A038305 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(13 + 24\,x + 12\,n x\right)\,P_{n}(x) = \left(169 + 624\,x + 432\,x^{2} + 156\,n x + 144\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 169 + 624\,x + 432\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	169	624	432															1225
1	2197	12168	16848	6912														38125
2	28561	210912	438048	359424	103680													1140625
3	371293	3427320	9491040	11681280	6739200	1492992												33203125
4	4826809	53466192	185075280	303713280	262828800	116453376	20901888											947265625
5	62748517	810903912	3368370096	6909477120	7972473600	5298628608	1902071808	286654464										26611328125
6	815730721	12047715264	58385081664	143717124096	207284313600	183685791744	98907734016	29812064256	3869835264									738525390625
7	10604499373	176197835736	975864936384	2802483919872	4850452938240	5372809408512	3857401626624	1744005758976	452770725888	51597803520								20294189453125
8	137858491849	2545079849520	15857805216240	52046129940480	105093146995200	139693044621312	125365552865280	75573582888960	29430097182720	6707714457600	681091006464							553131103515625
9	1792160394037	36394641848136	251962905102480	930324572686080	2146902860044800	3329350896807936	3585454811947008	2701755588280320	1402834632376320	479601583718400	97396013924352	8916100448256						14972686767578125
10	23298085122481	516142193482656	3930621319598688	16125625926558720	41864605770873600	74196962843148288	93221825110622208	84294774354345984	54710550662676480	24939282353356800	7596889086099456	1390911669927936	115909305827328					402927398681640625
11	302875106592253	7269002558214072	60388636637470752	272523078158842368	786124263919737600	1567410840061507584	2250641206242164736	2374302810980745216	1849216612398465024	1053684679429324800	427958085183602688	117532036108910592	19588672684818432	1497904875307008				10788440704345703125
12	3937376385699289	101766035814997008	915894322334973072	4509018202264482816	14307461603339224320	31696530321243820032	51202087442009247744	61731873085499375616	56092903909420105728	38354122331227422720	19472092875853922304	7130276857273909248	1782569214318477312	272618687305875456	19258776968232960			287592411041259765625
13	51185893014090757	1417455498851744040	13738414835024596080	73271545786797845760	253632273877377158400	618082341264254490624	1109378561243533701120	1504714406459047280640	1562588037476702945280	1246508975764891238400	759411622158302969856	347600996792103075840	115866998930701025280	26580322012322856960	3755461508805427200	246512345193381888		7636845111846923828125
14	665416609183179841	19655382917410850688	204113591834651141760	1172344732588765532160	4396292747207870745600	11687375180269539459072	23075074073865500983296	34775621838164648263680	40627288974394276577280	37039123851299625369600	26326269568154502955008	14460201466551487954944	6025083944396453314560	1842902326187718082560	390567996915764428800	51274567800223432704	3143032401215619072	202096998691558837890625

Technical details

Node id: n-a4f5036208268596 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-72578208ad49d2f0

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:           169,            624,             432;
  1:          2197,          12168,           16848,            6912;
  2:         28561,         210912,          438048,          359424,           103680;
  3:        371293,        3427320,         9491040,        11681280,          6739200,          1492992;
  4:       4826809,       53466192,       185075280,       303713280,        262828800,        116453376,         20901888;
  5:      62748517,      810903912,      3368370096,      6909477120,       7972473600,       5298628608,       1902071808,        286654464;
  6:     815730721,    12047715264,     58385081664,    143717124096,     207284313600,     183685791744,      98907734016,      29812064256,       3869835264;
  7:   10604499373,   176197835736,    975864936384,   2802483919872,    4850452938240,    5372809408512,    3857401626624,    1744005758976,     452770725888,     51597803520;
  8:  137858491849,  2545079849520,  15857805216240,  52046129940480,  105093146995200,  139693044621312,  125365552865280,   75573582888960,   29430097182720,   6707714457600,   681091006464;
  9: 1792160394037, 36394641848136, 251962905102480, 930324572686080, 2146902860044800, 3329350896807936, 3585454811947008, 2701755588280320, 1402834632376320, 479601583718400, 97396013924352, 8916100448256;

%F Formula

The row polynomials satisfy (13+(12*n+24)*x)*P(n,x) = (169+(156*n+624)*x+(144*n+432)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 624*x + 432*x^2.

%t Mathematica

P[0, x_] := 169 + 624*x + 432*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((169 + (156*n + 624)*x + (144*n + 432)*x^2)*P[n - 1, x])/(13 + (12*n + 24)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 169 + 624*x + 432*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((169 + (156*n + 624)*x + (144*n + 432)*x^2)*v[i - 1])/(13 + (12*n + 24)*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_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.