A089503 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + x - 3\,n x + 2\,n^{2} x\right)\,P_{n}(x) = \left(1 - 7\,x + 10\,x^{2} - 2\,x^{3} + 5\,n x - 11\,n x^{2} + n x^{3} + 2\,n^{2} x - 7\,n^{2} x^{2} + 6\,n^{2} x^{3} + 8\,n^{3} x^{2} - 7\,n^{3} x^{3} + 2\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(-x^{2} + 3\,x^{3} - x^{4} - 2\,n x^{3} - 2\,n^{2} x^{3} + 3\,n^{2} x^{4} - 2\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	8							9
3	1	24	90						115
4	1	48	504	1344					1897
5	1	80	1620	11520	25200				38421
6	1	120	3960	52800	297000	570240			924121
7	1	168	8190	174720	1801800	8648640	15135120		25768639
8	1	224	15120	470400	7644000	66044160	282522240	461260800	817956945

All 15 rows

Technical details

Node id: n-0b2758f959e92a63 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-4afa4cee4d09ae3f

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 -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: 1,   8;
   3: 1,  24,    90;
   4: 1,  48,   504,    1344;
   5: 1,  80,  1620,   11520,    25200;
   6: 1, 120,  3960,   52800,   297000,     570240;
   7: 1, 168,  8190,  174720,  1801800,    8648640,    15135120;
   8: 1, 224, 15120,  470400,  7644000,   66044160,   282522240,    461260800;
   9: 1, 288, 25704, 1096704, 25704000,  345461760,  2619751680,  10265149440,  15878903040;
  10: 1, 360, 41040, 2298240, 73256400, 1406522880, 16409433600, 112521830400, 411407942400, 609493248000;

%F Formula

The row polynomials satisfy (1+(2*n^2-3*n+1)*x)*P(n,x) = (1+(2*n^2+5*n-7)*x+(8*n^3-7*n^2-11*n+10)*x^2+(2*n^4-7*n^3+6*n^2+n-2)*x^3)*P(n-1,x) - (x^2+(2*n^2+2*n-3)*x^3+(2*n^3-3*n^2+1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (2*n^2 + 5*n - 7)*x + (8*n^3 - 7*n^2 - 11*n + 10)*x^2 + (2*n^4 - 7*n^3 + 6*n^2 + n - 2)*x^3)*P[n - 1, x] - (x^2 + (2*n^2 + 2*n - 3)*x^3 + (2*n^3 - 3*n^2 + 1)*x^4)*D[P[n - 1, x], x])/(1 + (2*n^2 - 3*n + 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + (2*n^2 + 5*n - 7)*x + (8*n^3 - 7*n^2 - 11*n + 10)*x^2 + (2*n^4 - 7*n^3 + 6*n^2 + n - 2)*x^3)*v[i - 1] - (x^2 + (2*n^2 + 2*n - 3)*x^3 + (2*n^3 - 3*n^2 + 1)*x^4)*deriv(v[i - 1], x))/(1 + (2*n^2 - 3*n + 1)*x)); 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 of numbers used for basis change between certain falling factorials.?
  - 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

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.