A139548 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 - x + 2\,n x\right)\,P_{n}(x) = \left(8 + 24\,x + 24\,x^{2} + 8\,x^{3} + 16\,n x + 32\,n x^{2} + 16\,n x^{3}\right)\,P_{n-1}(x)\)

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

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	8	32	24																											64
2	64	512	1152	1024	320																									3072
3	512	6144	23040	40960	38400	18432	3584																							131072
4	4096	65536	344064	917504	1433600	1376256	802816	262144	36864																					5242880
5	32768	655360	4423680	15728640	34406400	49545216	48168960	31457280	13271040	3276800	360448																			201326592
6	262144	6291456	51904512	230686720	648806400	1245708288	1695547392	1660944384	1167851520	576716800	190316544	37748736	3407872																	7516192768
7	2097152	58720256	572522496	3053453312	10496245760	25190989824	44084232192	57579405312	56679727104	41984983040	23091740672	9160359936	2480930816	411041792	31457280															274877906944
8	16777216	536870912	6039797760	37580963840	152672665600	439697276928	940463620096	1535450808320	1943304929280	1919313510400	1477871403008	879394553856	396948930560	131533373440	30198988800	4294967296	285212672													9895604649984
9	134217728	4831838208	61605937152	438086664192	2053531238400	6899864961024	17441325318144	34170759806976	52857894076416	65256659353600	64604092760064	51256139710464	32391032733696	16099684909056	6160593715200	1752346656768	349100310528	43486543872	2550136832											351843720888320
10	1073741824	42949672960	612032839680	4896262717440	26011395686400	99883759435776	291327631687680	665891729571840	1217333318123520	1803456767590400	2182182688784384	2164148121108480	1758370348400640	1165310526750720	624273496473600	266356691828736	88438745333760	22033182228480	3876207984640	429496729600	22548578304									12384898975268864
11	8589934592	377957122048	5952824672256	52913997086720	314176857702400	1357244025274368	4486445527990272	11719694440464384	24721230460354560	42728052647526400	61101115285962752	72715376869244928	72210408974319616	59819273706536960	41202050767257600	23439388880928768	10895653425119232	4071732075823104	1193872059269120	264569985433600	41669772705792	4157528342528	197568495616							432345564227567616
12	68719476736	3298534883328	56899726737408	556352883654656	3651065798983680	17525115835121664	64745566835310592	190272686209892352	454870640470523904	898509907102269440	1482541346718744576	2058404514452471808	2415766409322692608	2401471933527883776	2021647290980106240	1437615851363631104	859200098666545152	428113543972257792	175737967124414464	58417052783738880	15334476355731456	3059940860100608	436231238320128	39582418599936	1717986918400					14987979559889010688
13	549755813888	28587302322176	536011918540800	5717460464435200	41094247088128000	216977624625315840	885991967220039680	2893034995004211200	7729827877276876800	17177395282837504000	32121729178906132480	50970016548346920960	69021897409219788800	80049064450929459200	79640650856792064000	67960022064462561280	49642672367400386560	30919311509107507200	16318525518695628800	7232587487510528000	2657975901660119040	795584623626158080	189033536605388800	34304762786611200	4466765987840000	371634930188288	14843406974976			516508834063867445248
14	4398046511104	246290604621824	4987384743591936	57632001481506816	450250011574272000	2593440066667806720	11598440298153246720	41659703928060641280	123026313162554081280	303768674475442176000	634876529653674147840	1133333309133831536640	1739351814712338677760	2305413056186768424960	2646520090010320896000	2634757778499163914240	2274536988469981347840	1699999963700747304960	1096604914856346255360	607537348950884352000	287061397379292856320	114564185802166763520	38109160979646382080	10373760266671226880	2251250057871360000	374608009629794304	44886462692327424	3448068464705536	127543348822016	17708874310761169551360

Technical details

Node id: n-062516b812f0ea15 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c94e3f620e523ccf

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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:         1;
  1:         8,         32,          24;
  2:        64,        512,        1152,         1024,           320;
  3:       512,       6144,       23040,        40960,         38400,         18432,           3584;
  4:      4096,      65536,      344064,       917504,       1433600,       1376256,         802816,         262144,          36864;
  5:     32768,     655360,     4423680,     15728640,      34406400,      49545216,       48168960,       31457280,       13271040,        3276800,         360448;
  6:    262144,    6291456,    51904512,    230686720,     648806400,    1245708288,     1695547392,     1660944384,     1167851520,      576716800,      190316544,       37748736,        3407872;
  7:   2097152,   58720256,   572522496,   3053453312,   10496245760,   25190989824,    44084232192,    57579405312,    56679727104,    41984983040,    23091740672,     9160359936,     2480930816,      411041792,      31457280;
  8:  16777216,  536870912,  6039797760,  37580963840,  152672665600,  439697276928,   940463620096,  1535450808320,  1943304929280,  1919313510400,  1477871403008,   879394553856,   396948930560,   131533373440,   30198988800,    4294967296,    285212672;
  9: 134217728, 4831838208, 61605937152, 438086664192, 2053531238400, 6899864961024, 17441325318144, 34170759806976, 52857894076416, 65256659353600, 64604092760064, 51256139710464, 32391032733696, 16099684909056, 6160593715200, 1752346656768, 349100310528, 43486543872, 2550136832;

%F Formula

The row polynomials satisfy (1+(2*n-1)*x)*P(n,x) = (8+(16*n+24)*x+(32*n+24)*x^2+(16*n+8)*x^3)*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[((8 + (16*n + 24)*x + (32*n + 24)*x^2 + (16*n + 8)*x^3)*P[n - 1, x])/(1 + (2*n - 1)*x)]];
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] = ((8 + (16*n + 24)*x + (32*n + 24)*x^2 + (16*n + 8)*x^3)*v[i - 1])/(1 + (2*n - 1)*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 basis_map_A013609 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    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
    
    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
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from basis_map_A013609 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
    
    ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

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.