A360282 → θ+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2\,x\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	2														2
2	0	6	6													12
3	0	20	36	12												68
4	0	70	180	120	20											390
5	0	252	840	840	300	30										2262
6	0	924	3780	5040	2800	630	42									13216
7	0	3432	16632	27720	21000	7560	1176	56								77576
8	0	12870	72072	144144	138600	69300	17640	2016	72							456714
9	0	48620	308880	720720	840840	540540	194040	36960	3240	90						2693930
10	0	184756	1312740	3500640	4804800	3783780	1765764	480480	71280	4950	110					15909300
11	0	705432	5542680	16628040	26254800	24504480	14126112	5045040	1081080	128700	7260	132				94023756
12	0	2704156	23279256	77597520	138567000	149652360	102918816	45741696	12972960	2252250	220220	10296	156			555916686
13	0	10400600	97349616	356948592	711310600	872972100	698377680	372468096	132324192	30630600	4404400	360360	14196	182		3287561214
14	0	40116600	405623400	1622493600	3569485920	4908043140	4481256780	2793510720	1197218880	349188840	67387320	8168160	567840	19110	210	19443080520

Technical details

Node id: n-0f7e7e7ef029e461 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-04db0035d045da4f

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: 0,     2;
  2: 0,     6,      6;
  3: 0,    20,     36,     12;
  4: 0,    70,    180,    120,     20;
  5: 0,   252,    840,    840,    300,     30;
  6: 0,   924,   3780,   5040,   2800,    630,     42;
  7: 0,  3432,  16632,  27720,  21000,   7560,   1176,    56;
  8: 0, 12870,  72072, 144144, 138600,  69300,  17640,  2016,   72;
  9: 0, 48620, 308880, 720720, 840840, 540540, 194040, 36960, 3240, 90;

%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) = 1, P(1,x) = 2*x, P(2,x) = 6*x + 6*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[2, x_] := 6*x + 6*x^2;
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] = 1; v[2] = 2*x; v[3] = 6*x + 6*x^2; 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 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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 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

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.