A201730 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-2304 - 4512\,x - 1552\,x^{2} + 176\,x^{3} + 1152\,n + 3216\,n x + 232\,n x^{2} - 768\,n x^{3} - 480\,n^{2} x + 688\,n^{2} x^{2} + 748\,n^{2} x^{3} - 4\,n^{2} x^{4} - 208\,n^{3} x^{2} - 252\,n^{3} x^{3} + 10\,n^{3} x^{4} + 24\,n^{4} x^{3} - 8\,n^{4} x^{4} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-5760 - 12864\,x - 11104\,x^{2} - 3600\,x^{3} - 176\,x^{4} + 2880\,n + 7824\,n x + 5080\,n x^{2} + 1196\,n x^{3} + 20\,n x^{4} - 696\,n^{2} x + 1420\,n^{2} x^{2} + 1458\,n^{2} x^{3} + 300\,n^{2} x^{4} - 2\,n^{2} x^{5} - 592\,n^{3} x^{2} - 674\,n^{3} x^{3} - 153\,n^{3} x^{4} - 3\,n^{3} x^{5} + 48\,n^{4} x^{3} + 5\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(864 + 4752\,x + 5544\,x^{2} + 904\,x^{3} - 840\,x^{4} - 88\,x^{5} - 864\,n x - 1272\,n x^{2} - 140\,n x^{3} + 256\,n x^{4} - 12\,n x^{5} - 408\,n^{2} x^{2} - 740\,n^{2} x^{3} - 170\,n^{2} x^{4} + 160\,n^{2} x^{5} - 2\,n^{2} x^{6} + 72\,n^{3} x^{3} + 60\,n^{3} x^{4} - 16\,n^{3} x^{5} - 4\,n^{3} x^{6} + 6\,n^{4} x^{4} + 4\,n^{4} x^{5} - 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
1	1				1
2	2				2
3	5	4			9
4	14	24			38
5	41	104	9		154
6	122	400	90		612
7	365	1452	567	16	2400
8	1094	5096	2898	224	9312

All 15 rows

Technical details

Node id: n-160fd7a84ab6ab3d · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-9c06be62c07bae00

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    1;
   2:    2;
   3:    5,     4;
   4:   14,    24;
   5:   41,   104,     9;
   6:  122,   400,    90;
   7:  365,  1452,   567,    16;
   8: 1094,  5096,  2898,   224;
   9: 3281, 17488, 13158,  1856,  25;
  10: 9842, 59040, 55404, 11904, 450;

%F Formula

The row polynomials satisfy (1152*n-2304-(480*n^2-3216*n+4512)*x-(208*n^3-688*n^2-232*n+1552)*x^2+(24*n^4-252*n^3+748*n^2-768*n+176)*x^3+(2*n^5-8*n^4+10*n^3-4*n^2)*x^4)*P(n,x) = (2880*n-5760-(696*n^2-7824*n+12864)*x-(592*n^3-1420*n^2-5080*n+11104)*x^2+(48*n^4-674*n^3+1458*n^2+1196*n-3600)*x^3+(5*n^5-153*n^3+300*n^2+20*n-176)*x^4+(n^5-3*n^3-2*n^2)*x^5)*P(n-1,x) + (864-(864*n-4752)*x-(408*n^2+1272*n-5544)*x^2+(72*n^3-740*n^2-140*n+904)*x^3+(6*n^4+60*n^3-170*n^2+256*n-840)*x^4+(4*n^4-16*n^3+160*n^2-12*n-88)*x^5-(2*n^4+4*n^3+2*n^2)*x^6)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2880*n - 5760 - (696*n^2 - 7824*n + 12864)*x - (592*n^3 - 1420*n^2 - 5080*n + 11104)*x^2 + (48*n^4 - 674*n^3 + 1458*n^2 + 1196*n - 3600)*x^3 + (5*n^5 - 153*n^3 + 300*n^2 + 20*n - 176)*x^4 + (n^5 - 3*n^3 - 2*n^2)*x^5)*P[n - 1, x] + (864 - (864*n - 4752)*x - (408*n^2 + 1272*n - 5544)*x^2 + (72*n^3 - 740*n^2 - 140*n + 904)*x^3 + (6*n^4 + 60*n^3 - 170*n^2 + 256*n - 840)*x^4 + (4*n^4 - 16*n^3 + 160*n^2 - 12*n - 88)*x^5 - (2*n^4 + 4*n^3 + 2*n^2)*x^6)*D[P[n - 1, x], x])/(1152*n - 2304 - (480*n^2 - 3216*n + 4512)*x - (208*n^3 - 688*n^2 - 232*n + 1552)*x^2 + (24*n^4 - 252*n^3 + 748*n^2 - 768*n + 176)*x^3 + (2*n^5 - 8*n^4 + 10*n^3 - 4*n^2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((2880*n - 5760 - (696*n^2 - 7824*n + 12864)*x - (592*n^3 - 1420*n^2 - 5080*n + 11104)*x^2 + (48*n^4 - 674*n^3 + 1458*n^2 + 1196*n - 3600)*x^3 + (5*n^5 - 153*n^3 + 300*n^2 + 20*n - 176)*x^4 + (n^5 - 3*n^3 - 2*n^2)*x^5)*v[i - 1] + (864 - (864*n - 4752)*x - (408*n^2 + 1272*n - 5544)*x^2 + (72*n^3 - 740*n^2 - 140*n + 904)*x^3 + (6*n^4 + 60*n^3 - 170*n^2 + 256*n - 840)*x^4 + (4*n^4 - 16*n^3 + 160*n^2 - 12*n - 88)*x^5 - (2*n^4 + 4*n^3 + 2*n^2)*x^6)*deriv(v[i - 1], x))/(1152*n - 2304 - (480*n^2 - 3216*n + 4512)*x - (208*n^3 - 688*n^2 - 232*n + 1552)*x^2 + (24*n^4 - 252*n^3 + 748*n^2 - 768*n + 176)*x^3 + (2*n^5 - 8*n^4 + 10*n^3 - 4*n^2)*x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA …?
  - 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.