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	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

All 15 rows

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.