A257608 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-8 + 9\,n\right)\,P_{n}(x) = \left(8 - 272\,x - 136\,x^{2} + 72\,n - 2304\,n x - 999\,n x^{2} + 1296\,n^{2} x + 1944\,n^{2} x^{2} - 729\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(72\,x - 72\,x^{3} + 648\,n x - 81\,n x^{2} - 567\,n x^{3} - 729\,n^{2} x^{2} + 729\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-16 + 128\,x - 63\,n - 288\,n x + 162\,n^{2} x\right)\,P_{n-1}(x) + \left(-72\,x + 72\,x^{2} + 81\,n x - 81\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	40	3						44
3	1	438	657	4					1100
4	1	4436	24966	8872	5				38280
5	1	44434	660666	880888	111085	6			1697080
6	1	444432	15018741	49329728	25031235	1333296	7		91157440
7	1	4444430	316021887	2122156940	2652696175	632043774	15555505	8	5742918720

All 15 rows

Technical details

Node id: n-9f302d2b05fb08ea · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-a684a258783da613

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: 1,         2;
  2: 1,        40,            3;
  3: 1,       438,          657,             4;
  4: 1,      4436,        24966,          8872,              5;
  5: 1,     44434,       660666,        880888,         111085,              6;
  6: 1,    444432,     15018741,      49329728,       25031235,        1333296,             7;
  7: 1,   4444430,    316021887,    2122156940,     2652696175,      632043774,      15555505,            8;
  8: 1,  44444428,   6371081328,   78803070056,   196299516950,   118204605084,   14865856432,    177777712,          9;
  9: 1, 444444426, 125317210320, 2673698592288, 11794258655370, 14153110386444, 4678972536504, 334179227520, 1999999917, 10;

%F Formula

The row polynomials satisfy (9*n-8)*P(n,x) = (72*n+8+(1296*n^2-2304*n-272)*x-(729*n^3-1944*n^2+999*n+136)*x^2)*P(n-2,x) + ((648*n+72)*x-(729*n^2+81*n)*x^2+(729*n^2-567*n-72)*x^3)*d/dx P(n-2,x) - (63*n+16-(162*n^2-288*n+128)*x)*P(n-1,x) + ((81*n-72)*x-(81*n-72)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((72*n + 8 + (1296*n^2 - 2304*n - 272)*x - (729*n^3 - 1944*n^2 + 999*n + 136)*x^2)*P[n - 2, x] + ((648*n + 72)*x - (729*n^2 + 81*n)*x^2 + (729*n^2 - 567*n - 72)*x^3)*D[P[n - 2, x], x] - (63*n + 16 - (162*n^2 - 288*n + 128)*x)*P[n - 1, x] + ((81*n - 72)*x - (81*n - 72)*x^2)*D[P[n - 1, x], x])/(9*n - 8)]];
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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((72*n + 8 + (1296*n^2 - 2304*n - 272)*x - (729*n^3 - 1944*n^2 + 999*n + 136)*x^2)*v[i - 2] + ((648*n + 72)*x - (729*n^2 + 81*n)*x^2 + (729*n^2 - 567*n - 72)*x^3)*deriv(v[i - 2], x) - (63*n + 16 - (162*n^2 - 288*n + 128)*x)*v[i - 1] + ((81*n - 72)*x - (81*n - 72)*x^2)*deriv(v[i - 1], x))/(9*n - 8)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.?
    
    ↑ Liu–Wang theorem (above)

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.