A357613 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 6\,n x - 2\,n x^{2} - 2\,n^{2} - 18\,n^{2} x + 4\,n^{2} x^{2} - 8\,n^{3} + 12\,n^{3} x + 8\,n^{4}\right)\,P_{n}(x) = \left(-18 - 30\,x - 6\,x^{2} + 6\,x^{3} + 87\,n + 157\,n x + 44\,n x^{2} - 25\,n x^{3} - n x^{4} - 96\,n^{2} - 218\,n^{2} x - 99\,n^{2} x^{2} + 20\,n^{2} x^{3} + n^{2} x^{4} - 27\,n^{3} + 43\,n^{3} x + 69\,n^{3} x^{2} + 3\,n^{3} x^{3} + 54\,n^{4} + 48\,n^{4} x + 2\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-6\,x - 9\,x^{2} + 3\,x^{4} + 23\,n x + 42\,n x^{2} + 10\,n x^{3} - 10\,n x^{4} - n x^{5} - 9\,n^{2} x - 37\,n^{2} x^{2} - 31\,n^{2} x^{3} - 3\,n^{2} x^{4} - 18\,n^{3} x - 20\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	5	2							7
2	35	32	6						73
3	252	378	165	20					815
4	1815	3840	2772	784	70				9281
5	13013	35750	36630	17160	3570	252			106375
6	92820	314496	420420	280280	96525	15840	924		1221305
7	658920	2658936	4399668	3836560	1881880	510510	69069	3432	14018975

All 15 rows

Technical details

Node id: n-649fb22a61937772 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-93693dbe17940c80

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:        5,         2;
  2:       35,        32,         6;
  3:      252,       378,       165,        20;
  4:     1815,      3840,      2772,       784,        70;
  5:    13013,     35750,     36630,     17160,      3570,       252;
  6:    92820,    314496,    420420,    280280,     96525,     15840,      924;
  7:   658920,   2658936,   4399668,   3836560,   1881880,    510510,    69069,     3432;
  8:  4657983,  21829632,  43139880,  46667040,  29980860,  11583936,  2586584,   297440,   12870;
  9: 32807775, 175215150, 403038108, 521694096, 416340540, 210815640, 67016040, 12697776, 1268982, 48620;

%F Formula

The row polynomials satisfy (8*n^4-8*n^3-2*n^2+2*n+(12*n^3-18*n^2+6*n)*x+(4*n^2-2*n)*x^2)*P(n,x) = (54*n^4-27*n^3-96*n^2+87*n-18+(48*n^4+43*n^3-218*n^2+157*n-30)*x+(2*n^4+69*n^3-99*n^2+44*n-6)*x^2+(3*n^3+20*n^2-25*n+6)*x^3+(n^2-n)*x^4)*P(n-1,x) - ((18*n^3+9*n^2-23*n+6)*x+(20*n^3+37*n^2-42*n+9)*x^2+(2*n^3+31*n^2-10*n)*x^3+(3*n^2+10*n-3)*x^4+n*x^5)*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[((54*n^4 - 27*n^3 - 96*n^2 + 87*n - 18 + (48*n^4 + 43*n^3 - 218*n^2 + 157*n - 30)*x + (2*n^4 + 69*n^3 - 99*n^2 + 44*n - 6)*x^2 + (3*n^3 + 20*n^2 - 25*n + 6)*x^3 + (n^2 - n)*x^4)*P[n - 1, x] - ((18*n^3 + 9*n^2 - 23*n + 6)*x + (20*n^3 + 37*n^2 - 42*n + 9)*x^2 + (2*n^3 + 31*n^2 - 10*n)*x^3 + (3*n^2 + 10*n - 3)*x^4 + n*x^5)*D[P[n - 1, x], x])/(8*n^4 - 8*n^3 - 2*n^2 + 2*n + (12*n^3 - 18*n^2 + 6*n)*x + (4*n^2 - 2*n)*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] = ((54*n^4 - 27*n^3 - 96*n^2 + 87*n - 18 + (48*n^4 + 43*n^3 - 218*n^2 + 157*n - 30)*x + (2*n^4 + 69*n^3 - 99*n^2 + 44*n - 6)*x^2 + (3*n^3 + 20*n^2 - 25*n + 6)*x^3 + (n^2 - n)*x^4)*v[i - 1] - ((18*n^3 + 9*n^2 - 23*n + 6)*x + (20*n^3 + 37*n^2 - 42*n + 9)*x^2 + (2*n^3 + 31*n^2 - 10*n)*x^3 + (3*n^2 + 10*n - 3)*x^4 + n*x^5)*deriv(v[i - 1], x))/(8*n^4 - 8*n^3 - 2*n^2 + 2*n + (12*n^3 - 18*n^2 + 6*n)*x + (4*n^2 - 2*n)*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*n).?
  - 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.