A110608 → D+1

nonnegative coefficients

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

Recurrence

\(\left(36 + 12\,x^{2} + 78\,n + 24\,n x + 14\,n x^{2} + 66\,n^{2} + 28\,n^{2} x + 4\,n^{2} x^{2} + 26\,n^{3} + 8\,n^{3} x + 4\,n^{4}\right)\,P_{n}(x) = \left(166 + 49\,x + 60\,x^{2} + 25\,x^{3} + 367\,n + 232\,n x + 131\,n x^{2} + 35\,n x^{3} + 302\,n^{2} + 270\,n^{2} x + 97\,n^{2} x^{2} + 12\,n^{2} x^{3} + 110\,n^{3} + 109\,n^{3} x + 24\,n^{3} x^{2} + 15\,n^{4} + 12\,n^{4} x\right)\,P_{n-1}(x) + \left(6 + 47\,x - 25\,x^{2} - 15\,x^{3} - 13\,x^{4} + 9\,n + 86\,n x - 52\,n x^{2} - 35\,n x^{3} - 8\,n x^{4} + 5\,n^{2} + 46\,n^{2} x - 35\,n^{2} x^{2} - 16\,n^{2} x^{3} + n^{3} + 7\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 14 + 10\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	14	10	1															25
1	65	81	21	1														168
2	294	560	264	36	1													1155
3	1302	3450	2550	650	55	1												8008
4	5676	19602	20625	8360	1350	78	1											55692
5	24453	105105	147147	85995	22295	2499	105	1										387600
6	104390	539968	958048	754208	284200	51520	4256	136	1									2696727
7	442442	2685150	5823792	5877648	3007368	796824	107100	6804	171	1								18747300
8	1864356	13016900	33575850	41860800	27674640	10011708	1975050	205200	10350	210	1							130195065
9	7818538	61829306	185644250	277880130	228635550	107469054	29063958	4446750	368445	15125	253	1						903171360
10	32657884	288839520	992395536	1743883460	1734629985	1020314592	360008880	75742128	9267390	627440	21384	300	1					6258388500
11	135950700	1330831060	5160109110	10455369210	12287292875	8783356725	3894307560	1072345560	180862110	18125250	1022450	29406	351	1				43319602368
12	564306840	6060949650	26218184265	60361533960	82274302110	69829449690	37779016275	13114427040	2902159260	401711310	33612579	1605240	39494	406	1			299541298120
13	2336457645	27330908625	130638036375	337619475375	525692164725	519868894545	335252292375	142466556375	39846967875	7251299055	839431593	59575425	2441075	51975	465	1		2069204553504
14	9652643910	122196107520	640159920000	1838301091200	3228837612000	3663263036160	2762761921920	1403842440000	481919054760	111073248000	16932499584	1664976768	101556000	3610880	67200	528	1	14280709786431

Technical details

Node id: n-1cde98f92dda3d31 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-66f0156215487683

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:      14,       10,         1;
  1:      65,       81,        21,         1;
  2:     294,      560,       264,        36,         1;
  3:    1302,     3450,      2550,       650,        55,         1;
  4:    5676,    19602,     20625,      8360,      1350,        78,        1;
  5:   24453,   105105,    147147,     85995,     22295,      2499,      105,       1;
  6:  104390,   539968,    958048,    754208,    284200,     51520,     4256,     136,      1;
  7:  442442,  2685150,   5823792,   5877648,   3007368,    796824,   107100,    6804,    171,     1;
  8: 1864356, 13016900,  33575850,  41860800,  27674640,  10011708,  1975050,  205200,  10350,   210,   1;
  9: 7818538, 61829306, 185644250, 277880130, 228635550, 107469054, 29063958, 4446750, 368445, 15125, 253, 1;

%F Formula

The row polynomials satisfy (4*n^4+26*n^3+66*n^2+78*n+36+(8*n^3+28*n^2+24*n)*x+(4*n^2+14*n+12)*x^2)*P(n,x) = (15*n^4+110*n^3+302*n^2+367*n+166+(12*n^4+109*n^3+270*n^2+232*n+49)*x+(24*n^3+97*n^2+131*n+60)*x^2+(12*n^2+35*n+25)*x^3)*P(n-1,x) + (n^3+5*n^2+9*n+6+(7*n^3+46*n^2+86*n+47)*x-(8*n^3+35*n^2+52*n+25)*x^2-(16*n^2+35*n+15)*x^3-(8*n+13)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 10*x + x^2.

%t Mathematica

P[0, x_] := 14 + 10*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15*n^4 + 110*n^3 + 302*n^2 + 367*n + 166 + (12*n^4 + 109*n^3 + 270*n^2 + 232*n + 49)*x + (24*n^3 + 97*n^2 + 131*n + 60)*x^2 + (12*n^2 + 35*n + 25)*x^3)*P[n - 1, x] + (n^3 + 5*n^2 + 9*n + 6 + (7*n^3 + 46*n^2 + 86*n + 47)*x - (8*n^3 + 35*n^2 + 52*n + 25)*x^2 - (16*n^2 + 35*n + 15)*x^3 - (8*n + 13)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 26*n^3 + 66*n^2 + 78*n + 36 + (8*n^3 + 28*n^2 + 24*n)*x + (4*n^2 + 14*n + 12)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 14 + 10*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15*n^4 + 110*n^3 + 302*n^2 + 367*n + 166 + (12*n^4 + 109*n^3 + 270*n^2 + 232*n + 49)*x + (24*n^3 + 97*n^2 + 131*n + 60)*x^2 + (12*n^2 + 35*n + 25)*x^3)*v[i - 1] + (n^3 + 5*n^2 + 9*n + 6 + (7*n^3 + 46*n^2 + 86*n + 47)*x - (8*n^3 + 35*n^2 + 52*n + 25)*x^2 - (16*n^2 + 35*n + 15)*x^3 - (8*n + 13)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 26*n^3 + 66*n^2 + 78*n + 36 + (8*n^3 + 28*n^2 + 24*n)*x + (4*n^2 + 14*n + 12)*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] 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.