A360282 → D

nonnegative coefficients

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

Recurrence

\(\left(120\,n - 248\,n^{2} + 190\,n^{3} - 64\,n^{4} + 8\,n^{5}\right)\,P_{n}(x) = \left(1620\,x^{2} + 18\,x^{3} - 4968\,n x^{2} + 189\,n x^{3} + 5868\,n^{2} x^{2} - 383\,n^{2} x^{3} - 3336\,n^{3} x^{2} + 262\,n^{3} x^{3} + 912\,n^{4} x^{2} - 76\,n^{4} x^{3} - 96\,n^{5} x^{2} + 8\,n^{5} x^{3}\right)\,P_{n-3}(x) + \left(540\,x^{3} + 54\,x^{4} - 1656\,n x^{3} - 81\,n x^{4} + 1956\,n^{2} x^{3} + 39\,n^{2} x^{4} - 1112\,n^{3} x^{3} - 6\,n^{3} x^{4} + 304\,n^{4} x^{3} - 32\,n^{5} x^{3}\right)\,P_{n-3}'(x) + \left(1800 - 300\,x + 240\,x^{2} - 5640\,n + 1190\,n x - 936\,n x^{2} + 6848\,n^{2} - 1608\,n^{2} x + 1236\,n^{2} x^{2} - 4032\,n^{3} + 992\,n^{3} x - 750\,n^{3} x^{2} + 1152\,n^{4} - 288\,n^{4} x + 216\,n^{4} x^{2} - 128\,n^{5} + 32\,n^{5} x - 24\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(-1800\,x - 1140\,x^{2} - 96\,x^{3} + 5640\,n x + 3226\,n x^{2} + 144\,n x^{3} - 6848\,n^{2} x - 3648\,n^{2} x^{2} - 72\,n^{2} x^{3} + 4032\,n^{3} x + 2056\,n^{3} x^{2} + 12\,n^{3} x^{3} - 1152\,n^{4} x - 576\,n^{4} x^{2} + 128\,n^{5} x + 64\,n^{5} x^{2}\right)\,P_{n-2}'(x) + \left(-540 - 270\,x + 2016\,n + 873\,n x - 2760\,n^{2} - 1101\,n^{2} x + 1776\,n^{3} + 678\,n^{3} x - 544\,n^{4} - 204\,n^{4} x + 64\,n^{5} + 24\,n^{5} x\right)\,P_{n-1}(x) + \left(300\,x + 30\,x^{2} - 1040\,n x - 57\,n x^{2} + 1388\,n^{2} x + 33\,n^{2} x^{2} - 888\,n^{3} x - 6\,n^{3} x^{2} + 272\,n^{4} x - 32\,n^{5} x\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 3 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	3	4													7
3	10	24	9												43
4	35	120	90	16											261
5	126	560	630	240	25										1581
6	462	2520	3780	2240	525	36									9563
7	1716	11088	20790	16800	6300	1008	49								57751
8	6435	48048	108108	110880	57750	15120	1764	64							348169
9	24310	205920	540540	672672	450450	166320	32340	2880	81						2095513
10	92378	875160	2625480	3843840	3153150	1513512	420420	63360	4455	100					12591855
11	352716	3695120	12471030	21003840	20420400	12108096	4414410	960960	115830	6600	121				75549123
12	1352078	15519504	58198140	110853600	124710300	88216128	40023984	11531520	2027025	200200	9438	144			452642061
13	5200300	64899744	267711444	569048480	727476750	598609440	325909584	117621504	27567540	4004000	330330	13104	169		2708392389
14	20058300	270415600	1216870200	2855588736	4090035950	3841077240	2444321880	1064194560	314269956	61261200	7487480	524160	17745	196	16186123203

Technical details

Node id: n-c0ca490ee0c92fd2 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-555bf30a4e16df31

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     0;
  1:     1;
  2:     3,      4;
  3:    10,     24,      9;
  4:    35,    120,     90,     16;
  5:   126,    560,    630,    240,     25;
  6:   462,   2520,   3780,   2240,    525,     36;
  7:  1716,  11088,  20790,  16800,   6300,   1008,    49;
  8:  6435,  48048, 108108, 110880,  57750,  15120,  1764,   64;
  9: 24310, 205920, 540540, 672672, 450450, 166320, 32340, 2880, 81;

%F Formula

The row polynomials satisfy (8*n^5-64*n^4+190*n^3-248*n^2+120*n)*P(n,x) = -((96*n^5-912*n^4+3336*n^3-5868*n^2+4968*n-1620)*x^2-(8*n^5-76*n^4+262*n^3-383*n^2+189*n+18)*x^3)*P(n-3,x) - ((32*n^5-304*n^4+1112*n^3-1956*n^2+1656*n-540)*x^3+(6*n^3-39*n^2+81*n-54)*x^4)*d/dx P(n-3,x) - (128*n^5-1152*n^4+4032*n^3-6848*n^2+5640*n-1800-(32*n^5-288*n^4+992*n^3-1608*n^2+1190*n-300)*x+(24*n^5-216*n^4+750*n^3-1236*n^2+936*n-240)*x^2)*P(n-2,x) + ((128*n^5-1152*n^4+4032*n^3-6848*n^2+5640*n-1800)*x+(64*n^5-576*n^4+2056*n^3-3648*n^2+3226*n-1140)*x^2+(12*n^3-72*n^2+144*n-96)*x^3)*d/dx P(n-2,x) + (64*n^5-544*n^4+1776*n^3-2760*n^2+2016*n-540+(24*n^5-204*n^4+678*n^3-1101*n^2+873*n-270)*x)*P(n-1,x) - ((32*n^5-272*n^4+888*n^3-1388*n^2+1040*n-300)*x+(6*n^3-33*n^2+57*n-30)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 3 + 4*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((96*n^5 - 912*n^4 + 3336*n^3 - 5868*n^2 + 4968*n - 1620)*x^2 - (8*n^5 - 76*n^4 + 262*n^3 - 383*n^2 + 189*n + 18)*x^3)*P[n - 3, x] - ((32*n^5 - 304*n^4 + 1112*n^3 - 1956*n^2 + 1656*n - 540)*x^3 + (6*n^3 - 39*n^2 + 81*n - 54)*x^4)*D[P[n - 3, x], x] - (128*n^5 - 1152*n^4 + 4032*n^3 - 6848*n^2 + 5640*n - 1800 - (32*n^5 - 288*n^4 + 992*n^3 - 1608*n^2 + 1190*n - 300)*x + (24*n^5 - 216*n^4 + 750*n^3 - 1236*n^2 + 936*n - 240)*x^2)*P[n - 2, x] + ((128*n^5 - 1152*n^4 + 4032*n^3 - 6848*n^2 + 5640*n - 1800)*x + (64*n^5 - 576*n^4 + 2056*n^3 - 3648*n^2 + 3226*n - 1140)*x^2 + (12*n^3 - 72*n^2 + 144*n - 96)*x^3)*D[P[n - 2, x], x] + (64*n^5 - 544*n^4 + 1776*n^3 - 2760*n^2 + 2016*n - 540 + (24*n^5 - 204*n^4 + 678*n^3 - 1101*n^2 + 873*n - 270)*x)*P[n - 1, x] - ((32*n^5 - 272*n^4 + 888*n^3 - 1388*n^2 + 1040*n - 300)*x + (6*n^3 - 33*n^2 + 57*n - 30)*x^2)*D[P[n - 1, x], x])/(8*n^5 - 64*n^4 + 190*n^3 - 248*n^2 + 120*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 3 + 4*x; for(n = 3, N, my(i = n + 1); v[i] = (-((96*n^5 - 912*n^4 + 3336*n^3 - 5868*n^2 + 4968*n - 1620)*x^2 - (8*n^5 - 76*n^4 + 262*n^3 - 383*n^2 + 189*n + 18)*x^3)*v[i - 3] - ((32*n^5 - 304*n^4 + 1112*n^3 - 1956*n^2 + 1656*n - 540)*x^3 + (6*n^3 - 39*n^2 + 81*n - 54)*x^4)*deriv(v[i - 3], x) - (128*n^5 - 1152*n^4 + 4032*n^3 - 6848*n^2 + 5640*n - 1800 - (32*n^5 - 288*n^4 + 992*n^3 - 1608*n^2 + 1190*n - 300)*x + (24*n^5 - 216*n^4 + 750*n^3 - 1236*n^2 + 936*n - 240)*x^2)*v[i - 2] + ((128*n^5 - 1152*n^4 + 4032*n^3 - 6848*n^2 + 5640*n - 1800)*x + (64*n^5 - 576*n^4 + 2056*n^3 - 3648*n^2 + 3226*n - 1140)*x^2 + (12*n^3 - 72*n^2 + 144*n - 96)*x^3)*deriv(v[i - 2], x) + (64*n^5 - 544*n^4 + 1776*n^3 - 2760*n^2 + 2016*n - 540 + (24*n^5 - 204*n^4 + 678*n^3 - 1101*n^2 + 873*n - 270)*x)*v[i - 1] - ((32*n^5 - 272*n^4 + 888*n^3 - 1388*n^2 + 1040*n - 300)*x + (6*n^3 - 33*n^2 + 57*n - 30)*x^2)*deriv(v[i - 1], x))/(8*n^5 - 64*n^4 + 190*n^3 - 248*n^2 + 120*n)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, …?
  - 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
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

A367270 → x ↦ x+1 → θ+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
[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.