A038328 → 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

\(\left(11 + 4\,x + 4\,n\right)\,P_{n}(x) = \left(210 + 176\,x + 32\,x^{2} + 56\,n + 32\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 420 + 352\,x + 64\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	420	352	64															836
1	7448	10080	4224	512														22264
2	126224	238336	161280	45056	4096													574992
3	2074464	5048960	4766720	2150400	450560	32768												14523872
4	33345088	99574272	121175040	76267520	25804800	4325376	262144											360754240
5	527067520	1867324928	2788079616	2261934080	1067745280	289013760	40370176	2097152										8843632512
6	8222253312	33732321280	59754397696	59479031808	36190945280	13667139584	3082813440	369098752	16777216									214514778368
7	126917858816	592002238464	1214363566080	1434105544704	1070622572544	521149612032	164005675008	31708938240	3321888768	134217728								5158332112384
8	1942138397696	10153428705280	23680089538560	32383028428800	28682110894080	17129961160704	6948661493760	1874350571520	317089382400	29527900160	1073741824							123141460214784
9	29503974807552	170908178997248	446750863032320	694615959797760	712426625433600	504805151735808	251239430356992	87354601635840	20617856286720	3100429516800	259845521408	8589934592						2921591507056640
10	445452168663040	2832381581524992	8203592591867904	14296027617034240	16670783035146240	13678591208325120	8076882427772928	3445569330610176	1048255219630080	219923800391680	29764123361280	2267742732288	68719476736					68949559566536704
11	6689881660284928	46327025540956160	147283842239299584	284391209851420672	371696718042890240	346752287131041792	237095580944302080	119999396069769216	44792401297932288	12113171426836480	2287207524073472	281406257233920	19653770346496	549755813888				1619730331512201216
12	100008061430022144	749266745951911936	2594313430293544960	5498596776933851136	7962953875839778816	8326006484160741376	6472709359779446784	3793529295108833280	1679991544976769024	557416549485379584	135667519980568576	23287931154202624	2626458400849920	169324790677504	4398046511104			37896547756333088768
13	1489008914624774144	12000967371602657280	44956004757114716160	103772537211741798400	164957903308015534080	191110893020154691584	166520129683214827520	110960731881933373440	56902939426632499200	22399887266356920320	6688998593824555008	1480009308878929920	232879311542026240	24244231392460800	1451355348664320	35184372088832		883498620826750517248
14	22090669569209335808	190593141071971090432	768061911782570065920	1918122869636894556160	3320721190775737548800	4222922324685197672448	4077032384429966753792	3044939514207356846080	1775371710110933975040	809286249623217766400	286718557009368580096	77835620000867549184	15786765961375252480	2292965529029181440	221661544159641600	12384898975268864	281474976710656	20532010202311807795200

Technical details

Node id: n-b34ce0ed948ac46c · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-c09fbe1bbbcc2fe2

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:            420,             352,              64;
  1:           7448,           10080,            4224,             512;
  2:         126224,          238336,          161280,           45056,            4096;
  3:        2074464,         5048960,         4766720,         2150400,          450560,           32768;
  4:       33345088,        99574272,       121175040,        76267520,        25804800,         4325376,          262144;
  5:      527067520,      1867324928,      2788079616,      2261934080,      1067745280,       289013760,        40370176,        2097152;
  6:     8222253312,     33732321280,     59754397696,     59479031808,     36190945280,     13667139584,      3082813440,      369098752,       16777216;
  7:   126917858816,    592002238464,   1214363566080,   1434105544704,   1070622572544,    521149612032,    164005675008,    31708938240,     3321888768,     134217728;
  8:  1942138397696,  10153428705280,  23680089538560,  32383028428800,  28682110894080,  17129961160704,   6948661493760,  1874350571520,   317089382400,   29527900160,   1073741824;
  9: 29503974807552, 170908178997248, 446750863032320, 694615959797760, 712426625433600, 504805151735808, 251239430356992, 87354601635840, 20617856286720, 3100429516800, 259845521408, 8589934592;

%F Formula

The row polynomials satisfy (4*n+11+4*x)*P(n,x) = (56*n+210+(32*n+176)*x+32*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 420 + 352*x + 64*x^2.

%t Mathematica

P[0, x_] := 420 + 352*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((56*n + 210 + (32*n + 176)*x + 32*x^2)*P[n - 1, x])/(4*n + 11 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 420 + 352*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((56*n + 210 + (32*n + 176)*x + 32*x^2)*v[i - 1])/(4*n + 11 + 4*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)*12^(i-j)*2^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A038255 → basis A013611 → D+1

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.