A386789 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 2\,n x + 7\,n^{2} x - 7\,n^{3} x + n^{3} x^{2} - 2\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-4 + 2\,x + 4\,n + 23\,n x + 8\,n x^{2} + 2\,n x^{3} + 14\,n^{2} x - n^{2} x^{2} - 7\,n^{3} x - 21\,n^{3} x^{2} - 4\,n^{3} x^{3} - 3\,n^{4} x^{2} + n^{5} x^{2} + 2\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(8\,x + 14\,x^{2} + 8\,x^{3} + 2\,x^{4} - 14\,n^{2} x^{2} - 18\,n^{2} x^{3} - 4\,n^{2} x^{4} + 2\,n^{4} x^{3} + 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	8														8
2	0	12	54													66
3	0	16	180	320												516
4	0	20	405	1680	1750											3855
5	0	24	756	5376	12600	9072										27828
6	0	28	1260	13440	52500	83160	45276									195664
7	0	32	1944	28800	165000	427680	504504	219648								1347608
8	0	36	2835	55440	433125	1621620	3090087	2882880	1042470							9128493
9	0	40	3960	98560	1001000	5045040	13733720	20500480	15752880	4862000						60997680
10	0	44	5346	164736	2102100	13621608	49441392	104552448	127598328	83140200	22355476					402981678
11	0	48	7020	262080	4095000	33022080	152818848	427714560	734657040	755820000	426786360	101582208				2636765244
12	0	52	9009	400400	7507500	73513440	420251832	1489872384	3367178100	4849845000	4303429130	2141691552	457002364			17110700763
13	0	56	11340	591360	13090000	152681760	1052938656	4584222720	13054290480	24622290000	30455036920	23723352576	10546208400	2038517600		110243231868
14	0	60	14040	848640	21879000	299304720	2444321880	12770334720	44446750920	105172353000	169678062840	183573561600	127307801400	51108548400	9026235000	705850016220

Technical details

Node id: n-7c13c5ac4c438ba9 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-ccba683caf91532c

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, 12,   54;
  3: 0, 16,  180,   320;
  4: 0, 20,  405,  1680,    1750;
  5: 0, 24,  756,  5376,   12600,    9072;
  6: 0, 28, 1260, 13440,   52500,   83160,    45276;
  7: 0, 32, 1944, 28800,  165000,  427680,   504504,   219648;
  8: 0, 36, 2835, 55440,  433125, 1621620,  3090087,  2882880,  1042470;
  9: 0, 40, 3960, 98560, 1001000, 5045040, 13733720, 20500480, 15752880, 4862000;

%F Formula

The row polynomials satisfy (4*n-(7*n^3-7*n^2+2*n)*x+(n^5-2*n^4+n^3)*x^2)*P(n,x) = (4*n-4-(7*n^3-14*n^2-23*n-2)*x+(n^5-3*n^4-21*n^3-n^2+8*n)*x^2+(2*n^5-4*n^3+2*n)*x^3)*P(n-1,x) + (8*x-(14*n^2-14)*x^2+(2*n^4-18*n^2+8)*x^3+(2*n^4-4*n^2+2)*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[((4*n - 4 - (7*n^3 - 14*n^2 - 23*n - 2)*x + (n^5 - 3*n^4 - 21*n^3 - n^2 + 8*n)*x^2 + (2*n^5 - 4*n^3 + 2*n)*x^3)*P[n - 1, x] + (8*x - (14*n^2 - 14)*x^2 + (2*n^4 - 18*n^2 + 8)*x^3 + (2*n^4 - 4*n^2 + 2)*x^4)*D[P[n - 1, x], x])/(4*n - (7*n^3 - 7*n^2 + 2*n)*x + (n^5 - 2*n^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] = ((4*n - 4 - (7*n^3 - 14*n^2 - 23*n - 2)*x + (n^5 - 3*n^4 - 21*n^3 - n^2 + 8*n)*x^2 + (2*n^5 - 4*n^3 + 2*n)*x^3)*v[i - 1] + (8*x - (14*n^2 - 14)*x^2 + (2*n^4 - 18*n^2 + 8)*x^3 + (2*n^4 - 4*n^2 + 2)*x^4)*deriv(v[i - 1], x))/(4*n - (7*n^3 - 7*n^2 + 2*n)*x + (n^5 - 2*n^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) = binomial(n - 1, k - 1)*binomial(n + k, k).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

A357367 → (θ+1)²

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.