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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	12870	700128	9022104	50791104	152617500	263723040	263234664	141210432	31538727							912850569
9	48620	3023280	45349200	303878400	1119195000	2471182560	3363554040	2768832000	1265442750	246675000						11587180850
10	184756	12932920	221600610	1725905280	7550835600	20387256120	35366781450	39594297600	27693215550	11016505500	1904281665					145473797051
11	705432	54915168	1059078240	9414028800	47805615000	153067955040	324088324560	461491430400	438056318700	266112990000	93671772480	14540497920				1809363631740
12	2704156	231784800	4971783960	49717839600	287888225625	1068641093520	2679414437700	4649526161280	5604339569400	4612625160000	2475174770640	781046884800	109997436276			22323577851757
13	10400600	973496160	22998846780	255743167680	1664994581250	7041349715400	20445993247680	41964954071040	61561514654640	64252788600000	46647524523600	22407634319616	6405886200168	825479089568		273497844914182
14	40116600	4071834900	105089268375	1287019188000	9313954650000	44259912496800	146303599642200	347437119744000	602232208578000	764149235850000	702711637287660	456155412935040	198277430005200	51803024498400	6151837005000	3330191593100175

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.