A013612 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 10\,x + 5\,n x\right)\,P_{n}(x) = \left(1 + 20\,x + 75\,x^{2} + 5\,n x + 25\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 20\,x + 75\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	20	75								96
1	1	30	225	500							756
2	1	40	450	2000	3125						5616
3	1	50	750	5000	15625	18750					40176
4	1	60	1125	10000	46875	112500	109375				279936
5	1	70	1575	17500	109375	393750	765625	625000			1912896
6	1	80	2100	28000	218750	1050000	3062500	5000000	3515625		12877056
7	1	90	2700	42000	393750	2362500	9187500	22500000	31640625	19531250	85660416

All 15 rows

Technical details

Node id: n-83658f01107c8a31 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-85115ae2f0d3177d

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1,  20,   75;
  1: 1,  30,  225,   500;
  2: 1,  40,  450,  2000,    3125;
  3: 1,  50,  750,  5000,   15625,   18750;
  4: 1,  60, 1125, 10000,   46875,  112500,   109375;
  5: 1,  70, 1575, 17500,  109375,  393750,   765625,    625000;
  6: 1,  80, 2100, 28000,  218750, 1050000,  3062500,   5000000,   3515625;
  7: 1,  90, 2700, 42000,  393750, 2362500,  9187500,  22500000,  31640625,   19531250;
  8: 1, 100, 3375, 60000,  656250, 4725000, 22968750,  75000000, 158203125,  195312500,  107421875;
  9: 1, 110, 4125, 82500, 1031250, 8662500, 50531250, 206250000, 580078125, 1074218750, 1181640625, 585937500;

%F Formula

The row polynomials satisfy (1+(5*n+10)*x)*P(n,x) = (1+(5*n+20)*x+(25*n+75)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 20*x + 75*x^2.

%t Mathematica

P[0, x_] := 1 + 20*x + 75*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (5*n + 20)*x + (25*n + 75)*x^2)*P[n - 1, x])/(1 + (5*n + 10)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 20*x + 75*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (5*n + 20)*x + (25*n + 75)*x^2)*v[i - 1])/(1 + (5*n + 10)*x)); 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 of coefficients in expansion of (1+5x)^n.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.