A110608 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(48\,x + 12\,n + 104\,n x + 14\,n^{2} + 84\,n^{2} x + 4\,n^{3} + 30\,n^{3} x + 4\,n^{4} x\right)\,P_{n}(x) = \left(-2 + 140\,x + 162\,x^{2} + 30\,n + 374\,n x + 361\,n x^{2} + 42\,n^{2} + 349\,n^{2} x + 290\,n^{2} x^{2} + 14\,n^{3} + 130\,n^{3} x + 99\,n^{3} x^{2} + 15\,n^{4} x + 12\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(2 + 6\,x + 82\,x^{2} - 90\,x^{3} + 3\,n + 22\,n x + 102\,n x^{2} - 127\,n x^{3} + n^{2} + 13\,n^{2} x + 43\,n^{2} x^{2} - 57\,n^{2} x^{3} + n^{3} x + 7\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 6 + 16\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	6	16	3															25
1	20	90	54	4														168
2	70	448	504	128	5													1155
3	252	2100	3600	1800	250	6												8008
4	924	9504	22275	17600	4950	432	7											55692
5	3432	42042	126126	140140	63700	11466	686	8										387600
6	12870	183040	672672	978432	637000	188160	23520	1024	9									2696727
7	48620	787644	3436992	6237504	5397840	2313360	479808	44064	1458	10								18747300
8	184756	3359200	17005950	37209600	40698000	23442048	7122150	1094400	76950	2000	11							130195065
9	705432	14226212	82074300	211048200	281397600	206827236	85164156	19311600	2286900	127050	2662	12						903171360
10	2704156	59907456	388328688	1150603520	1820290725	1644686208	870566928	269305344	47338830	4452800	200376	3456	13					6258388500
11	10400600	251100200	1807921440	6076624840	11170266250	12063887550	7901493600	3160597440	761929740	106892500	8179600	304200	4394	14				43319602368
12	40116600	1048380480	8305139115	31266406080	65681165550	82965682800	65335475205	32509002240	10182151980	1967565600	225450225	14309568	447174	5488	15			299541298120
13	155117520	4362680250	37724352750	157417669500	372831322500	541351080270	501251000250	301308579000	117903357000	29718438750	4707400698	448893900	24014900	639450	6750	16		2069204553504
14	601080390	18103127040	169716816000	778116864000	2054714844000	3381473571840	3616298125440	2567026176000	1218334689000	385053926400	79824640896	10555324416	850813600	38886400	892800	8192	17	14280709786431

Technical details

Node id: n-7f9f22821e7542de · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-93d3345b2e8ce678

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:      6,       16,        3;
  1:     20,       90,       54,         4;
  2:     70,      448,      504,       128,         5;
  3:    252,     2100,     3600,      1800,       250,         6;
  4:    924,     9504,    22275,     17600,      4950,       432,        7;
  5:   3432,    42042,   126126,    140140,     63700,     11466,      686,        8;
  6:  12870,   183040,   672672,    978432,    637000,    188160,    23520,     1024,       9;
  7:  48620,   787644,  3436992,   6237504,   5397840,   2313360,   479808,    44064,    1458,     10;
  8: 184756,  3359200, 17005950,  37209600,  40698000,  23442048,  7122150,  1094400,   76950,   2000,   11;
  9: 705432, 14226212, 82074300, 211048200, 281397600, 206827236, 85164156, 19311600, 2286900, 127050, 2662, 12;

%F Formula

The row polynomials satisfy (4*n^3+14*n^2+12*n+(4*n^4+30*n^3+84*n^2+104*n+48)*x)*P(n,x) = (14*n^3+42*n^2+30*n-2+(15*n^4+130*n^3+349*n^2+374*n+140)*x+(12*n^4+99*n^3+290*n^2+361*n+162)*x^2)*P(n-1,x) + (n^2+3*n+2+(n^3+13*n^2+22*n+6)*x+(7*n^3+43*n^2+102*n+82)*x^2-(8*n^3+57*n^2+127*n+90)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + 16*x + 3*x^2.

%t Mathematica

P[0, x_] := 6 + 16*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^3 + 42*n^2 + 30*n - 2 + (15*n^4 + 130*n^3 + 349*n^2 + 374*n + 140)*x + (12*n^4 + 99*n^3 + 290*n^2 + 361*n + 162)*x^2)*P[n - 1, x] + (n^2 + 3*n + 2 + (n^3 + 13*n^2 + 22*n + 6)*x + (7*n^3 + 43*n^2 + 102*n + 82)*x^2 - (8*n^3 + 57*n^2 + 127*n + 90)*x^3)*D[P[n - 1, x], x])/(4*n^3 + 14*n^2 + 12*n + (4*n^4 + 30*n^3 + 84*n^2 + 104*n + 48)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 6 + 16*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n^3 + 42*n^2 + 30*n - 2 + (15*n^4 + 130*n^3 + 349*n^2 + 374*n + 140)*x + (12*n^4 + 99*n^3 + 290*n^2 + 361*n + 162)*x^2)*v[i - 1] + (n^2 + 3*n + 2 + (n^3 + 13*n^2 + 22*n + 6)*x + (7*n^3 + 43*n^2 + 102*n + 82)*x^2 - (8*n^3 + 57*n^2 + 127*n + 90)*x^3)*deriv(v[i - 1], x))/(4*n^3 + 14*n^2 + 12*n + (4*n^4 + 30*n^3 + 84*n^2 + 104*n + 48)*x)); 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 Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k).?
  - 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.