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	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

All 15 rows

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.