A357613 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + 2\,n x + 16\,n^{2} - 20\,n^{2} x - 42\,n^{3} + 76\,n^{3} x - 2\,n^{3} x^{2} + 36\,n^{4} - 124\,n^{4} x + 12\,n^{4} x^{2} + 72\,n^{5} x - 24\,n^{5} x^{2} + 16\,n^{6} x^{2}\right)\,P_{n}(x) = \left(6 + 18\,x + 18\,x^{2} + 6\,x^{3} - 63\,n - 173\,n x - 159\,n x^{2} - 45\,n x^{3} + 243\,n^{2} + 561\,n^{2} x + 411\,n^{2} x^{2} + 85\,n^{2} x^{3} - 405\,n^{3} - 594\,n^{3} x - 132\,n^{3} x^{2} + 36\,n^{3} x^{3} - n^{3} x^{4} + 243\,n^{4} - 216\,n^{4} x - 696\,n^{4} x^{2} - 212\,n^{4} x^{3} - 3\,n^{4} x^{4} + 486\,n^{5} x + 504\,n^{5} x^{2} + 50\,n^{5} x^{3} + 108\,n^{6} x^{2} + 96\,n^{6} x^{3} + 4\,n^{6} x^{4}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4} - 27\,n x - 73\,n x^{2} - 65\,n x^{3} - 19\,n x^{4} + 81\,n^{2} x + 171\,n^{2} x^{2} + 113\,n^{2} x^{3} + 23\,n^{2} x^{4} - 81\,n^{3} x - 54\,n^{3} x^{2} + 64\,n^{3} x^{3} + 36\,n^{3} x^{4} - n^{3} x^{5} - 162\,n^{4} x^{2} - 228\,n^{4} x^{3} - 70\,n^{4} x^{4} - 4\,n^{4} x^{5} - 36\,n^{5} x^{3} - 40\,n^{5} x^{4} - 4\,n^{5} x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	8														11
2	15	80	54													149
3	84	672	945	320												2021
4	495	5280	11340	8064	1750											26929
5	3003	40040	115830	126720	57750	9072										352415
6	18564	297024	1081080	1601600	1126125	370656	45276									4540325
7	116280	2170560	9523332	17821440	17017000	8648640	2207205	219648								57724105
8	735471	15690048	80582040	182327040	220447500	152373312	60035976	12446720	1042470							725680577
9	4686825	112483800	661923900	1757285376	2574148500	2256297120	1209815880	386979840	67343562	4862000						9035826803
10	30045015	801200400	5314859550	16197667200	27886608750	29654190720	20181324240	8786426880	2357024670	352716000	22355476					111584418901
11	193536720	5677077120	41912795925	144216072000	285427642500	357158561760	295151867010	163176499200	59573151000	13728792000	1799615818	101582208				1368117193261
12	1251677700	40053686400	325722299760	1248956966400	2794186395000	4023628408800	3916067255100	2630501153280	1219760266725	382444920000	77126392200	8986118400	457002364			16669142542129
13	8122425444	281577415392	2500852044600	10574173209600	26389538175000	42995343568320	48194126940960	38149958246400	21501264309525	8562829275000	2348498642490	420550341120	44068085100	2038517600		201972941196551
14	52860229080	1973448552320	19006475538960	87852153602304	241980591352500	440149959849600	558613744088400	509574442291200	338588330600520	163925601840000	57100751307600	13899807230400	2236455318825	212821237440	9026235000	2435176469274149

Technical details

Node id: n-6089882dd1c1a14b · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-00538ae0682bc638

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:       3,         8;
  2:      15,        80,        54;
  3:      84,       672,       945,        320;
  4:     495,      5280,     11340,       8064,       1750;
  5:    3003,     40040,    115830,     126720,      57750,       9072;
  6:   18564,    297024,   1081080,    1601600,    1126125,     370656,      45276;
  7:  116280,   2170560,   9523332,   17821440,   17017000,    8648640,    2207205,    219648;
  8:  735471,  15690048,  80582040,  182327040,  220447500,  152373312,   60035976,  12446720,  1042470;
  9: 4686825, 112483800, 661923900, 1757285376, 2574148500, 2256297120, 1209815880, 386979840, 67343562, 4862000;

%F Formula

The row polynomials satisfy (36*n^4-42*n^3+16*n^2-2*n+(72*n^5-124*n^4+76*n^3-20*n^2+2*n)*x+(16*n^6-24*n^5+12*n^4-2*n^3)*x^2)*P(n,x) = (243*n^4-405*n^3+243*n^2-63*n+6+(486*n^5-216*n^4-594*n^3+561*n^2-173*n+18)*x+(108*n^6+504*n^5-696*n^4-132*n^3+411*n^2-159*n+18)*x^2+(96*n^6+50*n^5-212*n^4+36*n^3+85*n^2-45*n+6)*x^3+(4*n^6-3*n^4-n^3)*x^4)*P(n-1,x) - ((81*n^3-81*n^2+27*n-3)*x+(162*n^4+54*n^3-171*n^2+73*n-9)*x^2+(36*n^5+228*n^4-64*n^3-113*n^2+65*n-9)*x^3+(40*n^5+70*n^4-36*n^3-23*n^2+19*n-3)*x^4+(4*n^5+4*n^4+n^3)*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[((243*n^4 - 405*n^3 + 243*n^2 - 63*n + 6 + (486*n^5 - 216*n^4 - 594*n^3 + 561*n^2 - 173*n + 18)*x + (108*n^6 + 504*n^5 - 696*n^4 - 132*n^3 + 411*n^2 - 159*n + 18)*x^2 + (96*n^6 + 50*n^5 - 212*n^4 + 36*n^3 + 85*n^2 - 45*n + 6)*x^3 + (4*n^6 - 3*n^4 - n^3)*x^4)*P[n - 1, x] - ((81*n^3 - 81*n^2 + 27*n - 3)*x + (162*n^4 + 54*n^3 - 171*n^2 + 73*n - 9)*x^2 + (36*n^5 + 228*n^4 - 64*n^3 - 113*n^2 + 65*n - 9)*x^3 + (40*n^5 + 70*n^4 - 36*n^3 - 23*n^2 + 19*n - 3)*x^4 + (4*n^5 + 4*n^4 + n^3)*x^5)*D[P[n - 1, x], x])/(36*n^4 - 42*n^3 + 16*n^2 - 2*n + (72*n^5 - 124*n^4 + 76*n^3 - 20*n^2 + 2*n)*x + (16*n^6 - 24*n^5 + 12*n^4 - 2*n^3)*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] = ((243*n^4 - 405*n^3 + 243*n^2 - 63*n + 6 + (486*n^5 - 216*n^4 - 594*n^3 + 561*n^2 - 173*n + 18)*x + (108*n^6 + 504*n^5 - 696*n^4 - 132*n^3 + 411*n^2 - 159*n + 18)*x^2 + (96*n^6 + 50*n^5 - 212*n^4 + 36*n^3 + 85*n^2 - 45*n + 6)*x^3 + (4*n^6 - 3*n^4 - n^3)*x^4)*v[i - 1] - ((81*n^3 - 81*n^2 + 27*n - 3)*x + (162*n^4 + 54*n^3 - 171*n^2 + 73*n - 9)*x^2 + (36*n^5 + 228*n^4 - 64*n^3 - 113*n^2 + 65*n - 9)*x^3 + (40*n^5 + 70*n^4 - 36*n^3 - 23*n^2 + 19*n - 3)*x^4 + (4*n^5 + 4*n^4 + n^3)*x^5)*deriv(v[i - 1], x))/(36*n^4 - 42*n^3 + 16*n^2 - 2*n + (72*n^5 - 124*n^4 + 76*n^3 - 20*n^2 + 2*n)*x + (16*n^6 - 24*n^5 + 12*n^4 - 2*n^3)*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 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 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.