A330798 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n^{2} - 2\,n^{2} x^{2} + 4\,n^{3} + 2\,n^{3} x - 4\,n^{3} x^{2} - 4\,n^{4} + 4\,n^{4} x + 8\,n^{5}\right)\,P_{n}(x) = \left(1 - 4\,x^{2} - 3\,x^{3} - 9\,n + 4\,n x + 23\,n x^{2} + 12\,n x^{3} + 3\,n^{2} - 14\,n^{2} x - 20\,n^{2} x^{2} - 3\,n^{2} x^{3} + 5\,n^{3} - 16\,n^{3} x - 29\,n^{3} x^{2} - 18\,n^{3} x^{3} + 20\,n^{4} + 38\,n^{4} x + 18\,n^{4} x^{2} + 28\,n^{5} + 36\,n^{5} x\right)\,P_{n-1}(x) + \left(-2 - x + 8\,x^{2} + 10\,x^{3} + 3\,x^{4} + 2\,n + n x - 16\,n x^{2} - 21\,n x^{3} - 6\,n x^{4} - 7\,n^{2} x - 20\,n^{2} x^{2} - 22\,n^{2} x^{3} - 9\,n^{2} x^{4} - 2\,n^{3} + 11\,n^{3} x + 22\,n^{3} x^{2} + 9\,n^{3} x^{3} + 2\,n^{4} + 20\,n^{4} x + 18\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	2							6
2	21	33	9						63
3	104	308	256	48					716
4	490	2220	3375	1925	275				8285
5	2232	13860	31680	32604	14196	1638			96210
6	9933	79079	245245	375830	293020	103292	9996		1116395
7	43472	424424	1677312	3465280	3998400	2511648	744192	62016	12926744

All 15 rows

Technical details

Node id: n-00ee928c36df0ff4 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-0e9e89899f02b109

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:      4,        2;
  2:     21,       33,        9;
  3:    104,      308,      256,        48;
  4:    490,     2220,     3375,      1925,       275;
  5:   2232,    13860,    31680,     32604,     14196,      1638;
  6:   9933,    79079,   245245,    375830,    293020,    103292,      9996;
  7:  43472,   424424,  1677312,   3465280,   3998400,   2511648,    744192,     62016;
  8: 187902,  2179944, 10525788,  27593244,  42732900,  39558456,  20817027,   5321349,   389367;
  9: 804440, 10833420, 62016000, 198063600, 387987600, 480507720, 371484960, 168245000, 37823500, 2466750;

%F Formula

The row polynomials satisfy (8*n^5-4*n^4+4*n^3+4*n^2+(4*n^4+2*n^3)*x-(4*n^3+2*n^2)*x^2)*P(n,x) = (28*n^5+20*n^4+5*n^3+3*n^2-9*n+1+(36*n^5+38*n^4-16*n^3-14*n^2+4*n)*x+(18*n^4-29*n^3-20*n^2+23*n-4)*x^2-(18*n^3+3*n^2-12*n+3)*x^3)*P(n-1,x) + (2*n^4-2*n^3+2*n-2+(20*n^4+11*n^3-7*n^2+n-1)*x+(18*n^4+22*n^3-20*n^2-16*n+8)*x^2+(9*n^3-22*n^2-21*n+10)*x^3-(9*n^2+6*n-3)*x^4)*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[((28*n^5 + 20*n^4 + 5*n^3 + 3*n^2 - 9*n + 1 + (36*n^5 + 38*n^4 - 16*n^3 - 14*n^2 + 4*n)*x + (18*n^4 - 29*n^3 - 20*n^2 + 23*n - 4)*x^2 - (18*n^3 + 3*n^2 - 12*n + 3)*x^3)*P[n - 1, x] + (2*n^4 - 2*n^3 + 2*n - 2 + (20*n^4 + 11*n^3 - 7*n^2 + n - 1)*x + (18*n^4 + 22*n^3 - 20*n^2 - 16*n + 8)*x^2 + (9*n^3 - 22*n^2 - 21*n + 10)*x^3 - (9*n^2 + 6*n - 3)*x^4)*D[P[n - 1, x], x])/(8*n^5 - 4*n^4 + 4*n^3 + 4*n^2 + (4*n^4 + 2*n^3)*x - (4*n^3 + 2*n^2)*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] = ((28*n^5 + 20*n^4 + 5*n^3 + 3*n^2 - 9*n + 1 + (36*n^5 + 38*n^4 - 16*n^3 - 14*n^2 + 4*n)*x + (18*n^4 - 29*n^3 - 20*n^2 + 23*n - 4)*x^2 - (18*n^3 + 3*n^2 - 12*n + 3)*x^3)*v[i - 1] + (2*n^4 - 2*n^3 + 2*n - 2 + (20*n^4 + 11*n^3 - 7*n^2 + n - 1)*x + (18*n^4 + 22*n^3 - 20*n^2 - 16*n + 8)*x^2 + (9*n^3 - 22*n^2 - 21*n + 10)*x^3 - (9*n^2 + 6*n - 3)*x^4)*deriv(v[i - 1], x))/(8*n^5 - 4*n^4 + 4*n^3 + 4*n^2 + (4*n^4 + 2*n^3)*x - (4*n^3 + 2*n^2)*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, interpolating between the central binomial coefficients and the central coefficients of the Cata…?
  - 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

Related ?

Same fingerprint ?

A104978 → D+1

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.