A181371 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(300\,n - 195\,n x + 147\,n x^{2} - 18\,n x^{3} - 375\,n^{2} x + 180\,n^{2} x^{2} - 36\,n^{2} x^{3} - 75\,n^{3} x^{2} + 15\,n^{3} x^{3} + 3\,n^{3} x^{4} + 45\,n^{4} x^{3} + 6\,n^{4} x^{4} + 3\,n^{5} x^{4}\right)\,P_{n}(x) = \left(200 - 580\,x + 548\,x^{2} - 164\,x^{3} + 4\,x^{4} - 8\,x^{5} + 700\,n - 430\,n x + 898\,n x^{2} - 34\,n x^{3} - 38\,n x^{4} - 16\,n x^{5} - 700\,n^{2} x - 440\,n^{2} x^{2} - 14\,n^{2} x^{3} - 68\,n^{2} x^{4} - 2\,n^{2} x^{5} - 250\,n^{3} x^{2} + 135\,n^{3} x^{3} + 65\,n^{3} x^{4} + 14\,n^{3} x^{5} + 95\,n^{4} x^{3} + 66\,n^{4} x^{4} + 10\,n^{4} x^{5} + 7\,n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(125 - 225\,x - 15\,x^{2} + 161\,x^{3} - 18\,x^{4} - 12\,x^{5} - 16\,x^{6} - 375\,n x + 405\,n x^{3} + 102\,n x^{4} - 84\,n x^{5} - 48\,n x^{6} - 50\,n^{2} x^{2} + 185\,n^{2} x^{3} + 70\,n^{2} x^{4} - 153\,n^{2} x^{5} - 52\,n^{2} x^{6} + 75\,n^{3} x^{3} + 15\,n^{3} x^{4} - 66\,n^{3} x^{5} - 24\,n^{3} x^{6} + 5\,n^{4} x^{4} - n^{4} x^{5} - 4\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	3				3
2	8	4			12
3	21	24			45
4	55	100	9		164
5	144	360	81		585
6	377	1200	459	16	2052
7	987	3816	2106	192	7101

All 15 rows

Technical details

Node id: n-cd00363a5a077fb1 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d0f3fab92b506d51

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:    3;
  2:    8,     4;
  3:   21,    24;
  4:   55,   100,     9;
  5:  144,   360,    81;
  6:  377,  1200,   459,   16;
  7:  987,  3816,  2106,  192;
  8: 2584, 11756,  8559, 1376,  25;
  9: 6765, 35400, 32157, 7680, 375;

%F Formula

The row polynomials satisfy (300*n-(375*n^2+195*n)*x-(75*n^3-180*n^2-147*n)*x^2+(45*n^4+15*n^3-36*n^2-18*n)*x^3+(3*n^5+6*n^4+3*n^3)*x^4)*P(n,x) = (700*n+200-(700*n^2+430*n+580)*x-(250*n^3+440*n^2-898*n-548)*x^2+(95*n^4+135*n^3-14*n^2-34*n-164)*x^3+(7*n^5+66*n^4+65*n^3-68*n^2-38*n+4)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) + (125-(375*n+225)*x-(50*n^2+15)*x^2+(75*n^3+185*n^2+405*n+161)*x^3+(5*n^4+15*n^3+70*n^2+102*n-18)*x^4-(n^4+66*n^3+153*n^2+84*n+12)*x^5-(4*n^4+24*n^3+52*n^2+48*n+16)*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[((700*n + 200 - (700*n^2 + 430*n + 580)*x - (250*n^3 + 440*n^2 - 898*n - 548)*x^2 + (95*n^4 + 135*n^3 - 14*n^2 - 34*n - 164)*x^3 + (7*n^5 + 66*n^4 + 65*n^3 - 68*n^2 - 38*n + 4)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] + (125 - (375*n + 225)*x - (50*n^2 + 15)*x^2 + (75*n^3 + 185*n^2 + 405*n + 161)*x^3 + (5*n^4 + 15*n^3 + 70*n^2 + 102*n - 18)*x^4 - (n^4 + 66*n^3 + 153*n^2 + 84*n + 12)*x^5 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(300*n - (375*n^2 + 195*n)*x - (75*n^3 - 180*n^2 - 147*n)*x^2 + (45*n^4 + 15*n^3 - 36*n^2 - 18*n)*x^3 + (3*n^5 + 6*n^4 + 3*n^3)*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] = ((700*n + 200 - (700*n^2 + 430*n + 580)*x - (250*n^3 + 440*n^2 - 898*n - 548)*x^2 + (95*n^4 + 135*n^3 - 14*n^2 - 34*n - 164)*x^3 + (7*n^5 + 66*n^4 + 65*n^3 - 68*n^2 - 38*n + 4)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] + (125 - (375*n + 225)*x - (50*n^2 + 15)*x^2 + (75*n^3 + 185*n^2 + 405*n + 161)*x^3 + (5*n^4 + 15*n^3 + 70*n^2 + 102*n - 18)*x^4 - (n^4 + 66*n^3 + 153*n^2 + 84*n + 12)*x^5 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(300*n - (375*n^2 + 195*n)*x - (75*n^3 - 180*n^2 - 147*n)*x^2 + (45*n^4 + 15*n^3 - 36*n^2 - 18*n)*x^3 + (3*n^5 + 6*n^4 + 3*n^3)*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 Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n h…?
  - 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.