A199478 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 + 32\,x + 224\,x^{2} + 576\,x^{3} - 8\,n - 4\,n x + 360\,n x^{2} + 1104\,n x^{3} - 5\,n^{2} - 50\,n^{2} x + 144\,n^{2} x^{2} + 832\,n^{2} x^{3} - n^{3} - 32\,n^{3} x - 26\,n^{3} x^{2} + 308\,n^{3} x^{3} - 6\,n^{4} x - 26\,n^{4} x^{2} + 56\,n^{4} x^{3} - 4\,n^{5} x^{2} + 4\,n^{5} x^{3}\right)\,P_{n}(x) = \left(-4 + 16\,x + 352\,x^{2} + 1472\,x^{3} + 2304\,x^{4} - 8\,n - 84\,n x + 256\,n x^{2} + 2344\,n x^{3} + 4992\,n x^{4} - 5\,n^{2} - 118\,n^{2} x - 282\,n^{2} x^{2} + 1028\,n^{2} x^{3} + 3856\,n^{2} x^{4} - n^{3} - 48\,n^{3} x - 272\,n^{3} x^{2} - 12\,n^{3} x^{3} + 1392\,n^{3} x^{4} - 6\,n^{4} x - 66\,n^{4} x^{2} - 84\,n^{4} x^{3} + 240\,n^{4} x^{4} - 4\,n^{5} x^{2} - 12\,n^{5} x^{3} + 16\,n^{5} x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 74\,x^{2} + 344\,x^{3} - 512\,x^{4} - 4608\,x^{5} + 4\,n x + 106\,n x^{2} + 712\,n x^{3} + 288\,n x^{4} - 5376\,n x^{5} + n^{2} x + 46\,n^{2} x^{2} + 422\,n^{2} x^{3} + 652\,n^{2} x^{4} - 2336\,n^{2} x^{5} + 6\,n^{3} x^{2} + 86\,n^{3} x^{3} + 248\,n^{3} x^{4} - 448\,n^{3} x^{5} + 4\,n^{4} x^{3} + 28\,n^{4} x^{4} - 32\,n^{4} x^{5}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	P_n(1)
0	1	4				5
1	1	12				13
2	1	20	18			39
3	1	28	72			101
4	1	36	162	64		263
5	1	44	288	320		653
6	1	52	450	896	200	1599
7	1	60	648	1920	1200	3829

All 15 rows

Technical details

Node id: n-8ca1e25abea620be · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-9a9a473010d76dc3

Observed on the stored rows ?

Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
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,  4;
  1: 1, 12;
  2: 1, 20,   18;
  3: 1, 28,   72;
  4: 1, 36,  162,   64;
  5: 1, 44,  288,  320;
  6: 1, 52,  450,  896,   200;
  7: 1, 60,  648, 1920,  1200;
  8: 1, 68,  882, 3520,  4000,  576;
  9: 1, 76, 1152, 5824, 10000, 4032;

%F Formula

The row polynomials satisfy -(n^3+5*n^2+8*n+4+(6*n^4+32*n^3+50*n^2+4*n-32)*x+(4*n^5+26*n^4+26*n^3-144*n^2-360*n-224)*x^2-(4*n^5+56*n^4+308*n^3+832*n^2+1104*n+576)*x^3)*P(n,x) = -(n^3+5*n^2+8*n+4+(6*n^4+48*n^3+118*n^2+84*n-16)*x+(4*n^5+66*n^4+272*n^3+282*n^2-256*n-352)*x^2+(12*n^5+84*n^4+12*n^3-1028*n^2-2344*n-1472)*x^3-(16*n^5+240*n^4+1392*n^3+3856*n^2+4992*n+2304)*x^4)*P(n-1,x) + ((n^2+4*n+4)*x+(6*n^3+46*n^2+106*n+74)*x^2+(4*n^4+86*n^3+422*n^2+712*n+344)*x^3+(28*n^4+248*n^3+652*n^2+288*n-512)*x^4-(32*n^4+448*n^3+2336*n^2+5376*n+4608)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 5*n^2 + 8*n + 4 + (6*n^4 + 48*n^3 + 118*n^2 + 84*n - 16)*x + (4*n^5 + 66*n^4 + 272*n^3 + 282*n^2 - 256*n - 352)*x^2 + (12*n^5 + 84*n^4 + 12*n^3 - 1028*n^2 - 2344*n - 1472)*x^3 - (16*n^5 + 240*n^4 + 1392*n^3 + 3856*n^2 + 4992*n + 2304)*x^4)*P[n - 1, x] + ((n^2 + 4*n + 4)*x + (6*n^3 + 46*n^2 + 106*n + 74)*x^2 + (4*n^4 + 86*n^3 + 422*n^2 + 712*n + 344)*x^3 + (28*n^4 + 248*n^3 + 652*n^2 + 288*n - 512)*x^4 - (32*n^4 + 448*n^3 + 2336*n^2 + 5376*n + 4608)*x^5)*D[P[n - 1, x], x])/(-n^3 - 5*n^2 - 8*n - 4 - (6*n^4 + 32*n^3 + 50*n^2 + 4*n - 32)*x - (4*n^5 + 26*n^4 + 26*n^3 - 144*n^2 - 360*n - 224)*x^2 + (4*n^5 + 56*n^4 + 308*n^3 + 832*n^2 + 1104*n + 576)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 + 5*n^2 + 8*n + 4 + (6*n^4 + 48*n^3 + 118*n^2 + 84*n - 16)*x + (4*n^5 + 66*n^4 + 272*n^3 + 282*n^2 - 256*n - 352)*x^2 + (12*n^5 + 84*n^4 + 12*n^3 - 1028*n^2 - 2344*n - 1472)*x^3 - (16*n^5 + 240*n^4 + 1392*n^3 + 3856*n^2 + 4992*n + 2304)*x^4)*v[i - 1] + ((n^2 + 4*n + 4)*x + (6*n^3 + 46*n^2 + 106*n + 74)*x^2 + (4*n^4 + 86*n^3 + 422*n^2 + 712*n + 344)*x^3 + (28*n^4 + 248*n^3 + 652*n^2 + 288*n - 512)*x^4 - (32*n^4 + 448*n^3 + 2336*n^2 + 5376*n + 4608)*x^5)*deriv(v[i - 1], x))/(-n^3 - 5*n^2 - 8*n - 4 - (6*n^4 + 32*n^3 + 50*n^2 + 4*n - 32)*x - (4*n^5 + 26*n^4 + 26*n^3 - 144*n^2 - 360*n - 224)*x^2 + (4*n^5 + 56*n^4 + 308*n^3 + 832*n^2 + 1104*n + 576)*x^3)); 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 Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.?
  - 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.