A038331 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(20 + 17\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 400 + 680\,x + 289\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	400	680	289															1369
1	8000	20400	17340	4913														50653
2	160000	544000	693600	393040	83521													1874161
3	3200000	13600000	23120000	19652000	8352100	1419857												69343957
4	64000000	326400000	693600000	786080000	501126000	170382840	24137569											2565726409
5	1280000000	7616000000	19420800000	27512800000	23385880000	11926798800	3379259660	410338673										94931877133
6	25600000000	174080000000	517888000000	880409600000	935435200000	636095936000	270340772800	65654187680	6975757441									3512479453921
7	512000000000	3916800000000	13317120000000	26412288000000	33675667200000	28624317120000	16220446368000	5908876891200	1255636339380	118587876497								129961739795077
8	10240000000000	87040000000000	332928000000000	754636800000000	1122522240000000	1144972684800000	811022318400000	393925126080000	125563633938000	23717575299400	2015993900449							4808584372417849
9	204800000000000	1914880000000000	8138240000000000	20752512000000000	35279270400000000	41982331776000000	35684982009600000	21665881934400000	9207999822120000	2608933282934000	443518658098780	34271896307633						177917621779460413
10	4096000000000000	41779200000000000	195317760000000000	553400320000000000	1058378112000000000	1439394232320000000	1427399280384000000	1039962332851200000	552479989327200000	208714662634720000	53222238971853600	8225255113831920	582622237229761					6582952005840035281
11	81920000000000000	905216000000000000	4616601600000000000	14388408320000000000	30575367680000000000	46780312550400000000	53017687557120000000	45065034423552000000	28728959445014400000	13566453071256800000	4612594044227312000	1069283164798149600	151481781679737860	9904578032905937				243569224216081305397
12	1638400000000000000	19496960000000000000	107720704000000000000	366250393600000000000	856110295040000000000	1455387501568000000000	1855619064499200000000	1802601376942080000000	1340684774100672000000	759721371990380800000	322881583095911840000	99799762047827296000	21207449435163300400	2773281849213662360	168377826559400929			9012061295995008299689
13	32768000000000000000	417792000000000000000	2485862400000000000000	9156259840000000000000	23348462592000000000000	43661625047040000000000	61853968816640000000000	67597551635328000000000	57457918890028800000000	37986068599519040000000	19372894985754710400000	7484982153587047200000	2120744943516330040000	415992277382049354000	50513347967820278700	2862423051509815793		333446267951815307088493
14	655360000000000000000	8912896000000000000000	56819712000000000000000	225384857600000000000000	622625669120000000000000	1270156365004800000000000	1979327002132480000000000	2403468502589440000000000	2298316755601152000000000	1736505993120870400000000	1033221065906917888000000	479038857829571020800000	169659595481306403200000	44372509587418597760000	8082135674851244592000	915975376483141053760	48661191875666868481	12337511914217166362274241

Technical details

Node id: n-2405be3a93998241 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-8c1ae502a0a48e51

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,              680,              289;
  1:            8000,            20400,            17340,              4913;
  2:          160000,           544000,           693600,            393040,             83521;
  3:         3200000,         13600000,         23120000,          19652000,           8352100,           1419857;
  4:        64000000,        326400000,        693600000,         786080000,         501126000,         170382840,          24137569;
  5:      1280000000,       7616000000,      19420800000,       27512800000,       23385880000,       11926798800,        3379259660,         410338673;
  6:     25600000000,     174080000000,     517888000000,      880409600000,      935435200000,      636095936000,      270340772800,       65654187680,       6975757441;
  7:    512000000000,    3916800000000,   13317120000000,    26412288000000,    33675667200000,    28624317120000,    16220446368000,     5908876891200,    1255636339380,     118587876497;
  8:  10240000000000,   87040000000000,  332928000000000,   754636800000000,  1122522240000000,  1144972684800000,   811022318400000,   393925126080000,  125563633938000,   23717575299400,   2015993900449;
  9: 204800000000000, 1914880000000000, 8138240000000000, 20752512000000000, 35279270400000000, 41982331776000000, 35684982009600000, 21665881934400000, 9207999822120000, 2608933282934000, 443518658098780, 34271896307633;

%F Formula

The row polynomials satisfy P(n,x) = (20+17*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 680*x + 289*x^2.

%t Mathematica

P[0, x_] := 400 + 680*x + 289*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 17*x)*P[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] = 400 + 680*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 17*x)*v[i - 1]); 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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 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 ?

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.