A013612 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 30\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 256 + 960\,x + 900\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	256	960	900								2116
1	4096	23040	43200	27000							97336
2	65536	491520	1382400	1728000	810000						4477456
3	1048576	9830400	36864000	69120000	64800000	24300000					205962976
4	16777216	188743680	884736000	2211840000	3110400000	2332800000	729000000				9474296896
5	268435456	3523215360	19818086400	61931520000	116121600000	130636800000	81648000000	21870000000			435817657216
6	4294967296	64424509440	422785843200	1585446912000	3715891200000	5573836800000	5225472000000	2799360000000	656100000000		20047612231936
7	68719476736	1159641169920	8697308774400	38050725888000	107017666560000	200658124800000	250822656000000	201553920000000	94478400000000	19683000000000	922190162669056

All 15 rows

Technical details

Node id: n-69d39bd7c8ec9d15 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-dd0a803b420669cd

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:            256,             960,              900;
  1:           4096,           23040,            43200,             27000;
  2:          65536,          491520,          1382400,           1728000,            810000;
  3:        1048576,         9830400,         36864000,          69120000,          64800000,           24300000;
  4:       16777216,       188743680,        884736000,        2211840000,        3110400000,         2332800000,          729000000;
  5:      268435456,      3523215360,      19818086400,       61931520000,      116121600000,       130636800000,        81648000000,        21870000000;
  6:     4294967296,     64424509440,     422785843200,     1585446912000,     3715891200000,      5573836800000,      5225472000000,      2799360000000,       656100000000;
  7:    68719476736,   1159641169920,    8697308774400,    38050725888000,   107017666560000,    200658124800000,    250822656000000,    201553920000000,     94478400000000,     19683000000000;
  8:  1099511627776,  20615843020800,  173946175488000,   869730877440000,  2853804441600000,   6421059993600000,  10032906240000000,  10749542400000000,   7558272000000000,   3149280000000000,    590490000000000;
  9: 17592186044416, 362838837166080, 3401614098432000, 19134079303680000, 71752797388800000, 188351093145600000, 353158299648000000, 472979865600000000, 443418624000000000, 277136640000000000, 103926240000000000, 17714700000000000;

%F Formula

The row polynomials satisfy P(n,x) = (16+30*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 960*x + 900*x^2.

%t Mathematica

P[0, x_] := 256 + 960*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 30*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] = 256 + 960*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 30*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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (1+5x)^n.?
    
    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.