A245962 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-100 - 60\,x + 4\,x^{2} + 12\,x^{3} + 100\,n - 65\,n x - 34\,n x^{2} - 5\,n x^{3} + 125\,n^{2} x - 45\,n^{2} x^{2} - 29\,n^{2} x^{3} - n^{2} x^{4} + 75\,n^{3} x^{2} + 7\,n^{3} x^{3} - n^{3} x^{4} + 15\,n^{4} x^{3} + n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-300 - 630\,x - 468\,x^{2} - 172\,x^{3} - 48\,x^{4} - 8\,x^{5} + 300\,n + 80\,n x - 472\,n x^{2} - 407\,n x^{3} - 124\,n x^{4} - 16\,n x^{5} + 550\,n^{2} x + 640\,n^{2} x^{2} + 173\,n^{2} x^{3} + 3\,n^{2} x^{4} - 2\,n^{2} x^{5} + 300\,n^{3} x^{2} + 351\,n^{3} x^{3} + 113\,n^{3} x^{4} + 14\,n^{3} x^{5} + 55\,n^{4} x^{3} + 53\,n^{4} x^{4} + 10\,n^{4} x^{5} + 3\,n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-125 - 450\,x - 685\,x^{2} - 594\,x^{3} - 326\,x^{4} - 108\,x^{5} - 16\,x^{6} - 375\,n x - 1200\,n x^{2} - 1530\,n x^{3} - 993\,n x^{4} - 336\,n x^{5} - 48\,n x^{6} - 300\,n^{2} x^{2} - 800\,n^{2} x^{3} - 773\,n^{2} x^{4} - 325\,n^{2} x^{5} - 52\,n^{2} x^{6} - 75\,n^{3} x^{3} - 165\,n^{3} x^{4} - 114\,n^{3} x^{5} - 24\,n^{3} x^{6} - 5\,n^{4} x^{4} - 9\,n^{4} x^{5} - 4\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	3	8							11
3	4	12							16
4	7	32	18						57
5	11	60	45						116
6	18	120	135	32					305
7	29	224	315	112					680
8	47	416	720	384	50				1617
9	76	756	1539	1056	225				3652
10	123	1360	3195	2720	875	72			8345
11	199	2420	6435	6512	2750	396			18712
12	322	4272	12690	14912	7875	1728	98		41897
13	521	7488	24570	32864	20800	6084	637		92964
14	843	13048	46872	70336	51975	19152	3087	128	205441

Technical details

Node id: n-76cae6339cf99ebf · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-3525ecf539972933

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:  1;
  2:  3,   8;
  3:  4,  12;
  4:  7,  32,   18;
  5: 11,  60,   45;
  6: 18, 120,  135,   32;
  7: 29, 224,  315,  112;
  8: 47, 416,  720,  384,  50;
  9: 76, 756, 1539, 1056, 225;

%F Formula

The row polynomials satisfy (100*n-100+(125*n^2-65*n-60)*x+(75*n^3-45*n^2-34*n+4)*x^2+(15*n^4+7*n^3-29*n^2-5*n+12)*x^3+(n^5+n^4-n^3-n^2)*x^4)*P(n,x) = (300*n-300+(550*n^2+80*n-630)*x+(300*n^3+640*n^2-472*n-468)*x^2+(55*n^4+351*n^3+173*n^2-407*n-172)*x^3+(3*n^5+53*n^4+113*n^3+3*n^2-124*n-48)*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+450)*x+(300*n^2+1200*n+685)*x^2+(75*n^3+800*n^2+1530*n+594)*x^3+(5*n^4+165*n^3+773*n^2+993*n+326)*x^4+(9*n^4+114*n^3+325*n^2+336*n+108)*x^5+(4*n^4+24*n^3+52*n^2+48*n+16)*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((300*n - 300 + (550*n^2 + 80*n - 630)*x + (300*n^3 + 640*n^2 - 472*n - 468)*x^2 + (55*n^4 + 351*n^3 + 173*n^2 - 407*n - 172)*x^3 + (3*n^5 + 53*n^4 + 113*n^3 + 3*n^2 - 124*n - 48)*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 + 450)*x + (300*n^2 + 1200*n + 685)*x^2 + (75*n^3 + 800*n^2 + 1530*n + 594)*x^3 + (5*n^4 + 165*n^3 + 773*n^2 + 993*n + 326)*x^4 + (9*n^4 + 114*n^3 + 325*n^2 + 336*n + 108)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(100*n - 100 + (125*n^2 - 65*n - 60)*x + (75*n^3 - 45*n^2 - 34*n + 4)*x^2 + (15*n^4 + 7*n^3 - 29*n^2 - 5*n + 12)*x^3 + (n^5 + n^4 - n^3 - n^2)*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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((300*n - 300 + (550*n^2 + 80*n - 630)*x + (300*n^3 + 640*n^2 - 472*n - 468)*x^2 + (55*n^4 + 351*n^3 + 173*n^2 - 407*n - 172)*x^3 + (3*n^5 + 53*n^4 + 113*n^3 + 3*n^2 - 124*n - 48)*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 + 450)*x + (300*n^2 + 1200*n + 685)*x^2 + (75*n^3 + 800*n^2 + 1530*n + 594)*x^3 + (5*n^4 + 165*n^3 + 773*n^2 + 993*n + 326)*x^4 + (9*n^4 + 114*n^3 + 325*n^2 + 336*n + 108)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(100*n - 100 + (125*n^2 - 65*n - 60)*x + (75*n^3 - 45*n^2 - 34*n + 4)*x^2 + (15*n^4 + 7*n^3 - 29*n^2 - 5*n + 12)*x^3 + (n^5 + n^4 - n^3 - 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 - 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 induced subgraphs of the Lucas cube Lambda(n) that are isomorphic to the…?
  - 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.