A272866 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(5 - 4\,x^{2} + 10\,n x + 8\,n x^{2}\right)\,P_{n}(x) = \left(25 + 100\,x + 140\,x^{2} + 80\,x^{3} + 16\,x^{4} + 50\,n x + 140\,n x^{2} + 120\,n x^{3} + 32\,n x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	5	20	12																											37
2	25	200	420	320	80																									1045
3	125	1500	5400	8800	7200	2880	448																							26353
4	625	10000	51000	128000	182000	153600	76160	20480	2304																					624169
5	3125	62500	412500	1400000	2850000	3720000	3192000	1792000	633600	128000	11264																			14204989
6	15625	375000	3037500	13000000	34500000	61200000	75320000	65280000	39744000	16640000	4561920	737280	53248																	314464573
7	78125	2187500	21000000	108500000	355250000	798000000	1283800000	1513600000	1320480000	851200000	400153600	133324800	29818880	4014080	245760															6821652745
8	390625	12500000	138750000	840000000	3272500000	8904000000	17718400000	26496000000	30232800000	26496000000	17819648000	9117696000	3485081600	963379200	181862400	20971520	1114112													145701093457
9	1953125	70312500	885937500	6150000000	27787500000	88830000000	210798000000	382464000000	540367200000	600880000000	528359040000	367165440000	200438784000	84897792000	27316224000	6448742400	1052835840	106168320	4980736											3074024910421
10	9765625	390625000	5507812500	43125000000	221718750000	814500000000	2245950000000	4795200000000	8088660000000	10920800000000	11893270400000	10483968000000	7477516800000	4296499200000	1971302400000	711720960000	197615616000	40697856000	5852364800	524288000	22020096									64214851858021
11	48828125	2148437500	33515625000	292187500000	1684375000000	6991875000000	21973875000000	53988000000000	105939900000000	168454000000000	219139712000000	234584448000000	207186636800000	150934784000000	90402048000000	44226969600000	17486684160000	5498634240000	1342308352000	245104640000	31488737280	2537553920	96468992							1330441377942817
12	244140625	11718750000	200390625000	1925000000000	12297656250000	56925000000000	200873750000000	558360000000000	1250073000000000	2289320000000000	3466620960000000	4372116480000000	4613197888000000	4080642048000000	3025414656000000	1875410944000000	967167590400000	411667660800000	142928445440000	39793459200000	8665458278400	1421030195200	164961976320	12079595520	419430400					27375210841081465
13	1220703125	63476562500	1180664062500	12390625000000	86886718750000	443746875000000	1737360625000000	5391100000000000	13564980000000000	28128100000000000	48627092800000000	70667001600000000	86818247360000000	90487013120000000	80139920640000000	60302508032000000	38477292339200000	20738285568000000	9383831142400000	3533111296000000	1093052792832000	272977362944000	53643994726400	7982599372800	845152256000	56706990080	1811939328			559968674523738733
14	6103515625	341796875000	6870117187500	78203125000000	597187500000000	3335718750000000	14350415625000000	49177700000000000	137409772500000000	318318000000000000	618862244000000000	1018827264000000000	1429523950400000000	1717180147200000000	1770902918400000000	1569993277440000000	1196401583104000000	782459338752000000	437839411609600000	208612884480000000	84049343938560000	28361605775360000	7910686457856000	1790850367488000	320612597760000	43664382361600	4248997724160	263066746880	7784628224	11396358518271662989

Technical details

Node id: n-515d78eac02765ac · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-0136bddfb709905a

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:       5,       20,        12;
  2:      25,      200,       420,        320,          80;
  3:     125,     1500,      5400,       8800,        7200,        2880,          448;
  4:     625,    10000,     51000,     128000,      182000,      153600,        76160,        20480,         2304;
  5:    3125,    62500,    412500,    1400000,     2850000,     3720000,      3192000,      1792000,       633600,       128000,        11264;
  6:   15625,   375000,   3037500,   13000000,    34500000,    61200000,     75320000,     65280000,     39744000,     16640000,      4561920,       737280,        53248;
  7:   78125,  2187500,  21000000,  108500000,   355250000,   798000000,   1283800000,   1513600000,   1320480000,    851200000,    400153600,    133324800,     29818880,     4014080,      245760;
  8:  390625, 12500000, 138750000,  840000000,  3272500000,  8904000000,  17718400000,  26496000000,  30232800000,  26496000000,  17819648000,   9117696000,   3485081600,   963379200,   181862400,   20971520,    1114112;
  9: 1953125, 70312500, 885937500, 6150000000, 27787500000, 88830000000, 210798000000, 382464000000, 540367200000, 600880000000, 528359040000, 367165440000, 200438784000, 84897792000, 27316224000, 6448742400, 1052835840, 106168320, 4980736;

%F Formula

The row polynomials satisfy (5+10*n*x+(8*n-4)*x^2)*P(n,x) = (25+(50*n+100)*x+(140*n+140)*x^2+(120*n+80)*x^3+(32*n+16)*x^4)*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[((25 + (50*n + 100)*x + (140*n + 140)*x^2 + (120*n + 80)*x^3 + (32*n + 16)*x^4)*P[n - 1, x])/(5 + 10*n*x + (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] = ((25 + (50*n + 100)*x + (140*n + 140)*x^2 + (120*n + 80)*x^3 + (32*n + 16)*x^4)*v[i - 1])/(5 + 10*n*x + (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 basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.