A330798 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-2\,n^{2} + 4\,n^{2} x - 2\,n^{3} - 10\,n^{3} x + 4\,n^{4} - 24\,n^{4} x - 8\,n^{4} x^{2} + 24\,n^{5} x - 8\,n^{5} x^{2} + 16\,n^{6} x^{2}\right)\,P_{n}(x) = \left(-12\,x - 24\,x^{2} - 12\,x^{3} + 4\,n + 44\,n x + 78\,n x^{2} + 36\,n x^{3} - 8\,n^{2} + 16\,n^{2} x + 74\,n^{2} x^{2} + 48\,n^{2} x^{3} - 4\,n^{3} - 126\,n^{3} x - 246\,n^{3} x^{2} - 120\,n^{3} x^{3} + 8\,n^{4} - 10\,n^{4} x - 130\,n^{4} x^{2} - 108\,n^{4} x^{3} + 76\,n^{5} x + 144\,n^{5} x^{2} + 84\,n^{5} x^{3} + 56\,n^{6} x^{2} + 72\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 18\,x^{2} + 27\,x^{3} + 12\,x^{4} - 5\,n x - 26\,n x^{2} - 33\,n x^{3} - 12\,n x^{4} + n^{2} - 7\,n^{2} x - 85\,n^{2} x^{2} - 149\,n^{2} x^{3} - 72\,n^{2} x^{4} + n^{3} + 7\,n^{3} x + 11\,n^{3} x^{2} - 19\,n^{3} x^{3} - 24\,n^{3} x^{4} + 6\,n^{4} x + 56\,n^{4} x^{2} + 110\,n^{4} x^{3} + 60\,n^{4} x^{4} + 4\,n^{5} x^{2} + 40\,n^{5} x^{3} + 36\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	8							10
2	6	60	81						147
3	20	336	1008	768					2132
4	70	1680	8100	13200	6875				29925
5	252	7920	53460	137280	150150	58968			408030
6	924	36036	315315	1121120	1911000	1559376	489804		5433575
7	3432	160160	1729728	7920640	18564000	23442048	15193920	3969024	70982952

All 15 rows

Technical details

Node id: n-88fe5d645a597110 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-afe77babea4b595f

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:     2,       8;
  2:     6,      60,       81;
  3:    20,     336,     1008,       768;
  4:    70,    1680,     8100,     13200,       6875;
  5:   252,    7920,    53460,    137280,     150150,      58968;
  6:   924,   36036,   315315,   1121120,    1911000,    1559376,     489804;
  7:  3432,  160160,  1729728,   7920640,   18564000,   23442048,   15193920,    3969024;
  8: 12870,  700128,  9022104,  50791104,  152617500,  263723040,  263234664,  141210432,   31538727;
  9: 48620, 3023280, 45349200, 303878400, 1119195000, 2471182560, 3363554040, 2768832000, 1265442750, 246675000;

%F Formula

The row polynomials satisfy (4*n^4-2*n^3-2*n^2+(24*n^5-24*n^4-10*n^3+4*n^2)*x+(16*n^6-8*n^5-8*n^4)*x^2)*P(n,x) = (8*n^4-4*n^3-8*n^2+4*n+(76*n^5-10*n^4-126*n^3+16*n^2+44*n-12)*x+(56*n^6+144*n^5-130*n^4-246*n^3+74*n^2+78*n-24)*x^2+(72*n^6+84*n^5-108*n^4-120*n^3+48*n^2+36*n-12)*x^3)*P(n-1,x) + (n^3+n^2+(6*n^4+7*n^3-7*n^2-5*n+3)*x+(4*n^5+56*n^4+11*n^3-85*n^2-26*n+18)*x^2+(40*n^5+110*n^4-19*n^3-149*n^2-33*n+27)*x^3+(36*n^5+60*n^4-24*n^3-72*n^2-12*n+12)*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[((8*n^4 - 4*n^3 - 8*n^2 + 4*n + (76*n^5 - 10*n^4 - 126*n^3 + 16*n^2 + 44*n - 12)*x + (56*n^6 + 144*n^5 - 130*n^4 - 246*n^3 + 74*n^2 + 78*n - 24)*x^2 + (72*n^6 + 84*n^5 - 108*n^4 - 120*n^3 + 48*n^2 + 36*n - 12)*x^3)*P[n - 1, x] + (n^3 + n^2 + (6*n^4 + 7*n^3 - 7*n^2 - 5*n + 3)*x + (4*n^5 + 56*n^4 + 11*n^3 - 85*n^2 - 26*n + 18)*x^2 + (40*n^5 + 110*n^4 - 19*n^3 - 149*n^2 - 33*n + 27)*x^3 + (36*n^5 + 60*n^4 - 24*n^3 - 72*n^2 - 12*n + 12)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 2*n^3 - 2*n^2 + (24*n^5 - 24*n^4 - 10*n^3 + 4*n^2)*x + (16*n^6 - 8*n^5 - 8*n^4)*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] = ((8*n^4 - 4*n^3 - 8*n^2 + 4*n + (76*n^5 - 10*n^4 - 126*n^3 + 16*n^2 + 44*n - 12)*x + (56*n^6 + 144*n^5 - 130*n^4 - 246*n^3 + 74*n^2 + 78*n - 24)*x^2 + (72*n^6 + 84*n^5 - 108*n^4 - 120*n^3 + 48*n^2 + 36*n - 12)*x^3)*v[i - 1] + (n^3 + n^2 + (6*n^4 + 7*n^3 - 7*n^2 - 5*n + 3)*x + (4*n^5 + 56*n^4 + 11*n^3 - 85*n^2 - 26*n + 18)*x^2 + (40*n^5 + 110*n^4 - 19*n^3 - 149*n^2 - 33*n + 27)*x^3 + (36*n^5 + 60*n^4 - 24*n^3 - 72*n^2 - 12*n + 12)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 2*n^3 - 2*n^2 + (24*n^5 - 24*n^4 - 10*n^3 + 4*n^2)*x + (16*n^6 - 8*n^5 - 8*n^4)*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, 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 → (θ+1)²

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.