A038334 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 12\,x + 6\,n x\right)\,P_{n}(x) = \left(100 + 480\,x + 432\,x^{2} + 120\,n x + 144\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 400 + 1920\,x + 1728\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	400	1920	1728															4048
1	8000	57600	103680	55296														224576
2	160000	1536000	4147200	4423680	1658880													11925760
3	3200000	38400000	138240000	221184000	165888000	47775744												614687744
4	64000000	921600000	4147200000	8847360000	9953280000	5733089280	1337720832											31004250112
5	1280000000	21504000000	116121600000	309657600000	464486400000	401316249600	187280916480	36691771392										1538338537472
6	25600000000	491520000000	3096576000000	9909043200000	18579456000000	21403533312000	14982473318400	5870683422720	990677827584									75349563080704
7	512000000000	11059200000000	79626240000000	297271296000000	668860416000000	963158999040000	898948399104000	528361508044800	178322008965120	26418075402240								3652538142556160
8	10240000000000	245760000000000	1990656000000000	8493465600000000	22295347200000000	38526359961600000	44947419955200000	35224100536320000	17832200896512000	5283615080448000	697437190619136							175546602420699136
9	204800000000000	5406720000000000	48660480000000000	233570304000000000	700710912000000000	1412633198592000000	1977686478028800000	1937325529497600000	1307694732410880000	581197658849280000	153436181936209920	18260173718028288						8376787169032798208
10	4096000000000000	117964800000000000	1167851520000000000	6228541440000000000	21021327360000000000	48433138237440000000	79107459121152000000	92991625415884800000	78461683944652800000	46495812707942400000	18412341832345190400	4382441692326789120	474764516668735488					397299048588412715008
11	81920000000000000	2555904000000000000	27603763200000000000	161942077440000000000	607282790400000000000	1574076992716800000000	2938277053071360000000	4029637101355008000000	4080007565121945600000	3022227826016256000000	1595736292136583168000	569717420002482585600	123438774333871226880	12270836738515009536				18744856316532821590016
12	1638400000000000000	55050240000000000000	644087808000000000000	4122161971200000000000	17003918131200000000000	48971284217856000000000	102839696857497600000000	161185484054200320000000	190400353039024128000000	169244758256910336000000	111701540449560821760000	53173625866898374656000	17281428406741971763200	3435834286784202670080	315535801847528816640			880376397787721283665920
13	32768000000000000000	1179648000000000000000	14863564800000000000000	103054049280000000000000	463743221760000000000000	1469138526535680000000000	3427989895249920000000000	6044455652032512000000000	8160015130243891200000000	8462237912845516800000000	6702092426973649305600000	3988021940017378099200000	1728142840674197176320000	515375143017630400512000	94660740554258644992000	8077716527296737705984		41183081176511930364329984
14	655360000000000000000	25165824000000000000000	339738624000000000000000	2536715059200000000000000	12366485913600000000000000	42738575317401600000000000	109695676647997440000000000	214913978738933760000000000	326400605209755648000000000	386845161730080768000000000	357444929438594629632000000	255233404161112198348800000	138251427253935774105600000	54973348588547242721280000	15145718488681383198720000	2584869288734956065914880	205981771446066811502592	1919702437416021466883817472

Technical details

Node id: n-1d55f39abb48a994 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-572def99d14f8fb6

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:             400,             1920,              1728;
  1:            8000,            57600,            103680,              55296;
  2:          160000,          1536000,           4147200,            4423680,            1658880;
  3:         3200000,         38400000,         138240000,          221184000,          165888000,            47775744;
  4:        64000000,        921600000,        4147200000,         8847360000,         9953280000,          5733089280,          1337720832;
  5:      1280000000,      21504000000,      116121600000,       309657600000,       464486400000,        401316249600,        187280916480,         36691771392;
  6:     25600000000,     491520000000,     3096576000000,      9909043200000,     18579456000000,      21403533312000,      14982473318400,       5870683422720,        990677827584;
  7:    512000000000,   11059200000000,    79626240000000,    297271296000000,    668860416000000,     963158999040000,     898948399104000,     528361508044800,     178322008965120,     26418075402240;
  8:  10240000000000,  245760000000000,  1990656000000000,   8493465600000000,  22295347200000000,   38526359961600000,   44947419955200000,   35224100536320000,   17832200896512000,   5283615080448000,    697437190619136;
  9: 204800000000000, 5406720000000000, 48660480000000000, 233570304000000000, 700710912000000000, 1412633198592000000, 1977686478028800000, 1937325529497600000, 1307694732410880000, 581197658849280000, 153436181936209920, 18260173718028288;

%F Formula

The row polynomials satisfy (5+(6*n+12)*x)*P(n,x) = (100+(120*n+480)*x+(144*n+432)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1920*x + 1728*x^2.

%t Mathematica

P[0, x_] := 400 + 1920*x + 1728*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((100 + (120*n + 480)*x + (144*n + 432)*x^2)*P[n - 1, x])/(5 + (6*n + 12)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 400 + 1920*x + 1728*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((100 + (120*n + 480)*x + (144*n + 432)*x^2)*v[i - 1])/(5 + (6*n + 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 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^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

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
[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.