A038305 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(25 + 12\,x + 12\,n\right)\,P_{n}(x) = \left(481 + 600\,x + 144\,x^{2} + 156\,n + 144\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 481 + 600\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	481	600	144															1225
1	8281	17316	10800	1728														38125
2	134017	397488	415584	172800	20736													1140625
3	2084953	8041020	11924640	8311680	2592000	248832												33203125
4	31559905	150116616	289476720	286191360	149610240	37324800	2985984											947265625
5	468200473	2651032020	6304897872	8105348160	6010018560	2513452032	522547200	35831808										26611328125
6	6839588353	44947245408	127249536960	201756731904	194528355840	115392356352	40215232512	7166361600	429981696									738525390625
7	98703417241	738675542124	2427151252032	4580983330560	5447431761408	4201812486144	2077062414336	620463587328	96745881600	5159780352								20294189453125
8	1410398416609	11844410068920	44320532527440	97086050081280	137429499916800	130738362273792	84036249722880	35606784245760	9306953809920	1289945088000	61917364224							553131103515625
9	19989481318105	186172590992388	781731064548720	1950103431207360	3203839652682240	3628138797803520	2876243970023424	1584683566202880	587511940055040	136501989212160	17027275161600	743008370688						14972686767578125
10	281369181863809	2878485309807120	13404426551451936	37523091098338560	70203723523464960	92270581997248512	87075331147284480	59168447383339008	28524304191651840	9400191040880640	1965628644655104	222902511206400	8916100448256					402927398681640625
11	3937376385699289	43893592370754204	224521854164955360	697030180675500672	1463400552835203840	2190356173932106752	2399035131928461312	1940535951282339840	1153784723975110656	494421272655298560	146642980237737984	27876188051472384	2897732645683200	106993205379072				10788440704345703125
12	54820394293197793	661479232797480552	3687061759143353136	12573223833237500160	29275267588371028224	49170258575262849024	61329972870098989056	57576843166283071488	40751254976929136640	21537314847535398912	8306277380609015808	2239638243630907392	390266632720613376	37447621882675200	1283918464548864			287592411041259765625
13	759913642439962777	9867670972775602740	59533130951773249680	221223705548601188160	565795072495687507200	1053909633181357016064	1475107757257885470720	1577056445231116861440	1295478971241369108480	815025099538582732800	387671667255637180416	135920902591783895040	33594573654463610880	5403691837670031360	481469424205824000	15407021574586368		7636845111846923828125
14	10493108067888605185	145903419348472853184	947296413386457863040	3810120380913487979520	10618737866332857031680	21726530783834400276480	33725108261803424514048	40460098484787715768320	37849354685546804674560	27636884719815874314240	15648481911140788469760	6766632737552939876352	2174734441468542320640	496166010896693329920	74107773773760430080	6162808629834547200	184884258895036416	202096998691558837890625

Technical details

Node id: n-d3760c0acdc98316 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-85b44f7305edc983

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:            481,             600,             144;
  1:           8281,           17316,           10800,             1728;
  2:         134017,          397488,          415584,           172800,            20736;
  3:        2084953,         8041020,        11924640,          8311680,          2592000,           248832;
  4:       31559905,       150116616,       289476720,        286191360,        149610240,         37324800,          2985984;
  5:      468200473,      2651032020,      6304897872,       8105348160,       6010018560,       2513452032,        522547200,         35831808;
  6:     6839588353,     44947245408,    127249536960,     201756731904,     194528355840,     115392356352,      40215232512,       7166361600,       429981696;
  7:    98703417241,    738675542124,   2427151252032,    4580983330560,    5447431761408,    4201812486144,    2077062414336,     620463587328,     96745881600,      5159780352;
  8:  1410398416609,  11844410068920,  44320532527440,   97086050081280,  137429499916800,  130738362273792,   84036249722880,   35606784245760,   9306953809920,   1289945088000,    61917364224;
  9: 19989481318105, 186172590992388, 781731064548720, 1950103431207360, 3203839652682240, 3628138797803520, 2876243970023424, 1584683566202880, 587511940055040, 136501989212160, 17027275161600, 743008370688;

%F Formula

The row polynomials satisfy (12*n+25+12*x)*P(n,x) = (156*n+481+(144*n+600)*x+144*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 481 + 600*x + 144*x^2.

%t Mathematica

P[0, x_] := 481 + 600*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((156*n + 481 + (144*n + 600)*x + 144*x^2)*P[n - 1, x])/(12*n + 25 + 12*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 481 + 600*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((156*n + 481 + (144*n + 600)*x + 144*x^2)*v[i - 1])/(12*n + 25 + 12*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 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
    
    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
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.