A119900 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 4\,n x + 12\,n^{2} + 2\,n^{2} x + 13\,n^{3} - 34\,n^{3} x + 4\,n^{3} x^{2} + 6\,n^{4} - 36\,n^{4} x + 16\,n^{4} x^{2} + n^{5} - 8\,n^{5} x + 16\,n^{5} x^{2}\right)\,P_{n}(x) = \left(4 - 12\,x + 12\,x^{2} - 4\,x^{3} + 8\,n + 6\,n x + 110\,n x^{2} - 20\,n x^{3} + n^{2} + 91\,n^{2} x - 104\,n^{2} x^{2} - 20\,n^{2} x^{3} - 7\,n^{3} + 127\,n^{3} x - 342\,n^{3} x^{2} + 124\,n^{3} x^{3} - 5\,n^{4} + 65\,n^{4} x - 212\,n^{4} x^{2} + 160\,n^{4} x^{3} - n^{5} + 11\,n^{5} x - 40\,n^{5} x^{2} + 48\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(8\,x - 24\,x^{2} + 24\,x^{3} - 8\,x^{4} + 24\,n x - 156\,n x^{2} + 188\,n x^{3} - 56\,n x^{4} + 26\,n^{2} x - 214\,n^{2} x^{2} + 340\,n^{2} x^{3} - 152\,n^{2} x^{4} + 12\,n^{3} x - 108\,n^{3} x^{2} + 224\,n^{3} x^{3} - 128\,n^{3} x^{4} + 2\,n^{4} x - 18\,n^{4} x^{2} + 48\,n^{4} x^{3} - 32\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	8							8
2	0	4	27						31
3	0	0	36	64					100
4	0	0	9	160	125				294
5	0	0	0	96	500	216			812
6	0	0	0	16	525	1260	343		2144
7	0	0	0	0	200	2016	2744	512	5472

All 15 rows

Technical details

Node id: n-d00b626a5326cb60 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-6c80528aee0f2569

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: 0, 8;
  2: 0, 4, 27;
  3: 0, 0, 36,  64;
  4: 0, 0,  9, 160, 125;
  5: 0, 0,  0,  96, 500,  216;
  6: 0, 0,  0,  16, 525, 1260,  343;
  7: 0, 0,  0,   0, 200, 2016, 2744,   512;
  8: 0, 0,  0,   0,  25, 1296, 6174,  5376,  729;
  9: 0, 0,  0,   0,   0,  360, 5880, 16128, 9720, 1000;

%F Formula

The row polynomials satisfy (n^5+6*n^4+13*n^3+12*n^2+4*n-(8*n^5+36*n^4+34*n^3-2*n^2-4*n)*x+(16*n^5+16*n^4+4*n^3)*x^2)*P(n,x) = -(n^5+5*n^4+7*n^3-n^2-8*n-4-(11*n^5+65*n^4+127*n^3+91*n^2+6*n-12)*x+(40*n^5+212*n^4+342*n^3+104*n^2-110*n-12)*x^2-(48*n^5+160*n^4+124*n^3-20*n^2-20*n-4)*x^3)*P(n-1,x) + ((2*n^4+12*n^3+26*n^2+24*n+8)*x-(18*n^4+108*n^3+214*n^2+156*n+24)*x^2+(48*n^4+224*n^3+340*n^2+188*n+24)*x^3-(32*n^4+128*n^3+152*n^2+56*n+8)*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[(-(n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4 - (11*n^5 + 65*n^4 + 127*n^3 + 91*n^2 + 6*n - 12)*x + (40*n^5 + 212*n^4 + 342*n^3 + 104*n^2 - 110*n - 12)*x^2 - (48*n^5 + 160*n^4 + 124*n^3 - 20*n^2 - 20*n - 4)*x^3)*P[n - 1, x] + ((2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x - (18*n^4 + 108*n^3 + 214*n^2 + 156*n + 24)*x^2 + (48*n^4 + 224*n^3 + 340*n^2 + 188*n + 24)*x^3 - (32*n^4 + 128*n^3 + 152*n^2 + 56*n + 8)*x^4)*D[P[n - 1, x], x])/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (8*n^5 + 36*n^4 + 34*n^3 - 2*n^2 - 4*n)*x + (16*n^5 + 16*n^4 + 4*n^3)*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] = (-(n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4 - (11*n^5 + 65*n^4 + 127*n^3 + 91*n^2 + 6*n - 12)*x + (40*n^5 + 212*n^4 + 342*n^3 + 104*n^2 - 110*n - 12)*x^2 - (48*n^5 + 160*n^4 + 124*n^3 - 20*n^2 - 20*n - 4)*x^3)*v[i - 1] + ((2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x - (18*n^4 + 108*n^3 + 214*n^2 + 156*n + 24)*x^2 + (48*n^4 + 224*n^3 + 340*n^2 + 188*n + 24)*x^3 - (32*n^4 + 128*n^3 + 152*n^2 + 56*n + 8)*x^4)*deriv(v[i - 1], x))/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (8*n^5 + 36*n^4 + 34*n^3 - 2*n^2 - 4*n)*x + (16*n^5 + 16*n^4 + 4*n^3)*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 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 Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.