;

%F Formula

The row polynomials satisfy (8+(12*n+26)*x+(9*n^2+29*n+20)*x^2+(24*n^2+4*n)*x^3+(16*n^2-16*n)*x^4)*P(n,x) = -(24*n-32+(54*n^2+48*n-172)*x+(27*n^3+195*n^2-128*n-304)*x^2+(108*n^3+182*n^2-402*n-148)*x^3+(144*n^3-48*n^2-296*n+88)*x^4+(64*n^3-96*n^2-32*n+64)*x^5)*P(n-1,x) + (8+(24*n+52)*x+(18*n^2+146*n+116)*x^2+(102*n^2+326*n+80)*x^3+(212*n^2+316*n-56)*x^4+(192*n^2+112*n-96)*x^5+(64*n^2-32)*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[(-(24*n - 32 + (54*n^2 + 48*n - 172)*x + (27*n^3 + 195*n^2 - 128*n - 304)*x^2 + (108*n^3 + 182*n^2 - 402*n - 148)*x^3 + (144*n^3 - 48*n^2 - 296*n + 88)*x^4 + (64*n^3 - 96*n^2 - 32*n + 64)*x^5)*P[n - 1, x] + (8 + (24*n + 52)*x + (18*n^2 + 146*n + 116)*x^2 + (102*n^2 + 326*n + 80)*x^3 + (212*n^2 + 316*n - 56)*x^4 + (192*n^2 + 112*n - 96)*x^5 + (64*n^2 - 32)*x^6)*D[P[n - 1, x], x])/(8 + (12*n + 26)*x + (9*n^2 + 29*n + 20)*x^2 + (24*n^2 + 4*n)*x^3 + (16*n^2 - 16*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] = (-(24*n - 32 + (54*n^2 + 48*n - 172)*x + (27*n^3 + 195*n^2 - 128*n - 304)*x^2 + (108*n^3 + 182*n^2 - 402*n - 148)*x^3 + (144*n^3 - 48*n^2 - 296*n + 88)*x^4 + (64*n^3 - 96*n^2 - 32*n + 64)*x^5)*v[i - 1] + (8 + (24*n + 52)*x + (18*n^2 + 146*n + 116)*x^2 + (102*n^2 + 326*n + 80)*x^3 + (212*n^2 + 316*n - 56)*x^4 + (192*n^2 + 112*n - 96)*x^5 + (64*n^2 - 32)*x^6)*deriv(v[i - 1], x))/(8 + (12*n + 26)*x + (9*n^2 + 29*n + 20)*x^2 + (24*n^2 + 4*n)*x^3 + (16*n^2 - 16*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 basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.?
  - Geometric basis map (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

Related ?

Same fingerprint ?

A064861 → θ+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.