A110608 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(48\,x + 12\,n + 104\,n x + 14\,n^{2} + 84\,n^{2} x + 4\,n^{3} + 30\,n^{3} x + 4\,n^{4} x\right)\,P_{n}(x) = \left(-2 + 140\,x + 162\,x^{2} + 30\,n + 374\,n x + 361\,n x^{2} + 42\,n^{2} + 349\,n^{2} x + 290\,n^{2} x^{2} + 14\,n^{3} + 130\,n^{3} x + 99\,n^{3} x^{2} + 15\,n^{4} x + 12\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(2 + 6\,x + 82\,x^{2} - 90\,x^{3} + 3\,n + 22\,n x + 102\,n x^{2} - 127\,n x^{3} + n^{2} + 13\,n^{2} x + 43\,n^{2} x^{2} - 57\,n^{2} x^{3} + n^{3} x + 7\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 6 + 16\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	6	16	3								25
1	20	90	54	4							168
2	70	448	504	128	5						1155
3	252	2100	3600	1800	250	6					8008
4	924	9504	22275	17600	4950	432	7				55692
5	3432	42042	126126	140140	63700	11466	686	8			387600
6	12870	183040	672672	978432	637000	188160	23520	1024	9		2696727
7	48620	787644	3436992	6237504	5397840	2313360	479808	44064	1458	10	18747300

All 15 rows

Technical details

Node id: n-7f9f22821e7542de · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-93d3345b2e8ce678

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:      6,       16,        3;
  1:     20,       90,       54,         4;
  2:     70,      448,      504,       128,         5;
  3:    252,     2100,     3600,      1800,       250,         6;
  4:    924,     9504,    22275,     17600,      4950,       432,        7;
  5:   3432,    42042,   126126,    140140,     63700,     11466,      686,        8;
  6:  12870,   183040,   672672,    978432,    637000,    188160,    23520,     1024,       9;
  7:  48620,   787644,  3436992,   6237504,   5397840,   2313360,   479808,    44064,    1458,     10;
  8: 184756,  3359200, 17005950,  37209600,  40698000,  23442048,  7122150,  1094400,   76950,   2000,   11;
  9: 705432, 14226212, 82074300, 211048200, 281397600, 206827236, 85164156, 19311600, 2286900, 127050, 2662, 12;

%F Formula

The row polynomials satisfy (4*n^3+14*n^2+12*n+(4*n^4+30*n^3+84*n^2+104*n+48)*x)*P(n,x) = (14*n^3+42*n^2+30*n-2+(15*n^4+130*n^3+349*n^2+374*n+140)*x+(12*n^4+99*n^3+290*n^2+361*n+162)*x^2)*P(n-1,x) + (n^2+3*n+2+(n^3+13*n^2+22*n+6)*x+(7*n^3+43*n^2+102*n+82)*x^2-(8*n^3+57*n^2+127*n+90)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + 16*x + 3*x^2.

%t Mathematica

P[0, x_] := 6 + 16*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^3 + 42*n^2 + 30*n - 2 + (15*n^4 + 130*n^3 + 349*n^2 + 374*n + 140)*x + (12*n^4 + 99*n^3 + 290*n^2 + 361*n + 162)*x^2)*P[n - 1, x] + (n^2 + 3*n + 2 + (n^3 + 13*n^2 + 22*n + 6)*x + (7*n^3 + 43*n^2 + 102*n + 82)*x^2 - (8*n^3 + 57*n^2 + 127*n + 90)*x^3)*D[P[n - 1, x], x])/(4*n^3 + 14*n^2 + 12*n + (4*n^4 + 30*n^3 + 84*n^2 + 104*n + 48)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 6 + 16*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n^3 + 42*n^2 + 30*n - 2 + (15*n^4 + 130*n^3 + 349*n^2 + 374*n + 140)*x + (12*n^4 + 99*n^3 + 290*n^2 + 361*n + 162)*x^2)*v[i - 1] + (n^2 + 3*n + 2 + (n^3 + 13*n^2 + 22*n + 6)*x + (7*n^3 + 43*n^2 + 102*n + 82)*x^2 - (8*n^3 + 57*n^2 + 127*n + 90)*x^3)*deriv(v[i - 1], x))/(4*n^3 + 14*n^2 + 12*n + (4*n^4 + 30*n^3 + 84*n^2 + 104*n + 48)*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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k).?
  - 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.